# Patent

## Title
Magnetic noise reduction method for AC rotary electric machine, and motor control apparatus and AC rotary electric machine apparatus using the same

## Abstract

The invention provides techniques for reducing or altering the magnetic noise of an AC rotary electric machine. A magnetic noise reducing harmonic current of order n, whose frequency is n times the frequency of the fundamental frequency component of a polyphase AC current fed to an armature of a polyphase AC rotary electric machine, is superimposed on the polyphase AC current, thereby reducing or altering a harmonic component having a frequency (n−1) times the frequency of the fundamental frequency component and occurring due to a radial magnetic excitation force acting radially on an iron core of the AC rotary electric machine. Magnetic noise is caused by a vibration whose energy is the sum of the circumferential and radial vibrations of the iron core occurring due to the magnetomotive force of the rotor, and altering the radial vibration is particularly effective in altering the magnetic noise; as the harmonic component of the radially acting magnetic excitation force, occurring due to harmonic components having frequencies 3, 5, 7, and 13 times the fundamental frequency, has a frequency 6 or 12 times the fundamental frequency, the magnetomotive force of the rotor can be effectively reduced or altered when a current having a frequency 7 or 13 times the fundamental frequency is superimposed on the stator current.

## Background
CROSS REFERENCE TO RELATED APPLICATION
This application is based upon and claims the benefit of priority from the prior Japanese Patent Application No. 2002-303650, filed on Oct. 17, 2002, the prior Japanese Patent Application No. 2002-303651, filed on Oct. 17, 2002, the prior Japanese Patent Application No. 2003-323779, filed on Sep. 16, 2003, and the prior Japanese Patent Application No. 2003-323782, filed on Sep. 16, 2003, the entire contents of which are incorporated herein by reference, and a continuation of PCT/JP03/13303 filed as Oct. 17, 2003.
TECHNICAL FIELD
The present invention relates to a magnetic noise reduction method for an AC rotary electric machine, and to a motor control apparatus and AC rotary electric machine apparatus using the same.
BACKGROUND ART
Electric vehicles, hybrid vehicles, fuel-cell vehicles, etc. are currently in the development or in the practical implementation stage. These vehicles employ high-output AC rotary electric machines as primary means for generating vehicle driving power, but such high-output AC rotary electric machines have had the problem that their magnetic noise (magnetic sound) is large. Furthermore, quiet AC motors are needed in various applications, and a variety of techniques have been proposed for reduction of magnetic noise of AC rotary electric machines. For example, Japanese Patent No. 2928594 proposes that voltage components that cancel out the harmonics contained in the induced voltage of an electric motor be superimposed on the output voltage of an inverter, thereby bringing the waveform of the electric current to the motor closer to a sinusoidal wave and thus achieving a reduction in torque pulsation and its associated noise.
On the other hand, Japanese Unexamined Patent Publication No. 11-55986, for example, proposes to reduce the torque pulsation of an electric motor by deliberately superimposing a harmonic component (hereinafter sometimes referred to as the harmonic current) on the fundamental frequency component of the electric current fed to the motor.
Further, Japanese Unexamined Patent Publication No. H04-200294 and Japanese Examined Patent Publication No. H07-89753 disclose that the carrier frequency of a PWM control inverter is altered in a prescribed pattern to alter the electromagnetic sound of a motor.
DISCLOSURE OF THE INVENTION
However, the above prior art magnetic noise reduction techniques of the harmonic superimposing type, despite its name, have fallen far short of being able to reduce the noise of AC rotary electric machines to a satisfactory level. The reasons for this will be described below.
As noise is generated due to mechanical vibrations of a vibrating member (in this case, the iron core of the AC rotary electric machine), if some kind of vibrating means is added and vibration energy equal in frequency and amplitude but opposite in phase to the vibration energy of the frequency to be attenuated is applied, it should be possible to cancel out the vibrations of the vibrating member at that frequency. Further, the notion itself that the magnetic noise can be reduced by adding a certain harmonic component to the fundamental frequency component of the armature current has been presented in various ways in the prior art as described above.
However, the above prior art documents do not provide concrete descriptions about the numerical relations between the acoustic energy (vibration energy) to be reduced and the electric current (or power) to be supplied, that is, the parameters defining the frequency, phase, and amplitude (magnitude) of the electric current or power that should be applied to the armature coil of the AC rotary electric machine in order to reduce the magnetic noise (acoustic energy) at a designated frequency, and it is therefore not known what electric current should be applied to reduce the magnetic noise of a particular frequency; as a result, its practical implementation has been difficult. Furthermore, the relationship between the magnetic noise and the electric current (or power) applied to suppress it should vary in various ways depending on the structure and the use conditions of the AC rotary electric machine and, while the magnetic noise reduction techniques of the harmonic superimposing type described above are known as a concept, none of them have been practically implemented as yet.
On the other hand, Japanese Unexamined Patent Publication No. H04-200294 and Japanese Examined Patent Publication No. H07-89753 propose that the frequency of the electromagnetic sound of a motor be altered by altering the frequency of the PWM carrier signal of the motor, but since there is a limit to the range over which the frequency of the PWM carrier signal can be altered, there has been the problem that it is not easy to obtain a sound of the intended frequency.
In view of the above problems, it is an object of the present invention to provide a magnetic noise reduction method for an AC rotary electric machine that can reduce the magnetic noise of various kinds of AC rotary electric machines in a simple and reliable manner, and a motor control apparatus using the same.
It is another object of the present invention to provide an AC rotary electric machine apparatus that can alter the electromagnetic sound of various kinds of AC rotary electric machines as desired.
A magnetic noise reduction method for an AC rotary electric machine according to claim 1 is characterized in that a radial vibration reducing harmonic current, whose frequency is n times (n is a natural number) the frequency of the fundamental frequency component of a polyphase AC current fed to an armature of a polyphase AC rotary electric machine, is superimposed on the polyphase AC current, thereby attenuating an (n−1)th harmonic component, having a frequency (n−1) times the frequency of the fundamental frequency component, that is contained in a radial vibration occurring radially about an axial center of a rotating shaft of the AC rotary electric machine due to an excitation force generated by the AC rotary electric machine or externally applied to the AC rotary electric machine. This has made it possible, for the first time, to effectively reduce the magnetic noise of AC rotary electric machines of various sizes and in any output condition. In open-loop control, the phase and amplitude of the radial vibration reducing harmonic current may be set as predetermined values, or the value of the radial vibration reducing harmonic current detected in feedback control may be set as a predetermined value. However, these predetermined values can be changed according to the operating condition, based on a prestored map defining correspondences between the amplitude of the fundamental current and the number of revolutions and the phase and amplitude.
A further detailed description will be given below.
Magnetic noise is caused by a vibration (also known as magnetic vibration) induced by the magnetic force (magnetic excitation force) of the iron core of the AC rotary electric machine, and this magnetic vibration is the sum of circumferential and radial vibrations.
The circumferential vibration of the iron core causes torque ripple but, as the iron core of the stator or the iron core of the rotor is substantially cylindrical or columnar in shape, if the iron core periodically vibrates in the circumferential direction, the noise caused by the vibration of the air surrounding the vibrating iron core is small. On the other hand, the radial vibration of the iron core causes the outer or inner circumferential surface of the iron core of the stator or rotor to vibrate in the radial direction; here, since the outer or inner circumferential surface directly contacts the air, the radial vibration of the iron core of the stator or rotor causes the outer or inner circumferential surface to vibrate in the radial direction, which causes a large amount of noise.
That is, the torque pulsation is reduced by reducing the circumferential component of the magnetic excitation force, and the magnetic noise is reduced by reducing the radial component of the magnetic excitation force.
In the present invention, in order to reduce a given order harmonic component of the radial component of the magnetic excitation force (also called the radially acting magnetic excitation force) induced by the rotor magnetomotive force and the stator current (fundamental frequency component), usually a radial vibration reducing harmonic current whose order is higher by one than that given order is superimposed on the stator current (polyphase AC current) thereby applying a magnetic excitation force of that given order that has a phase and amplitude such that the amplitude of its vector sum with the harmonic component becomes smaller. It has been found that this can effectively reduce the magnetic noise. It will be noted here that, by superimposing the above-described radial vibration reducing harmonic current, not only can the radial component of the magnetic excitation force induced by the rotor magnetomotive force and the stator current (fundamental frequency component) be reduced, but also other radiation vibrations of the AC rotary electric machine, such as an externally applied radial vibration, can likewise be reduced.
That is, according to the present invention, the radial vibration reducing harmonic current, whose frequency is n times the fundamental frequency, is superimposed on the energizing current supplied for generating torque. It has been found that this can reduce the harmonic component of the radial vibration that has a frequency (n−1) times the fundamental frequency of the energizing current. The reason why this can be accomplished will be described later.
In a preferred mode (claim 2), a radial vibration reducing harmonic current, whose frequency is 6k+1 times (k is a natural number) the frequency of the fundamental frequency component of a stator current of a three-phase AC rotary electric machine as the AC rotary electric machine, is superimposed on the stator current, thereby reducing the radial vibration of the AC rotary electric machine that has a frequency 6k times the frequency of the fundamental frequency component, as compared to the case where the superimposition is not performed. In this way, the 6k-th harmonic component of the magnetic noise, which constitutes the dominant magnetic noise component in the three-phase AC rotary electric machine, can be effectively reduced or canceled.
In a preferred mode (claim 3), by superimposing the radial vibration reducing harmonic current of order 7, the sixth harmonic component of the radial vibration is attenuated as compared to the case where the superimposition is not performed. In this way, the sixth harmonic component of the radial vibration, which has previously been the magnetic noise component having the largest amplitude, can be effectively reduced.
In a preferred mode (claim 4), by superimposing the radial vibration reducing harmonic current of order 13, the 12th harmonic component of the radial vibration is attenuated as compared to the case where the superimposition is not performed. It has been found that, by so doing, the 12th harmonic component of the radial vibration, which has previously been the second largest magnetic noise component uncomfortable to the ear, can be effectively reduced.
In a preferred mode (claim 5), by superimposing the radial vibration reducing harmonic currents of orders 7 and 13, the sixth and 12th harmonic components of the radial vibration are simultaneously attenuated as compared to the case where the superimposition is not performed. It has been found that, by so doing, the sixth and 12th harmonic components of the radial vibration, which have previously been the largest and the second largest magnetic noise components uncomfortable to the ear, can be effectively reduced.
In a preferred mode (claim 6), the phases and amplitudes of the radial vibration reducing seventh and 13th harmonic currents are set so that the amplitude of a vector sum, taken between the sixth and 12th harmonic components of the radial vibration occurring due to the fundamental wave current and the radial vibration reducing seventh and 13th harmonic currents and the sixth and 12th harmonic components of the radial vibration, becomes smaller than a predetermined value. The phases and amplitudes of the seventh and 13th radial vibration reducing harmonic currents may be obtained by calculating mathematical equations or by computation using a finite element method, or may be determined by experiment. It has been found that, by so doing, the sixth and 12th harmonic components of the radial vibration, which have previously been the largest and the second largest magnetic noise components uncomfortable to the ear, can be effectively reduced.
In a preferred mode (claim 7), by superimposing the radial vibration reducing harmonic current of order 19, the 18th harmonic component of the radial vibration is attenuated as compared to the case where the superimposition is not performed. This achieves a quiet operation of the rotary electric machine.
In a preferred mode (claim 8), by superimposing the radial vibration reducing harmonic current of order 25, the 24th harmonic component of the radial vibration is attenuated as compared to the case where the superimposition is not performed. This achieves a quiet operation of the rotary electric machine.
In a preferred mode (claim 9), when I1 denotes the amplitude of the fundamental frequency, In denotes the amplitude of the harmonic current, and t, x, and y respectively denote prescribed phase angles, the fundamental frequency component of the polyphase AC current contains a first phase fundamental frequency component Iu1 (=I1 sin(θ)), a second phase fundamental frequency component Iv1 (=I1 sin(θ−x)), and a third phase fundamental frequency component Iw1 (=I1 sin(θ−y)), and the radial vibration reducing harmonic current whose frequency is n times contains a first phase harmonic component Iun (=In sin n(θ+t)), a second phase harmonic component Ivn (=In sin {n(θ+t)−x}), and a third phase harmonic component Iwn (=In sin {n(θ+t)−y}), wherein the first phase harmonic component Iun is superimposed on the first phase fundamental frequency component Iu1, the second phase harmonic component Ivn is superimposed on the second phase fundamental frequency component Iv1, and the third phase harmonic component Iwn is superimposed on the third phase fundamental frequency component Iw1. By so doing, as the rotating sequence, i.e., the phase sequence, of the fundamental frequency component of each phase coincides with the rotating sequence of the n-th harmonic component of the vibration reducing current of each phase, the (n−1)th component of the radial vibration can be effectively reduced.
The above description has been given dealing with the case where the harmonic phase current whose frequency is n times that of the fundamental frequency component is superimposed on the phase current of the fundamental frequency in order to reduce the radial vibration whose frequency is (n−1) times that of the fundamental frequency component of the stator current. It will, however, be appreciated that a harmonic current whose order is higher by one than the above may be additionally superimposed in order to reduce a higher order radial vibration component.
A motor control apparatus according to the present invention (claim 10) comprises: rotational position detecting means for detecting the rotational position of an automotive M-phase synchronous AC rotary electric machine (M is a positive integer not smaller than 3); and motor current control means for applying, based on the detected rotational position, a prescribed phase current having a prescribed fundamental frequency and amplitude to each individual phase winding of an armature of the AC rotary electric machine, wherein the motor current control means superimposes, on the polyphase AC current, a radial vibration reducing harmonic current of order n having a frequency n times the frequency of the fundamental frequency component of the phase current, and thereby attenuates an (n−1)th harmonic component, having a frequency (n−1) times the frequency of the fundamental frequency component, that is contained in a radial vibration occurring radially about an axial center of a rotating shaft of the AC rotary electric machine due to an excitation force generated by the AC rotary electric machine or externally applied to the AC rotary electric machine. Thus, an AC rotary electric machine quiet in operation can be achieved using the magnetic noise reduction method of the present invention.
Usually, a rotation angle sensor such as a resolver for detecting the rotation angle of the rotor is employed for use as the rotational position detecting means, but alternatively, a circuit generally known as a sensorless type may be employed which estimates the rotation angle of the rotor based on the phase of the motor current or motor voltage.
In a preferred mode (claim 11), the motor current control means superimposes at least either one of the seventh and 13th harmonic components on the phase current, and thereby attenuates the sixth or 12th harmonic component of the radial vibration occurring in a three-phase AC rotary electric machine as the AC rotary electric machine, as compared to the case where the superimposition is not performed. According to this mode, as the motor control using the above-described magnetic noise reduction method is performed, the magnetic noise can be reduced effectively and reliably.
In a preferred mode (claim 12), by superimposing the radial vibration reducing harmonic current of order 19, the motor current control means attenuates the 18th harmonic component of the radial vibration as compared to the case where the superimposition is not performed. This achieves a quiet operation of the rotary electric machine.
In a preferred mode (claim 13), by superimposing the radial vibration reducing harmonic current of order 25, the motor current control means attenuates the 24th harmonic component of the radial vibration as compared to the case where the superimposition is not performed. This achieves a quiet operation of the rotary electric machine.
In a preferred mode (claim 14), the AC rotary electric machine is coupled to an automotive engine and/or to an axle, and the motor current control means superimposes the radial vibration reducing harmonic current on the polyphase AC current when the automotive engine is in a stopped condition. This serves to enhance comfort by reducing vehicle noise during stoppage of the engine when the magnetic noise of the rotary electric machine becomes noticeable.
In a preferred mode (claim 15), the AC rotary electric machine is coupled to an automotive engine and/or to an axle so as to be capable of regenerative braking, and the motor current control means superimposes the radial vibration reducing harmonic current on the polyphase AC current at least either when the automotive engine is in a stopped condition or when regenerative braking is being imposed. This serves to enhance comfort by reducing vehicle noise during stoppage of the engine when the magnetic noise of the rotary electric machine becomes noticeable, or during regenerative braking when the engine noise is small.
In a preferred mode (claim 16), the motor current control means obtains the radial vibration reducing harmonic current of order n that reduces or minimizes the radial vibration having a frequency (n−1) times the frequency of the fundamental frequency component and occurring due to the application of the phase current containing at least the fundamental frequency component and a harmonic current whose frequency is n times the frequency of the fundamental frequency component (but not containing the radial vibration reducing harmonic current of the present invention), and the motor current control means superimposes the thus obtained radial vibration reducing harmonic current on the phase current. Here, the radial vibration reducing harmonic current may be obtained from a prestored map or computed using a mathematical equation. The important point here is that the phase current contains not only the fundamental frequency component but also the harmonic current whose frequency is n times that of the fundamental frequency component. Then, in this mode, the radial vibration reducing harmonic current, that is, the harmonic current that reduces or minimizes the radial vibration occurring due to the phase current (actual phase current) which also contains the above-stated harmonic current, is superimposed on the phase current. The phase and amplitude of the radial vibration reducing harmonic current of the prescribed order to be superimposed vary according to the frequency, amplitude, etc. of the fundamental frequency component; therefore, these parameters are determined in advance by experiment or by a finite element method, and are stored in the form of a map so that the desired phase and amplitude can be read out or computed by entering the detected frequency and amplitude of the fundamental frequency component into the map. In this way, the radial vibration occurring due to the actual harmonic-containing phase current can be effectively reduced.
Alternatively, assuming that the armature current does not contain the n-th harmonic current, a magnetic noise reducing harmonic current of order n that reduces or minimizes the radial vibration having a frequency (n−1) times that of the detected or obtained fundamental frequency component of the armature current may be computed and, after subtracting the said harmonic current currently contained in the armature current from the magnetic noise reducing harmonic current, the resulting magnetic noise reducing harmonic current may be superimposed on the armature current. For example, in open-loop control, the magnetic noise reducing harmonic current of order n is obtained that minimizes the magnetic noise of order (n−1) which occurs due to the fundamental frequency component of the armature current to be supplied. Further, the n-th harmonic current that is expected to occur due to the fundamental frequency component of the armature current is obtained as an introduced harmonic current from the prestored map or from a mathematical equation, and the value obtained by subtracting the introduced harmonic current from the magnetic noise reducing harmonic current is superimposed on the fundamental frequency component of the armature current to be supplied and is used as the target value for the armature current control.
In a preferred mode (claim 17), the motor current control means detects as an introduced harmonic current a harmonic current contained in the phase current and having a frequency n times the frequency of the fundamental frequency component, obtains the radial vibration reducing harmonic current of order n as a feedback control target value for reducing or minimizing the (n−1)th harmonic component of the radial vibration for the case where the phase current does not contain the introduced harmonic current but contains only the fundamental frequency component, and performs feedback control to asymptotically reduce the difference between the introduced harmonic current and the target value to zero. This serves to further simplify circuit processing for magnetic noise reduction.
To describe more specifically, the magnetic noise reducing harmonic current is obtained that minimizes the magnetic noise for the case where the detected armature current contains only its fundamental frequency component. Next, feedback control is performed so that the difference between the harmonic current of the detected armature current and the magnetic noise reducing harmonic current is reduced to zero.
Likewise, from the detected phase current, the harmonic current (referred to as the introduced harmonic current), excluding its fundamental frequency component, is extracted. Next, radial vibration reducing harmonic currents of one or more particular orders are obtained that minimize radial vibrations of one or more particular orders for the case where the phase current contains only its fundamental frequency component, and the resulting value is set as the feedback target value. Then, feedback control is performed so as to asymptotically reduce the difference between the introduced harmonic current and the feedback target value to zero. In this case, and as, in the end, only the fundamental frequency component of the phase current and the radial vibration reducing harmonic current are contained in the detected phase current, the radial vibration can be minimized by using simple circuitry.
An AC rotary electric machine apparatus according to claim 18 comprises: an AC rotary electric machine having M-phase stator coils (M is a positive integer not smaller than 3); an inverter having a transistor device for controlling on and off a stator current for each phase of the synchronous machine; and an inverter control circuit which controls the transistor device on and off based on a detected or estimated rotation angle of the AC rotary electric machine, wherein the inverter control circuit includes harmonic PWM signal generating means for performing processing to output to the inverter a harmonic PWM signal having a frequency n times (n is an integer not smaller than 2) the frequency of the fundamental frequency component of the stator current, and thereby altering the amplitude of the magnetic noise of the AC rotary electric machine that has a frequency (n−1) times the frequency of the fundamental frequency component, as compared to the case where the processing is not performed.
That is, according to this invention, the harmonic PWM signal that has a frequency n times that of the fundamental frequency component of the stator current is input to the inverter that controls the stator current of the AC rotary electric machine operating in motor mode or in generator mode, and the magnetic sound (for example, the radially acting magnetic excitation force) of the AC rotary electric machine that has a frequency (n−1) times that of the fundamental frequency component is thus increased or reduced as compared to the normal case, thereby increasing or reducing the magnetic sound of the AC rotary electric machine. This can achieve an AC rotary electric machine that is extremely quiet in operation or that produces a desired magnetic sound.
For example, in the case of a hybrid vehicle, when the vehicle is running by operating the AC rotary electric machine with the engine stopped, if a sound similar to engine rotating sound is generated, the driver will feel secure. Here, a sound similar to engine rotating sound or unique to the AC rotary electric machine may be produced. Further, when a fault occurs in the vehicle or the AC rotary electric machine or when the driving condition changes, the magnetic noise can be varied according to the situation to alert the driver. The level, frequency, etc. of the magnetic noise can be preset by the driver to match the driver's preference for each driving condition range (rpm, output level, etc.) of the AC rotary electric machine. It is preferable that the level of the harmonic PWM signal be set within such a range that the resulting torque ripple does not exceed a predetermined level.
As earlier described, the circumferential vibration of the iron core causes torque ripple, while the radial vibration of the iron core causes magnetic sound on the outer or inner circumferential surface of the stator or rotor iron core. Accordingly, by altering the circumferential component of the magnetic excitation force, torque pulsation is altered, and by altering the radial component of the magnetic excitation force, the magnetic sound can be altered.
In the present invention, in order to alter a given order harmonic component of the radial component of the magnetic excitation force (also called the radially acting magnetic excitation force) induced by the rotor magnetomotive force and the stator current (fundamental frequency component), the harmonic current whose order is higher by one than that given order is superimposed on the stator current (polyphase AC current) thereby applying a magnetic excitation force of that given order that has a phase and amplitude such that the amplitude of its vector sum, with the harmonic component, changes. It has been found that this can effectively alter the magnetic sound.
That is, according to this invention, the harmonic current, whose frequency is n times the fundamental frequency of the stator current, is superimposed on the original stator current. It has been found that this can reduce or increase the radially acting magnetic excitation force that has a frequency (n−1) times the fundamental frequency of the energizing current. The reason why this can be accomplished will be described later.
In a preferred mode (claim 19), the harmonic PWM signal generating means performs processing to output to the inverter a first harmonic PWM signal having a frequency n1 times the frequency of the fundamental frequency component of the stator current and a second harmonic PWM signal having a frequency n2 times the frequency of the fundamental frequency component, and thereby increases (or reduces) the magnetic sound of the AC rotary electric machine that has a frequency (n1−1) times the frequency of the fundamental frequency component, as compared to the case where the processing is not performed, while at the same time, reducing (or increasing) the magnetic noise of the AC rotary electric machine that has a frequency (n2−1) times the frequency of the fundamental frequency component, as compared to the case where the processing is not performed. This makes it possible to alter and control each frequency component of the magnetic sound that the AC rotary electric machine generates.
In a preferred mode (claim 20), the inverter control circuit includes fundamental PWM signal generating means for outputting to the inverter a fundamental PWM signal for feeding the fundamental frequency component to each of the stator coils. In this way, the fundamental current having a frequency (fundamental frequency) corresponding to the fundamental PWM signal can be fed to the stator coil of the AC rotary electric machine. For example, when in motor mode, the electric machine can produce rotation at the rpm given by the fundamental frequency, and when in generator mode, the stator current having the fundamental frequency can be synchronously rectified.
In a preferred mode (claim 21), the harmonic PWM signal generating means increases the noise generated by the AC rotary electric machine when the harmonic PWM signal generating means is in an operating condition, as compared to when the harmonic PWM signal generating means is in an stopped condition. In this way, the AC rotary electric machine can generate larger magnetic sound than usual, for example, when in a prescribed operating condition of the AC rotary electric machine, or in accordance with the driver's taste.
In a preferred mode (claim 22), the harmonic PWM signal generating means reduces the magnetic sound of the AC rotary electric machine when the harmonic PWM signal generating means is in an operating condition, as compared to when the harmonic PWM signal generating means is in an stopped condition. This achieves an AC rotary electric machine that is extremely quiet in operation compared to the prior art.
In a preferred mode (claim 23), the harmonic PWM signal generating means increases or reduces the magnetic sound as compared to when the harmonic PWM signal generating means is in an stopped condition, by altering the phase and/or amplitude of the harmonic current to be added to the stator current in accordance with the number of revolutions. As the amplitude of the magnetic sound producing harmonic current contained in the stator current varies with the number of revolutions, i.e., the frequency of the fundamental current (fundamental frequency) contained in the stator current, the harmonic PWM signal is controlled accordingly to adjust the phase or amplitude of the harmonic current. With this arrangement, the magnetic sound reducing effect constantly works regardless of variations in the number of revolutions of the AC rotary electric machine, and thus an AC rotary electric machine quiet in operation, or capable of producing magnetic sound of desired frequency, can be achieved.
In a preferred mode (claim 24), the harmonic PWM signal generating means in the inverter control circuit performs processing to output to the inverter a harmonic PWM signal having a frequency n=6k+1 times (k is a natural number) the frequency of the fundamental frequency component of the stator current, and thereby alters the magnetic sound of a three-phase AC rotary electric machine as the AC rotary electric machine that has a frequency 6k times the frequency of the fundamental frequency component, as compared to the case where the processing is not performed. In this way, the 6k-th harmonic component, which is particularly noticeable in the three-phase AC rotary electric machine, can be easily altered, that is, reduced or increased.
In a preferred mode (claim 25), the harmonic PWM signal generating means outputs a harmonic PWM signal for superimposing a seventh harmonic current on the fundamental frequency component of the stator current, and thereby alters the sixth harmonic component of the magnetic sound as compared to the case where the superimposition is not implemented. In this way, the sixth harmonic component of the magnetic sound of the three-phase AC rotary electric machine can be effectively altered. For example, the sixth harmonic component of the radially acting magnetic excitation force, which has previously been the cause of the largest amplitude magnetic sound component, can be effectively reduced.
In a preferred mode (claim 26), the harmonic PWM signal generating means outputs a harmonic PWM signal for superimposing a 13th harmonic current on the fundamental frequency component of the stator current, and thereby alters the 12th harmonic component of the magnetic sound as compared to the case where the superimposition is not implemented. In this way, the 12th harmonic component of the magnetic noise of the three-phase AC rotary electric machine can be effectively altered. For example, the 12th harmonic component of the radial acting magnetic excitation force, which has previously been the cause of the second largest magnetic sound component uncomfortable to the ear, can be effectively reduced.
In a preferred mode (claim 27), the harmonic PWM signal generating means outputs a harmonic PWM signal for superimposing seventh and 13th harmonic currents on the fundamental frequency component of the stator current, and thereby alters the sixth and 12th harmonic components of the magnetic sound as compared to the case where the superimposition is not implemented. In this way, the sixth and 12th harmonic components of the magnetic sound of the three-phase AC rotary electric machine can be simultaneously altered. For example, the sixth and 12th harmonic components of the radial acting magnetic excitation force, which have previously been the causes of the largest and the second largest magnetic sound components uncomfortable to the ear, can be effectively reduced.
In a preferred mode (claim 28), the harmonic PWM signal generating means outputs a harmonic PWM signal for superimposing 19th and/or 25th harmonic currents on the fundamental frequency component of the stator current, and thereby alters the 18th and/or 24th harmonic components of the magnetic sound as compared to the case where the superimposition is not implemented. In this way, the 18th and/or 24th harmonic components of the magnetic sound of the three-phase AC rotary electric machine can be altered.
In a preferred mode (claim 29), n1 is 6k1+1 (k1 is a natural number), n2 is 6k2+1 (k2 is a natural number), and the harmonic PWM signal generating means, as compared to the case where the processing is not performed, reduces the magnetic sound of the three-phase AC rotary electric machine as the AC rotary electric machine that has a frequency 6k1 times the frequency of the fundamental frequency component and, at the same time, increases the magnetic sound of the three-phase AC rotary electric machine as the AC rotary electric machine that has a frequency 6k2 times (k2 is a natural number) the frequency of the fundamental frequency component.
In this way, magnetic sound of the desired frequency can be output while reducing the inherent magnetic sound of the three-phase AC rotary electric machine.
In a preferred mode (claim 30), k1 is 1. In this case, magnetic sound of the desired frequency can be produced while reducing the sixth harmonic of the magnetic noise.
In a preferred mode (claim 31), k1 is 2. In this case, magnetic sound of the desired frequency can be produced while reducing the 12th harmonic of the magnetic noise.
In a preferred mode (claim 32), k1 is 3. In this case, magnetic sound of the desired frequency can be produced while reducing the 18th harmonic of the magnetic noise.
In a preferred mode (claim 33), k1 is 4. In this case, magnetic sound of the desired frequency can be produced while reducing the 24th harmonic of the magnetic noise.
In a preferred mode (claim 34), the harmonic PWM signal generating means performs processing to output to the inverter a harmonic PWM signal having frequencies 7 and 13 times the frequency of the fundamental frequency component of the stator current and a harmonic PWM signal having at least either one of frequencies 19 and 25 times the frequency of the fundamental frequency component, and thereby attenuates the magnetic sound of the three-phase AC rotary electric machine as the AC rotary electric machine that has frequencies 6 and 12 times the frequency of the fundamental frequency component and the magnetic sound of the three-phase AC rotary electric machine as the AC rotary electric machine that has at least either one of frequencies 19 and 25 times the frequency of the fundamental frequency component, as compared to the case where the processing is not performed. This achieves an AC rotary electric machine that is quieter in operation.
In a preferred mode (claim 35), when I1 denotes the amplitude of the fundamental frequency of the stator current, In denotes the amplitude of the harmonic current of the stator current, and t, x, and y respectively denote prescribed phase angles, the fundamental frequency component of the stator current contains at least a first phase fundamental frequency component Iu1 (=I1 sin(θ)), a second phase fundamental frequency component Iv1 (=I1 sin(θ−x)), and a third phase fundamental frequency component Iw1 (=I1 sin(θ−y)), and the harmonic current whose frequency is n times contains at least a first phase harmonic component Iun (=In sin n(θ+t)), a second frequency harmonic component Ivn (=In sin{n(θ+t)−x}), and a third harmonic component Iwn (=In sin{n(θ+t)−y}), wherein the first phase harmonic component Iun is superimposed on the first phase fundamental frequency component Iu1, the second phase harmonic component Ivn is superimposed on the second phase fundamental frequency component Iv1, and the third phase harmonic component Iwn is superimposed on the third phase fundamental frequency component Iw1. By so doing, as the rotating sequence, i.e., the phase sequence, of the fundamental frequency component of each phase coincides with the rotating sequence of the n-th harmonic component of the vibration reducing current of each phase, the (n−1)th component of the radial vibration can be effectively reduced.
In a preferred mode (claim 36), the AC rotary electric machine is coupled to an automotive engine so as to be decouplable therefrom, and the inverter control circuit causes the magnetic sound to vary by performing processing to output the harmonic PWM signal to the inverter when the automotive engine is in a stopped condition. This serves to enhance comfort by reducing vehicle noise during stoppage of the engine when the magnetic noise of the rotary electric machine becomes noticeable.
In a preferred mode (claim 37), the AC rotary electric machine is coupled to an automotive engine and/or to an axle so as to be capable of regenerative braking, and the inverter control circuit causes the magnetic sound to vary by performing processing to output the harmonic PWM signal to the inverter at least either when the automotive engine is in a stopped condition or when regenerative braking is being imposed. This serves to enhance comfort by reducing vehicle noise during stoppage of the engine when the magnetic noise of the rotary electric machine becomes noticeable, or during regenerative braking when the engine noise is small.
(Modified Modes)
1. It will be appreciated that n, which denotes the order of the radial vibration reducing harmonic current (and the radial vibration altering harmonic current) (order n means that the frequency of the radial vibration reducing harmonic current (or the radial vibration altering harmonic current) is the n-th multiple of that of the fundamental frequency component), can contain manufacturing tolerances of the harmonic current generating circuit. For example, n may be set within a range of (n)−0.1 to (n)+0.1.
2. Each of the above configurations has been described for the case of reducing or altering the radial vibration, but at the same time, the a harmonic current for reducing or altering the circumferentially acting magnetic excitation force (torque ripple) may be additionally superimposed.
3. Various types of synchronous machines can be advantageously employed for use as the AC rotary electric machine, and either operating mode, the motor mode or the generator mode, may be employed as its operating mode.
4. Further, the radial vibration reducing harmonic current (and the radial vibration altering harmonic current) may be superimposed in all rpm ranges, or the radial vibration reducing harmonic current (and the radial vibration altering harmonic current) may be superimposed only in rpm ranges where the magnetic noise becomes a particular problem.
A radial vibration component of a given order may be reduced (or altered) by superimposing a radial vibration reducing harmonic current (or a radial vibration altering harmonic current) of a given order, or a plurality of radial vibration components of given orders may be reduced or altered by superimposing a plurality of radial vibration reducing harmonic currents (radial vibration altering harmonic currents) of given orders. Further, the harmonic current superimposing operation may be switched, according to the rpm range, between the superimposition of the radial vibration reducing (or altering) harmonic current and the superimposition of the torque ripple reducing (or altering) harmonic current.
5. The phases and amplitudes of the harmonic currents to be superimposed may be obtained by calculating the mathematical equations to be given later or by computation using a finite element method, or may be determined by experiment.
6. For rotational position detection, a rotation angle sensor such as a resolver for detecting the rotation angle of the rotor is usually employed, but alternatively, a circuit generally known as a sensorless type may be employed which estimates the rotation angle of the rotor based on the phase of the motor current or motor voltage.
7. The armature current of the rotary electric machine contains harmonic currents of various orders in addition to its fundamental frequency component, and the magnitude of each harmonic current varies depending on such conditions as the operating condition of the rotary electric machine. If magnetic noise reducing (or altering) harmonic currents were to be calculated for the armature current having such a complex waveform, the calculation procedures would become very complex, and the amount of circuitry required would increase. In view of this, assuming that the armature current does not contain the n-th harmonic current, a magnetic noise reducing (or altering) harmonic current of order n that reduces or minimizes the radial vibration having a frequency (n−1) times that of the detected or obtained fundamental frequency component of the armature current is computed and, after subtracting the said harmonic current currently contained in the armature current from the magnetic noise reducing (or altering) harmonic current, the resulting magnetic noise reducing (or altering) harmonic current is superimposed on the armature current. In this way, even if the armature current contains harmonic components in addition to its fundamental frequency component, the calculation and superimposition process of the radial vibration reducing harmonic current (and the radial vibration altering harmonic current) can be simplified. For example, in open-loop control, the magnetic noise reducing (or altering) harmonic current of order n is obtained that minimizes the magnetic noise of order (n−1) which occurs due to the fundamental frequency component of the armature current to be supplied. Further, the n-th harmonic current that is expected to occur due to the fundamental frequency component of the armature current is obtained as an introduced harmonic current from the prestored map or from a mathematical equation, and the value obtained by subtracting the introduced harmonic current from the magnetic noise reducing (or altering) harmonic current is superimposed on the fundamental frequency component of the armature current to be supplied and is used as the target value for the armature current control.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a diagram schematically showing a magnetic circuit for one phase of a polyphase AC rotary electric machine.
FIG. 2 is an equivalent circuit diagram of the magnetic circuit of FIG. 1.
FIG. 3 is a block circuit diagram showing an example of a motor control circuit which employs a magnetic sound altering method according to the present invention.
FIG. 4 is a block circuit diagram showing an example of a motor control circuit which employs a magnetic sound altering method according to the present invention.
FIG. 5 is a block circuit diagram showing an example of a motor control circuit which employs a magnetic sound altering method according to the present invention.
FIG. 6 is a block circuit diagram showing an example of a motor control circuit which employs a magnetic sound altering method according to the present invention.
FIG. 7 is a block circuit diagram showing an example of a motor control circuit which employs a magnetic sound altering method according to the present invention.
FIG. 8 is a schematic cross-sectional view, taken in a radial direction, of a three-phase synchronous machine used in an experiment.
FIG. 9 is a diagram showing the waveforms (magnetic noise non-reduction mode) of the respective phase currents of the three-phase synchronous machine depicted in FIG. 8.
FIG. 10 is a diagram showing the waveforms (magnetic noise reduction mode) of the respective phase currents of the three-phase synchronous machine depicted in FIG. 8.
FIG. 11 is a diagram showing the frequency spectra (magnetic noise non-reduction mode versus magnetic noise reduction mode) of the phase currents of the three-phase synchronous machine depicted in FIG. 8.
FIG. 12 is a diagram showing the waveforms (magnetic noise non-reduction mode versus magnetic noise reduction mode) of radially acting excitation forces in the three-phase synchronous machine depicted in FIG. 8.
FIG. 13 is a diagram showing the frequency spectra (magnetic noise non-reduction mode versus magnetic noise reduction mode) of the radially acting excitation forces in the three-phase synchronous machine depicted in FIG. 8.
FIG. 14 is a diagram showing the result of the measurement (in motor mode) taken of the magnetic noise (sound) of the three-phase synchronous machine depicted in FIG. 8.
FIG. 15 is a diagram showing the result of the measurement (in generator mode) taken of the magnetic noise (sound) of the three-phase synchronous machine depicted in FIG. 8.
FIG. 16 is a flowchart for compensating for temperature variations of rotor magnetic flux.
FIG. 17 is a flowchart for reducing the amplitude of magnetic sound in accordance with the operation mode of an AC rotary electric machine.
FIG. 18 is a flowchart for reducing the magnetic noise of the three-phase AC rotary electric machine and producing magnetic sound of a designated frequency.
FIG. 19 is a flowchart for reducing, under advantageous conditions, the inherent magnetic noise of the three-phase AC rotary electric machine mounted in an engine-powered vehicle.
FIG. 20 is a block circuit diagram showing another circuit example.
FIG. 21 is a block circuit diagram showing one example of the circuit shown in FIG. 20.
FIG. 22 is a waveform diagram showing signal waveforms (in a coordinate system rotating at the fundamental frequency) in various parts in FIG. 20.
FIG. 23 is a waveform diagram showing signal waveforms (in a stationary coordinate system) in various parts in FIG. 20.
FIG. 24 is a block circuit diagram showing another embodiment.
FIG. 25 is a block circuit diagram showing another embodiment.
FIG. 26 is a flowchart showing another embodiment.



BEST MODE FOR CARRYING OUT THE INVENTION
The preferred modes of the present invention will be described below with reference to the accompanying drawings.
(Description of the Basic Principle)
The basic principle will be described below for the case where the present invention is applied to a polyphase AC rotary electric machine.
FIG. 1 is a diagram schematically showing a magnetic circuit for one phase of an N-phase AC rotary electric machine, and FIG. 2 is an equivalent circuit diagram of the magnetic circuit of FIG. 1. In a synchronous machine, magnetic flux φ is formed by the magnetic poles of a rotor (formed from a coil or a permanent magnet), and rotor magnetomotive force Fmag represents the magnetomotive force, i.e., the magnetic field intensity, of the rotor's magnetic poles in the magnetic circuit, while stator magnetomotive force Fcoil represents the magnetomotive force, i.e., the magnetic field intensity, formed in the magnetic circuit by a stator current. Rg denotes the magnetic resistance of a gap between the stator and the rotor.
As seen from FIGS. 1 and 2, the magnetic flux φ, magnetic energy W, and radially acting magnetic excitation force f, per phase, are defined respectively by




  
    
      
        Magnetic
        ⁢
        
          
        
        ⁢
        flux
      
    
    
      
        
      
    
  
  
    
      
        φ
        =
        
          
            
              
                F
                mag
              
              +
              
                F
                coil
              
            
            
              2
              ⁢
              
                R
                g
              
            
          
          =
          
            
              
                F
                mag
              
              +
              
                NI
                coil
              
            
            
              2
              ⁢
              
                R
                g
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  
  
    
      
        Magnetic
        ⁢
        
          
        
        ⁢
        energy
      
    
    
      
        
      
    
  
  
    
      
        W
        =
        
          
            
              1
              2
            
            ⁢
            
              φ
              2
            
            ⁢
            
              R
              g
            
          
          =
          
            
              
                1
                
                  8
                  ⁢
                  
                    R
                    g
                  
                
              
              ⁢
              
                
                  (
                  
                    
                      F
                      mag
                    
                    +
                    
                      NI
                      coil
                    
                  
                  )
                
                2
              
            
            =
            
              
                1
                
                  8
                  ⁢
                  x
                
              
              ⁢
              
                
                  (
                  
                    
                      F
                      mag
                    
                    +
                    
                      NI
                      coil
                    
                  
                  )
                
                2
              
              ⁢
              
                μ
                0
              
              ⁢
              S
            
          
        
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        Magnetic
        ⁢
        
          
        
        ⁢
        excitation
        ⁢
        
          
        
        ⁢
        force
      
    
    
      
        
      
    
  
  
    
      
        f
        =
        
          
            ∂
            W
          
          
            ∂
            x
          
        
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                -
                
                  1
                  
                    8
                    ⁢
                    
                      x
                      2
                    
                  
                
              
              ⁢
              
                μ
                0
              
              ⁢
              
                
                  S
                  ⁡
                  
                    (
                    
                      
                        F
                        mag
                      
                      +
                      
                        NI
                        coil
                      
                    
                    )
                  
                
                2
              
            
            ∝
            
              
                (
                
                  
                    F
                    mag
                  
                  +
                  
                    NI
                    coil
                  
                
                )
              
              2
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              F
              mag
              2
            
            +
            
              2
              ⁢
              
                NF
                mag
              
              ⁢
              
                I
                coil
              
            
            +
            
              
                N
                2
              
              ⁢
              
                I
                coil
                2
              
            
          
        
      
    
    
      
        
      
    
  




That is, the magnetic excitation force f is defined as the sum of the square of the rotor magnetomotive force, the square of the stator magnetomotive force, and the product of the rotor magnetomotive force and the stator magnetomotive force.

In the above figures and equations, Icoil is the stator current (the phase current of the armature), x is the gap width, S is the opposing area size of the gap, μ0 is the permeability of air, N is the number of turns in each phase coil of the armature, and M is the number of phases.
The rotor magnetomotive force Fmag and the stator current (phase current) Icoil for each of the first phase (U phase in the case of three phases), second phase (V phase in the case of three phases), and M-th phase (W phase in the case of three phases) of the polyphase AC rotary electric machine are shown below.




  
    
      
        U
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      j
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      j
                      ⁡
                      
                        (
                        
                          θ
                          +
                          β
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      k
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      l
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  Fundamental
                  ⁢
                  
                    
                  
                  ⁢
                  j
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  k
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  L
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  (
                  
                    first
                    ⁢
                    
                      
                    
                    ⁢
                    harmonic
                  
                  )
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          s
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      m
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      m
                      ⁡
                      
                        (
                        
                          θ
                          +
                          t
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      n
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      n
                      ⁡
                      
                        (
                        
                          θ
                          +
                          u
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  Fundamental
                  ⁢
                  
                    
                  
                  ⁢
                  m
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  n
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  (
                  
                    first
                    ⁢
                    
                      
                    
                    ⁢
                    harmonic
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  
  
    
      
        
          
            
              V
              ⁢
              
                
              
              ⁢
              phase
            
          
          
            
              
            
          
          
            
              x
              =
              
                360
                /
                M
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          x
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      j
                      ⁡
                      
                        (
                        
                          θ
                          +
                          β
                          -
                          x
                        
                        )
                      
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      k
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                          -
                          x
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      l
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                          -
                          x
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          s
                          -
                          x
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      m
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      {
                      
                        
                          m
                          ⁡
                          
                            (
                            
                              θ
                              +
                              t
                            
                            )
                          
                        
                        -
                        x
                      
                      }
                    
                  
                  +
                  
                    
                      I
                      n
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      {
                      
                        
                          n
                          ⁡
                          
                            (
                            
                              θ
                              +
                              u
                            
                            )
                          
                        
                        -
                        x
                      
                      }
                    
                  
                
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          
            
              W
              ⁢
              
                
              
              ⁢
              phase
            
          
          
            
              
            
          
          
            
              y
              =
              
                360
                ⁢
                
                  
                    (
                    
                      M
                      -
                      1
                    
                    )
                  
                  /
                  M
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          y
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      j
                      ⁡
                      
                        (
                        
                          θ
                          +
                          β
                          -
                          y
                        
                        )
                      
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      k
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                          -
                          y
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      l
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                          -
                          y
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          s
                          -
                          y
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      m
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      {
                      
                        
                          m
                          ⁡
                          
                            (
                            
                              θ
                              +
                              t
                            
                            )
                          
                        
                        -
                        y
                      
                      }
                    
                  
                  +
                  
                    
                      I
                      n
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      {
                      
                        
                          n
                          ⁡
                          
                            (
                            
                              θ
                              +
                              u
                            
                            )
                          
                        
                        -
                        y
                      
                      }
                    
                  
                
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




The rotor magnetomotive forces given above are expressed by trigonometric functions to express the rotor rotation. The rotor magnetomotive forces Fmag and the stator currents Icoil for the remaining phases when the number of phases is larger than three will not be given herein, since they can be easily derived by a person skilled in the art. As a matter of course, x is 360/M=120 degrees in the case of three phases.
Here, Fi (i is a subscript) represents the amplitude of the i-th harmonic of the rotor magnetomotive force, Ii represents the amplitude of the i-th harmonic of the stator current, θ denotes the rotation angle of the rotor, α, β, γ, δ, s, t, and u respectively denote phase angles, and j, k, L, m, and n are integer values.
The above equations have assumed that the rotor magnetomotive force Fmag has only the j-th, k-th, and L-th harmonics as the harmonic components, and that the stator current Icoil has only the m-th and n-th harmonics as the harmonic components, but it is apparent that they may further include other harmonics. When the application is limited to a three-phase AC rotary electric machine for simplicity, the equations (4), (5), and (6) can be written as shown below.




  
    
      
        U
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      j
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      j
                      ⁡
                      
                        (
                        
                          θ
                          +
                          β
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      k
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      l
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  Fundamental
                  ⁢
                  
                    
                  
                  ⁢
                  j
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  k
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  L
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  (
                  
                    first
                    ⁢
                    
                      
                    
                    ⁢
                    harmonic
                  
                  )
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          s
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      m
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      m
                      ⁡
                      
                        (
                        
                          θ
                          +
                          t
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      n
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      n
                      ⁡
                      
                        (
                        
                          θ
                          +
                          u
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  Fundamental
                  ⁢
                  
                    
                  
                  ⁢
                  m
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  n
                  ⁢
                  
                    -
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  (
                  
                    first
                    ⁢
                    
                      
                    
                    ⁢
                    harmonic
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        V
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          120
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      j
                      ⁡
                      
                        (
                        
                          θ
                          +
                          β
                          -
                          120
                        
                        )
                      
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      k
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                          -
                          120
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      l
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                          -
                          120
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          s
                          -
                          120
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      m
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      {
                      
                        
                          m
                          ⁡
                          
                            (
                            
                              θ
                              +
                              t
                            
                            )
                          
                        
                        -
                        120
                      
                      }
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    I
                    n
                  
                  ⁢
                  
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    {
                    
                      
                        n
                        ⁡
                        
                          (
                          
                            θ
                            +
                            u
                          
                          )
                        
                      
                      -
                      120
                    
                    }
                  
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  
  
    
      
        W
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          240
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      j
                      ⁡
                      
                        (
                        
                          θ
                          +
                          β
                          -
                          240
                        
                        )
                      
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      k
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                          -
                          240
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      l
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                          -
                          240
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          s
                          -
                          240
                        
                        )
                      
                    
                  
                  +
                  
                    
                      I
                      m
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      {
                      
                        
                          m
                          ⁡
                          
                            (
                            
                              θ
                              +
                              t
                            
                            )
                          
                        
                        -
                        240
                      
                      }
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    I
                    n
                  
                  ⁢
                  
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    {
                    
                      
                        n
                        ⁡
                        
                          (
                          
                            θ
                            +
                            u
                          
                          )
                        
                      
                      -
                      240
                    
                    }
                  
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




The U-phase excitation force fu is found from the following equation (10) which is obtained by substituting the equation (7) into the equation (3).




  
    
      
        
          
            
              
                U
                ⁢
                
                  -
                
                ⁢
                phase
                ⁢
                
                  
                
                ⁢
                excitation
                ⁢
                
                  
                
                ⁢
                force
                ⁢
                
                  
                
                ⁢
                fu
              
              ∝
            
          
        
        
          
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      F
                      1
                      2
                    
                    +
                    
                      F
                      j
                      2
                    
                    +
                    
                      F
                      k
                      2
                    
                    +
                    
                      F
                      l
                      2
                    
                  
                  )
                
              
              -
              
                
                  
                    F
                    1
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    α
                  
                  )
                
              
              -
              
                
                  
                    F
                    j
                    2
                  
                  2
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    j
                    ⁢
                    
                      (
                      
                        θ
                        +
                        β
                      
                      )
                    
                  
                  
                    
                      -
                      
                        
                      
                      ⁢
                      
                        ·
                        
                          
                        
                        ⁢
                        
                          -
                          
                            
                          
                          ⁢
                          
                            ·
                            
                              
                            
                            ⁢
                            
                              -
                              
                                
                              
                              ⁢
                              
                                ·
                                
                                  
                                
                                ⁢
                                
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    ·
                                    
                                      
                                    
                                    ⁢
                                    
                                      -
                                      
                                        
                                      
                                      ·
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                  
                
              
              -
            
          
        
        
          
            
              
                
                  
                    F
                    k
                    2
                  
                  2
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    k
                    ⁢
                    
                      (
                      
                        θ
                        +
                        γ
                      
                      )
                    
                  
                  
                    
                      -
                      
                        
                      
                      ⁢
                      
                        ·
                        
                          
                        
                        ⁢
                        
                          -
                          
                            
                          
                          ⁢
                          
                            ·
                            
                              
                            
                            ⁢
                            
                              -
                              
                                
                              
                              ⁢
                              
                                ·
                                
                                  
                                
                                ⁢
                                
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    ·
                                    
                                      
                                    
                                    ⁢
                                    
                                      -
                                      
                                        
                                      
                                      ·
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                  
                
              
              -
              
                
                  
                    F
                    l
                    2
                  
                  2
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    l
                    ⁢
                    
                      (
                      
                        θ
                        +
                        δ
                      
                      )
                    
                  
                  
                    -
                    
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ·
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              -
            
          
        
        
          
            
              
                
                  F
                  1
                
                ⁢
                
                  
                    F
                    j
                  
                  ⁡
                  
                    [
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            α
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                          
                          }
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            -
                            α
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                          
                          }
                        
                      
                    
                    ]
                  
                
              
              -
            
          
        
        
          
            
              
                
                  
                    F
                    1
                  
                  ⁢
                  
                    
                      F
                      k
                    
                    [
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              α
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                            
                            }
                          
                        
                        _
                      
                      -
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              -
                              α
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                            
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        --
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    }
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  l
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          α
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                        
                        }
                      
                    
                    _
                  
                  -
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  l
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            -
                            α
                            +
                            
                              l
                              ⁢
                              
                                
                              
                              ⁢
                              δ
                            
                          
                          }
                        
                      
                      ]
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    
                      F
                      j
                    
                    ⁢
                    
                      
                        F
                        k
                      
                      [
                      
                        c
                        ⁢
                        
                          
                            os
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      k
                                      +
                                      j
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                +
                                
                                  k
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                              
                            
                          
                          _
                        
                      
                      }
                    
                  
                  -
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                k
                                -
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                        
                        }
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                
                  
                    
                      F
                      j
                    
                    ⁢
                    
                      
                        F
                        l
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    +
                                    j
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                j
                                ⁢
                                
                                  
                                
                                ⁢
                                β
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                            
                          
                        
                        _
                      
                      }
                    
                  
                  -
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                -
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                        
                        }
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                
                  F
                  k
                
                ⁢
                
                  
                    F
                    l
                  
                  [
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  l
                                  +
                                  k
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              k
                              ⁢
                              
                                
                              
                              ⁢
                              γ
                            
                            +
                            
                              l
                              ⁢
                              
                                
                              
                              ⁢
                              δ
                            
                          
                          }
                        
                      
                      _
                    
                    -
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  l
                                  -
                                  k
                                
                                )
                              
                              ⁢
                              θ
                            
                            -
                            
                              k
                              ⁢
                              
                                
                              
                              ⁢
                              γ
                            
                            +
                            
                              l
                              ⁢
                              
                                
                              
                              ⁢
                              δ
                            
                          
                          }
                        
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  ]
                
              
              +
            
          
        
        
          
            
              
                2
                ⁢
                N
                ⁢
                
                  〈
                  
                    
                      
                        
                          
                            
                              -
                              
                                
                                  
                                    F
                                    1
                                  
                                  ⁢
                                  
                                    I
                                    1
                                  
                                
                                2
                              
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        α
                                        +
                                        s
                                      
                                      )
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      ...
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ..
                                  
                                
                                -
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      α
                                      -
                                      s
                                    
                                    )
                                  
                                
                              
                              }
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      F
                                      1
                                    
                                    ⁢
                                    
                                      I
                                      m
                                    
                                  
                                  2
                                
                                [
                                
                                  
                                    cos
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            (
                                            
                                              m
                                              +
                                              1
                                            
                                            )
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        α
                                        +
                                        mt
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                ...
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ..
                                  
                                
                                }
                              
                              -
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            m
                                            -
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      -
                                      α
                                      +
                                      mt
                                    
                                    }
                                  
                                
                                _
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      F
                                      1
                                    
                                    ⁢
                                    
                                      I
                                      n
                                    
                                  
                                  2
                                
                                [
                                
                                  
                                    cos
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            (
                                            
                                              n
                                              +
                                              1
                                            
                                            )
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        α
                                        +
                                        nu
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                ...
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    .
                                  
                                
                                }
                              
                              -
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            n
                                            -
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      -
                                      α
                                      +
                                      nu
                                    
                                    }
                                  
                                
                                _
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  j
                                
                                ⁢
                                
                                  I
                                  1
                                
                              
                              2
                            
                            ⁡
                            
                              [
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            j
                                            +
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        j
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        β
                                      
                                      +
                                      s
                                    
                                    }
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            j
                                            -
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        j
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        β
                                      
                                      -
                                      s
                                    
                                    }
                                  
                                
                              
                              ]
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  j
                                
                                ⁢
                                
                                  I
                                  m
                                
                              
                              2
                            
                            ⁡
                            
                              [
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            j
                                            +
                                            m
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        j
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        β
                                      
                                      +
                                      mt
                                    
                                    }
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            j
                                            -
                                            m
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        j
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        β
                                      
                                      -
                                      mt
                                    
                                    }
                                  
                                
                              
                              ]
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  j
                                
                                ⁢
                                
                                  I
                                  n
                                
                              
                              2
                            
                            ⁡
                            
                              [
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            j
                                            +
                                            n
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        j
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        β
                                      
                                      +
                                      nu
                                    
                                    }
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            j
                                            -
                                            n
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        j
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        β
                                      
                                      -
                                      nu
                                    
                                    }
                                  
                                
                              
                              ]
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                k
                              
                              ⁢
                              
                                I
                                1
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    +
                                    s
                                  
                                  }
                                
                              
                              _
                            
                            -
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            k
                                            -
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        k
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        γ
                                      
                                      -
                                      s
                                    
                                    }
                                  
                                
                                ]
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      F
                                      k
                                    
                                    ⁢
                                    
                                      I
                                      m
                                    
                                  
                                  2
                                
                                [
                                
                                  
                                    cos
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            (
                                            
                                              k
                                              +
                                              m
                                            
                                            )
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        
                                          k
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          γ
                                        
                                        +
                                        mt
                                      
                                    
                                  
                                  _
                                
                                }
                              
                              -
                              
                                c
                                ⁢
                                
                                  
                                    os
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            (
                                            
                                              k
                                              -
                                              m
                                            
                                            )
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        
                                          k
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          γ
                                        
                                        -
                                        mt
                                      
                                      }
                                    
                                  
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    -
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      F
                                      k
                                    
                                    ⁢
                                    
                                      I
                                      n
                                    
                                  
                                  2
                                
                                [
                                
                                  
                                    cos
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            (
                                            
                                              k
                                              +
                                              n
                                            
                                            )
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        
                                          k
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          γ
                                        
                                        +
                                        nu
                                      
                                    
                                  
                                  _
                                
                                }
                              
                              -
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            k
                                            -
                                            n
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        k
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        γ
                                      
                                      -
                                      nu
                                    
                                    }
                                  
                                
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                l
                              
                              ⁢
                              
                                I
                                1
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    +
                                    s
                                  
                                  }
                                
                              
                              _
                            
                            -
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            l
                                            -
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        l
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        δ
                                      
                                      -
                                      s
                                    
                                    }
                                  
                                
                                ]
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      F
                                      l
                                    
                                    ⁢
                                    
                                      I
                                      m
                                    
                                  
                                  2
                                
                                [
                                
                                  
                                    cos
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            (
                                            
                                              l
                                              +
                                              m
                                            
                                            )
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        
                                          l
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          δ
                                        
                                        +
                                        mt
                                      
                                    
                                  
                                  _
                                
                                }
                              
                              -
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            l
                                            -
                                            m
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        l
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        δ
                                      
                                      -
                                      mt
                                    
                                    }
                                  
                                
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                l
                              
                              ⁢
                              
                                I
                                n
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          +
                                          n
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    +
                                    nu
                                  
                                  }
                                
                              
                              _
                            
                            -
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          -
                                          n
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    -
                                    nu
                                  
                                  }
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                    
                  
                  〉
                
              
              +
            
          
        
        
          
            
              
                N
                2
              
              ⁢
              
                〈
                
                  
                    
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                I
                                1
                                2
                              
                              +
                              
                                I
                                m
                                2
                              
                              +
                              
                                I
                                n
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          
                            
                              I
                              1
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              2
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  s
                                
                                )
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          ...
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              ...
                            
                          
                        
                        -
                        
                          
                            
                              I
                              m
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              2
                              ⁢
                              m
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  t
                                
                                )
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            ...
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              .
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              I
                              n
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              2
                              ⁢
                              
                                n
                                ⁡
                                
                                  (
                                  
                                    θ
                                    +
                                    u
                                  
                                  )
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            ...
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              ...
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            I
                            1
                          
                          ⁢
                          
                            
                              I
                              m
                            
                            [
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            m
                                            +
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      s
                                      +
                                      mt
                                    
                                    }
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              ...
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  .
                                
                              
                              -
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            m
                                            -
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      -
                                      s
                                      +
                                      mt
                                    
                                    }
                                  
                                
                                _
                              
                            
                            ]
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              
                                
                                  I
                                  1
                                
                                ⁢
                                
                                  
                                    I
                                    n
                                  
                                  [
                                  
                                    
                                      cos
                                      ⁢
                                      
                                        {
                                        
                                          
                                            
                                              (
                                              
                                                n
                                                +
                                                1
                                              
                                              )
                                            
                                            ⁢
                                            θ
                                          
                                          +
                                          s
                                          +
                                          nu
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                ...
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ..
                                    
                                  
                                  }
                                
                              
                              -
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            n
                                            -
                                            1
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      -
                                      s
                                      +
                                      nu
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            ...
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ..
                                
                              
                            
                            }
                          
                          ]
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          I
                          m
                        
                        ⁢
                        
                          
                            I
                            n
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            m
                                            +
                                            n
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      mt
                                      +
                                      nu
                                    
                                    }
                                  
                                
                                _
                              
                              -
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            n
                                            -
                                            m
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      -
                                      mt
                                      +
                                      nu
                                    
                                    }
                                  
                                
                                _
                              
                            
                            ]
                          
                        
                      
                    
                  
                
                〉
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




The V-phase excitation force fv is found from the following equation (11) which is obtained by substituting the equation (8) into the equation (3).




  
    
      
        
          
            
              
                V
                ⁢
                
                  -
                
                ⁢
                phase
                ⁢
                
                  
                
                ⁢
                excitation
                ⁢
                
                  
                
                ⁢
                force
                ⁢
                
                  
                
                ⁢
                fv
              
              ∝
            
          
        
        
          
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      F
                      1
                      2
                    
                    +
                    
                      F
                      j
                      2
                    
                    +
                    
                      F
                      k
                      2
                    
                    +
                    
                      F
                      l
                      2
                    
                  
                  )
                
              
              -
              
                
                  
                    F
                    1
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    α
                    -
                    120
                  
                  )
                
              
              -
            
          
        
        
          
            
              
                
                  
                    
                      F
                      j
                      2
                    
                    2
                  
                  ⁢
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        j
                        (
                        
                          θ
                          +
                          β
                          -
                          120
                        
                      
                    
                    
                      -
                      
                        
                      
                      ⁢
                      
                        ·
                        
                          
                        
                        ⁢
                        
                          -
                          
                            
                          
                          ⁢
                          
                            ·
                            
                              
                            
                            ⁢
                            
                              -
                              
                                
                              
                              ⁢
                              
                                ·
                                
                                  
                                
                                ⁢
                                
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    ·
                                    
                                      
                                    
                                    ⁢
                                    
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        ·
                                        
                                          
                                        
                                        ⁢
                                        
                                          -
                                          
                                            
                                          
                                          ⁢
                                          
                                            ·
                                            
                                              
                                            
                                            ⁢
                                            
                                              -
                                              
                                                
                                              
                                              ⁢
                                              
                                                ·
                                                
                                                  
                                                
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                )
              
              -
              
                
                  
                    F
                    k
                    2
                  
                  2
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                          -
                          120
                        
                        )
                      
                    
                  
                  
                    -
                    
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          ⁢
                                          
                                            -
                                            
                                              
                                            
                                            ⁢
                                            
                                              ·
                                              
                                                
                                              
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              -
            
          
        
        
          
            
              
                
                  
                    F
                    l
                    2
                  
                  2
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                          -
                          120
                        
                        )
                      
                    
                  
                  
                    -
                    
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          ⁢
                                          
                                            -
                                            
                                              
                                            
                                            ⁢
                                            
                                              ·
                                              
                                                
                                              
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              -
            
          
        
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  j
                
                [
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              j
                              +
                              1
                            
                            )
                          
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        -
                        
                          120
                          ⁢
                          
                            (
                            
                              j
                              +
                              1
                            
                            )
                          
                        
                      
                      }
                    
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            j
                            -
                            1
                          
                          )
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      -
                      
                        120
                        ⁢
                        
                          (
                          
                            j
                            -
                            1
                          
                          )
                        
                      
                    
                    }
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  k
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                k
                                +
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          α
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          -
                          
                            120
                            ⁢
                            
                              (
                              
                                k
                                +
                                1
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                k
                                -
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          α
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          -
                          
                            120
                            ⁢
                            
                              (
                              
                                k
                                -
                                1
                              
                              )
                            
                          
                        
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  }
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  l
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          α
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            120
                            ⁢
                            
                              (
                              
                                l
                                +
                                1
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              l
                              -
                              1
                            
                            )
                          
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          l
                          ⁢
                          
                            
                          
                          ⁢
                          δ
                        
                        -
                        
                          120
                          ⁢
                          
                            (
                            
                              l
                              -
                              1
                            
                            )
                          
                        
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                j
              
              ⁢
              
                
                  F
                  k
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                k
                                +
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          -
                          
                            120
                            ⁢
                            
                              (
                              
                                k
                                +
                                j
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              k
                              -
                              j
                            
                            )
                          
                          ⁢
                          θ
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          k
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                        -
                        
                          120
                          ⁢
                          
                            (
                            
                              k
                              -
                              j
                            
                            )
                          
                        
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                j
              
              ⁢
              
                
                  F
                  l
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            120
                            ⁢
                            
                              (
                              
                                l
                                +
                                j
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                -
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            120
                            ⁢
                            
                              (
                              
                                l
                                -
                                j
                              
                              )
                            
                          
                        
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  }
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                k
              
              ⁢
              
                
                  F
                  l
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                k
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            120
                            ⁢
                            
                              (
                              
                                l
                                +
                                k
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              l
                              -
                              k
                            
                            )
                          
                          ⁢
                          θ
                        
                        -
                        
                          k
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                        +
                        
                          l
                          ⁢
                          
                            
                          
                          ⁢
                          δ
                        
                        -
                        
                          120
                          ⁢
                          
                            (
                            
                              l
                              -
                              k
                            
                            )
                          
                        
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                2
                ⁢
                N
                ⁢
                
                  〈
                  
                    
                      
                        
                          
                            
                              -
                              
                                
                                  
                                    F
                                    1
                                  
                                  ⁢
                                  
                                    I
                                    1
                                  
                                
                                2
                              
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        α
                                        +
                                        s
                                        -
                                        240
                                      
                                      )
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              ...
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                
                                -
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      α
                                      -
                                      s
                                    
                                    )
                                  
                                
                              
                              }
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                1
                              
                              ⁢
                              
                                I
                                m
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    α
                                    +
                                    mt
                                    -
                                    240
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    
                                                                      ...
                                                                      ...
                                                                    
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                .
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    -
                                    α
                                    +
                                    mt
                                  
                                  }
                                
                              
                              _
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                1
                              
                              ⁢
                              
                                I
                                n
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          n
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    α
                                    +
                                    nu
                                    -
                                    240
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    ...
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          n
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    -
                                    α
                                    +
                                    nu
                                  
                                  }
                                
                              
                              _
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                j
                              
                              ⁢
                              
                                I
                                1
                              
                            
                            2
                          
                          [
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  +
                                  s
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  -
                                  s
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                j
                              
                              ⁢
                              
                                I
                                m
                              
                            
                            2
                          
                          [
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        +
                                        m
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  +
                                  mt
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        -
                                        m
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  -
                                  mt
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                j
                              
                              ⁢
                              
                                I
                                n
                              
                            
                            2
                          
                          [
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        +
                                        n
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  +
                                  nu
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        -
                                        n
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  -
                                  nu
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  k
                                
                                ⁢
                                
                                  I
                                  1
                                
                              
                              2
                            
                            [
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    +
                                    s
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              _
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    -
                                    s
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          --
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                k
                              
                              ⁢
                              
                                I
                                m
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          +
                                          m
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    +
                                    mt
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              _
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          -
                                          m
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    -
                                    mt
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            --
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  k
                                
                                ⁢
                                
                                  I
                                  n
                                
                              
                              2
                            
                            ⁢
                            
                              
                                [
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            k
                                            +
                                            n
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        k
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        γ
                                      
                                      +
                                      nu
                                      -
                                      
                                        120
                                        ⁢
                                        
                                          (
                                          
                                            k
                                            +
                                            1
                                          
                                          )
                                        
                                      
                                    
                                    }
                                  
                                
                              
                              _
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          -
                                          n
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    -
                                    nu
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              ]
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          -
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                l
                              
                              ⁢
                              
                                I
                                1
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    +
                                    s
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              _
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    -
                                    s
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        --
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  l
                                
                                ⁢
                                
                                  I
                                  m
                                
                              
                              2
                            
                            ⁢
                            
                              
                                [
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            l
                                            +
                                            m
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        l
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        δ
                                      
                                      +
                                      mt
                                      -
                                      
                                        120
                                        ⁢
                                        
                                          (
                                          
                                            l
                                            +
                                            1
                                          
                                          )
                                        
                                      
                                    
                                    }
                                  
                                
                              
                              _
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          -
                                          m
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    -
                                    mt
                                    -
                                    
                                      120
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              ]
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          -
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  l
                                
                                ⁢
                                
                                  I
                                  n
                                
                              
                              2
                            
                            ⁢
                            
                              
                                [
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            l
                                            +
                                            n
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        l
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        δ
                                      
                                      +
                                      nu
                                      -
                                      
                                        120
                                        ⁢
                                        
                                          (
                                          
                                            l
                                            +
                                            1
                                          
                                          )
                                        
                                      
                                    
                                    }
                                  
                                
                              
                              _
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        l
                                        -
                                        n
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    l
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    δ
                                  
                                  -
                                  nu
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        l
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        -
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                    
                  
                  〉
                
              
              +
            
          
        
        
          
            
              
                N
                2
              
              ⁢
              
                〈
                
                  
                    
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                I
                                1
                                2
                              
                              +
                              
                                I
                                m
                                2
                              
                              +
                              
                                I
                                n
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          
                            
                              I
                              1
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              2
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  s
                                  -
                                  120
                                
                                )
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    ...
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              ..
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              I
                              m
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ⁡
                                      
                                        (
                                        
                                          θ
                                          +
                                          t
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  240
                                
                                }
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        ...
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              .
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              I
                              n
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ⁡
                                      
                                        (
                                        
                                          θ
                                          +
                                          u
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  240
                                
                                }
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        ...
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              ..
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          I
                          1
                        
                        ⁢
                        
                          
                            I
                            m
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    s
                                    +
                                    mt
                                    -
                                    240
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    ...
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                            
                            -
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        m
                                        -
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  -
                                  s
                                  +
                                  mt
                                
                                }
                              
                            
                            ]
                          
                          _
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            I
                            1
                          
                          ⁢
                          
                            I
                            n
                          
                          ⁢
                          
                            
                              [
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          n
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    s
                                    +
                                    nu
                                    -
                                    240
                                  
                                  }
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    ...
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              .
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        n
                                        -
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  -
                                  s
                                  +
                                  nu
                                
                                }
                              
                            
                            _
                          
                          ]
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          I
                          m
                        
                        ⁢
                        
                          
                            I
                            n
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          +
                                          n
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    mt
                                    +
                                    nu
                                    -
                                    240
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    
                                                                      ...
                                                                      ...
                                                                    
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                            
                            -
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      n
                                      -
                                      m
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                -
                                mt
                                +
                                nu
                              
                              }
                            
                          
                          _
                        
                        ]
                      
                    
                  
                
                〉
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




The W-phase excitation force fw is found from the following equation (12) which is obtained by substituting the equation (9) into the equation (3).




  
    
      
        
          
            
              
                W
                ⁢
                
                  -
                
                ⁢
                phase
                ⁢
                
                  
                
                ⁢
                excitation
                ⁢
                
                  
                
                ⁢
                force
                ⁢
                
                  
                
                ⁢
                fw
              
              ∝
            
          
        
        
          
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      F
                      1
                      2
                    
                    +
                    
                      F
                      j
                      2
                    
                    +
                    
                      F
                      k
                      2
                    
                    +
                    
                      F
                      l
                      2
                    
                  
                  )
                
              
              -
              
                
                  
                    F
                    1
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    α
                    -
                    240
                  
                  )
                
              
              -
            
          
        
        
          
            
              
                
                  
                    
                      F
                      j
                      2
                    
                    2
                  
                  ⁢
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        j
                        (
                        
                          θ
                          +
                          β
                          -
                          240
                        
                      
                    
                    
                      
                        -
                        
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          ⁢
                                          
                                            -
                                            
                                              
                                            
                                            ⁢
                                            
                                              ·
                                              
                                                
                                              
                                              ⁢
                                              
                                                -
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  ·
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    -
                                                    
                                                      
                                                    
                                                    ⁢
                                                    
                                                      ·
                                                      -
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                
                )
              
              -
              
                
                  
                    F
                    k
                    2
                  
                  2
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      k
                      ⁡
                      
                        (
                        
                          θ
                          +
                          γ
                          -
                          240
                        
                        )
                      
                    
                  
                  
                    
                      -
                      
                        
                      
                      ⁢
                      
                        ·
                        
                          
                        
                        ⁢
                        
                          -
                          
                            
                          
                          ⁢
                          
                            ·
                            
                              
                            
                            ⁢
                            
                              -
                              
                                
                              
                              ⁢
                              
                                ·
                                
                                  
                                
                                ⁢
                                
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    ·
                                    
                                      
                                    
                                    ⁢
                                    
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        ·
                                        
                                          
                                        
                                        ⁢
                                        
                                          -
                                          
                                            
                                          
                                          ⁢
                                          
                                            ·
                                            
                                              
                                            
                                            ⁢
                                            
                                              -
                                              
                                                
                                              
                                              ⁢
                                              
                                                ·
                                                
                                                  
                                                
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                  
                
              
              -
            
          
        
        
          
            
              
                
                  
                    F
                    l
                    2
                  
                  2
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      l
                      ⁡
                      
                        (
                        
                          θ
                          +
                          δ
                          -
                          240
                        
                        )
                      
                    
                  
                  
                    
                      -
                      
                        
                      
                      ⁢
                      
                        ·
                        
                          
                        
                        ⁢
                        
                          -
                          
                            
                          
                          ⁢
                          
                            ·
                            
                              
                            
                            ⁢
                            
                              -
                              
                                
                              
                              ⁢
                              
                                ·
                                
                                  
                                
                                ⁢
                                
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    ·
                                    
                                      
                                    
                                    ⁢
                                    
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        ·
                                        
                                          
                                        
                                        ⁢
                                        
                                          -
                                          
                                            
                                          
                                          ⁢
                                          
                                            ·
                                            
                                              
                                            
                                            ⁢
                                            
                                              -
                                              
                                                
                                              
                                              ⁢
                                              
                                                ·
                                                
                                                  
                                                
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                  
                
              
              -
            
          
        
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  j
                
                [
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              j
                              +
                              1
                            
                            )
                          
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        -
                        
                          240
                          ⁢
                          
                            (
                            
                              j
                              +
                              1
                            
                            )
                          
                        
                      
                      }
                    
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            j
                            -
                            1
                          
                          )
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      -
                      
                        240
                        ⁢
                        
                          (
                          
                            j
                            -
                            1
                          
                          )
                        
                      
                    
                    }
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                
                  F
                  1
                
                ⁢
                
                  
                    F
                    k
                  
                  [
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                k
                                +
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          α
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                k
                                +
                                1
                              
                              )
                            
                          
                        
                      
                    
                    _
                  
                  }
                
              
              -
            
          
        
        
          
            
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              k
                              -
                              1
                            
                            )
                          
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          k
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                        -
                        
                          240
                          ⁢
                          
                            (
                            
                              k
                              -
                              1
                            
                            )
                          
                        
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  l
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          α
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                l
                                +
                                1
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                -
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          α
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                l
                                -
                                1
                              
                              )
                            
                          
                        
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  }
                
                ]
              
              -
            
          
        
        
          
            
              
                
                  F
                  j
                
                ⁢
                
                  
                    F
                    k
                  
                  [
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                k
                                +
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                k
                                +
                                j
                              
                              )
                            
                          
                        
                      
                    
                    _
                  
                  }
                
              
              -
            
          
        
        
          
            
              
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                k
                                -
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                k
                                -
                                j
                              
                              )
                            
                          
                        
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                --
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  }
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                j
              
              ⁢
              
                
                  F
                  l
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                l
                                +
                                j
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                -
                                j
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            β
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                l
                                -
                                j
                              
                              )
                            
                          
                        
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  }
                
                ]
              
              -
            
          
        
        
          
            
              
                F
                k
              
              ⁢
              
                
                  F
                  l
                
                [
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                k
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                l
                                +
                                k
                              
                              )
                            
                          
                        
                        }
                      
                    
                    _
                  
                  -
                
              
            
          
        
        
          
            
              
                
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                -
                                k
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                          -
                          
                            240
                            ⁢
                            
                              (
                              
                                l
                                -
                                k
                              
                              )
                            
                          
                        
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  }
                
                ]
              
              +
            
          
        
        
          
            
              
                2
                ⁢
                N
                ⁢
                
                  〈
                  
                    
                      
                        
                          
                            
                              -
                              
                                
                                  
                                    F
                                    1
                                  
                                  ⁢
                                  
                                    I
                                    1
                                  
                                
                                2
                              
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        α
                                        +
                                        s
                                        -
                                        480
                                      
                                      )
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              ...
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ..
                                  
                                
                                -
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      α
                                      -
                                      s
                                    
                                    )
                                  
                                
                              
                              }
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                1
                              
                              ⁢
                              
                                I
                                m
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    α
                                    +
                                    mt
                                    -
                                    480
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    ...
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    -
                                    α
                                    +
                                    mt
                                  
                                  }
                                
                              
                              _
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                1
                              
                              ⁢
                              
                                I
                                n
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          n
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    α
                                    +
                                    nu
                                    -
                                    480
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    ...
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ..
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          n
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    -
                                    α
                                    +
                                    nu
                                  
                                  }
                                
                              
                              _
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                j
                              
                              ⁢
                              
                                I
                                1
                              
                            
                            2
                          
                          [
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  +
                                  s
                                  -
                                  
                                    240
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  -
                                  s
                                  -
                                  
                                    240
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                j
                              
                              ⁢
                              
                                I
                                m
                              
                            
                            2
                          
                          [
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        +
                                        m
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  +
                                  mt
                                  -
                                  
                                    240
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        -
                                        m
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  -
                                  mt
                                  -
                                  
                                    240
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                j
                              
                              ⁢
                              
                                I
                                n
                              
                            
                            2
                          
                          [
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        +
                                        n
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  +
                                  nu
                                  -
                                  
                                    240
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        j
                                        -
                                        n
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    β
                                  
                                  -
                                  nu
                                  -
                                  
                                    240
                                    ⁢
                                    
                                      (
                                      
                                        j
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                }
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  k
                                
                                ⁢
                                
                                  I
                                  1
                                
                              
                              2
                            
                            [
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    +
                                    s
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              _
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    -
                                    s
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          --
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  k
                                
                                ⁢
                                
                                  I
                                  m
                                
                              
                              2
                            
                            [
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          k
                                          +
                                          m
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      k
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    +
                                    mt
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              _
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            k
                                            -
                                            m
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        k
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        γ
                                      
                                      -
                                      mt
                                      -
                                      
                                        240
                                        ⁢
                                        
                                          (
                                          
                                            k
                                            -
                                            1
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              -
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                              }
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      F
                                      k
                                    
                                    ⁢
                                    
                                      I
                                      n
                                    
                                  
                                  2
                                
                                ⁢
                                
                                  
                                    [
                                    
                                      cos
                                      ⁢
                                      
                                        {
                                        
                                          
                                            
                                              (
                                              
                                                k
                                                +
                                                n
                                              
                                              )
                                            
                                            ⁢
                                            θ
                                          
                                          +
                                          
                                            k
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            γ
                                          
                                          +
                                          nu
                                          -
                                          
                                            240
                                            ⁢
                                            
                                              (
                                              
                                                k
                                                +
                                                1
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  _
                                
                              
                              )
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    cos
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            (
                                            
                                              k
                                              -
                                              n
                                            
                                            )
                                          
                                          ⁢
                                          θ
                                        
                                        +
                                        
                                          k
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          γ
                                        
                                        -
                                        nu
                                        -
                                        
                                          240
                                          ⁢
                                          
                                            (
                                            
                                              k
                                              -
                                              1
                                            
                                          
                                        
                                      
                                    
                                  
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              --
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                )
                              
                              }
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                F
                                l
                              
                              ⁢
                              
                                I
                                1
                              
                            
                            2
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    +
                                    s
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              _
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          -
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    -
                                    s
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  }
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        --
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  l
                                
                                ⁢
                                
                                  I
                                  m
                                
                              
                              2
                            
                            [
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          +
                                          m
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    +
                                    mt
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              _
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    {
                                    
                                      
                                        
                                          (
                                          
                                            l
                                            -
                                            m
                                          
                                          )
                                        
                                        ⁢
                                        θ
                                      
                                      +
                                      
                                        l
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        δ
                                      
                                      -
                                      mt
                                      -
                                      
                                        240
                                        ⁢
                                        
                                          (
                                          
                                            l
                                            -
                                            1
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            -
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                              }
                            
                            ]
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  F
                                  l
                                
                                ⁢
                                
                                  I
                                  n
                                
                              
                              2
                            
                            [
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          +
                                          n
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    +
                                    nu
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              _
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          l
                                          -
                                          n
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      l
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      δ
                                    
                                    -
                                    nu
                                    -
                                    
                                      240
                                      ⁢
                                      
                                        (
                                        
                                          l
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          -
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            }
                          
                          ]
                        
                      
                    
                  
                  〉
                
              
              +
            
          
        
        
          
            
              
                N
                2
              
              ⁢
              
                〈
                
                  
                    
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                I
                                1
                                2
                              
                              +
                              
                                I
                                m
                                2
                              
                              +
                              
                                I
                                n
                                2
                              
                            
                            )
                          
                        
                        -
                        
                          
                            
                              I
                              1
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              2
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  s
                                  -
                                  240
                                
                                )
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    ...
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              .
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              I
                              m
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ⁡
                                      
                                        (
                                        
                                          θ
                                          +
                                          t
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  480
                                
                                }
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      ...
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              ...
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              I
                              n
                              2
                            
                            2
                          
                          ⁢
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ⁡
                                      
                                        (
                                        
                                          θ
                                          +
                                          u
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  480
                                
                                }
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      ...
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              ...
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          I
                          1
                        
                        ⁢
                        
                          
                            I
                            m
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    s
                                    +
                                    mt
                                    -
                                    480
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    ...
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ..
                              
                            
                            -
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        m
                                        -
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  -
                                  s
                                  +
                                  mt
                                
                                }
                              
                            
                            ]
                          
                          _
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            I
                            1
                          
                          ⁢
                          
                            I
                            n
                          
                          ⁢
                          
                            
                              [
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          n
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    s
                                    +
                                    nu
                                    -
                                    480
                                  
                                  }
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  ...
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                              ..
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        n
                                        -
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  -
                                  s
                                  +
                                  nu
                                
                                }
                              
                            
                            _
                          
                          ]
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          I
                          m
                        
                        ⁢
                        
                          
                            I
                            n
                          
                          [
                          
                            
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      
                                        (
                                        
                                          m
                                          +
                                          n
                                        
                                        )
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    mt
                                    +
                                    nu
                                    -
                                    480
                                  
                                  }
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    
                                                                      ...
                                                                      ...
                                                                    
                                                                    ...
                                                                  
                                                                  ...
                                                                
                                                                ...
                                                              
                                                              ...
                                                            
                                                            ...
                                                          
                                                          ...
                                                        
                                                        ...
                                                      
                                                      ...
                                                    
                                                    ...
                                                  
                                                  ...
                                                
                                                ...
                                              
                                              ...
                                            
                                            ...
                                          
                                          ...
                                        
                                        ...
                                      
                                      ...
                                    
                                    ...
                                  
                                  ...
                                
                                ...
                              
                            
                            -
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        n
                                        -
                                        m
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  -
                                  mt
                                  +
                                  nu
                                
                              
                            
                            _
                          
                          }
                        
                        ]
                      
                    
                  
                
                〉
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




In the equations (4) to (6), θ is the angle of the fundamental wave. It is apparent that, when the angular velocity of the fundamental wave is denoted by ω, θ is equal to ωt and, when the frequency of the fundamental wave (the fundamental frequency) is denoted by f, θ is equal to 2πft. Further, in the equations (4) to (6), it is convenient to represent the numeric values 360, 120, and 240 as 2π, 2π/3, and 4π/3, respectively, in actual calculations.
Magnetic sound has positive correlation with the vector sum of the above phase excitation forces which, as shown by the equations (10), (11), and (12), are expressed by linear functions (equations containing sums or differences) comprising large numbers of terms. The sum of the phase excitation forces is a linear function (an equation containing sums or differences) comprising terms (hereinafter sometimes referred to as the vector sum terms) each obtained by vector summing the terms of the same order (same frequency) in the respective phases. Magnetic sound becomes a problem when the terms constituting each vector sum term are in phase (similar in phase) and are therefore reinforcing each other, but there is hardly any problem when the terms constituting each vector sum term greatly differ in phase because, in that case, the amplitude of the vector sum term is small.
More specifically, when the equations (10), (11), and (12) are summed together, the terms constituting each vector sum term are in phase and the amplitude of the vector sum term becomes much larger than the amplitude of each constituent term when the order is (m−1), (n−1), and (n−m).
From the equations (10), (11), and (12), it can be seen that when the m-th and n-th harmonic current components are superimposed on the stator current Icoil, excitation force components of orders (m−1), (n−1), and (n−m) are always markedly generated.
This means that, when the x-th harmonic current component having a certain phase is superimposed, an excitation force of order (x−1) can be induced.
Accordingly, if the x-th harmonic current that induces an excitation force equal in amplitude but opposite in phase to the (x−1)th component of the magnetic sound is superimposed in order to cancel the (x−1)th component which is responsible for the currently occurring magnetic sound, then the magnetic sound can be canceled. Here, if the phase is not exactly opposite, or if the amplitude is not exactly equal, the magnetic sound can be greatly reduced by reducing the amplitude of their vector sum.
Further, if the x-th harmonic current that induces the (x−1)th component of the magnetic sound is superimposed on the fundamental frequency component of the stator current in order to alter (increase or reduce) the same component as the (x−1)th component which is responsible for the currently occurring magnetic sound, then the magnetic sound can be altered (increased or reduced). That is, the amplitude of their vector sum can be increased or reduced.
The notion that the x-th harmonic current having a suitable phase and suitable amplitude should be superimposed to increase or reduce the (x−1)th component of the magnetic noise of the AC rotary electric machine has not been known in the prior art, and is expected to play a very important role in the future development of low-noise motors.
Likewise, as can be seen from the equations (10), (11), and (12), it is also possible to simultaneously alter (increase or reduce) the excitation force components of orders (m−1), (n−1), and (n−m) by superimposing the m-th and n-th harmonic current components on the stator current Icoil. In this case, however, as the excitation force components of orders (m−1), (n−1), and (n−m) are induced by the phases and amplitudes of the superimposed m-th and n-th harmonic current components, it is not easy to make all the vector sums equal to zero between these excitation forces and the corresponding (m−1)th, (n−1)th, and (n−m)th components inherently contained in the magnetic sound. However, the phases and amplitudes of the respectively superimposed currents can be adjusted so that the amplitude of the vector sum with the excitation force of each order becomes as small as possible or becomes equal to the desired magnitude.
The notion that the m-th and n-th harmonic current components each having a suitable phase and suitable amplitude should be superimposed to simultaneously alter (increase or reduce) the (m−1)th, (n−1)th, and (n−m)th components of the magnetic sound of the AC rotary electric machine has not been known in the prior art, and is expected to play a very important role in the future development of low-noise motors.
The equations (10), (11), and (12) are simplified as shown below by eliminating those terms for which the vector sum is zero.




  
    
      
        
          U
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          fu
        
        ∝
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          
            1
            2
          
          ⁢
          
            (
            
              
                F
                1
                2
              
              +
              
                F
                j
                2
              
              +
              
                F
                k
                2
              
              +
              
                F
                l
                2
              
            
            )
          
        
        -
        
          
            
              F
              j
              2
            
            2
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          2
          ⁢
          
            j
            ⁡
            
              (
              
                θ
                +
                β
              
              )
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              F
              k
              2
            
            2
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          2
          ⁢
          
            
          
          ⁢
          
            k
            ⁡
            
              (
              
                θ
                +
                γ
              
              )
            
          
        
        -
        
          
            
              F
              l
              2
            
            2
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          2
          ⁢
          
            
          
          ⁢
          
            l
            ⁡
            
              (
              
                θ
                +
                δ
              
              )
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              j
            
            ⁡
            
              [
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            j
                            +
                            1
                          
                          )
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                    
                    }
                  
                
                -
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            j
                            -
                            1
                          
                          )
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                    
                    }
                  
                
              
              ]
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  k
                
                [
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              k
                              +
                              1
                            
                            )
                          
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          k
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                      
                    
                  
                  _
                
                }
              
            
            -
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          k
                          -
                          1
                        
                        )
                      
                      ⁢
                      θ
                    
                    -
                    α
                    +
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                  
                  }
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                -
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  F
                  1
                
                ⁢
                
                  
                    F
                    l
                  
                  [
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          α
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                        
                      
                    
                    _
                  
                  }
                
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            l
                            -
                            1
                          
                          )
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        l
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                -
                              
                            
                          
                        
                      
                    
                  
                
              
            
            }
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                F
                j
              
              ⁢
              
                
                  F
                  k
                
                [
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              k
                              +
                              j
                            
                            )
                          
                          ⁢
                          θ
                        
                        +
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          k
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                      
                    
                  
                  _
                
                }
              
            
            -
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          k
                          -
                          j
                        
                        )
                      
                      ⁢
                      θ
                    
                    -
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      β
                    
                    +
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                  
                  }
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                -
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              F
              j
            
            ⁢
            
              
                F
                l
              
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              l
                              +
                              j
                            
                            )
                          
                          ⁢
                          θ
                        
                        +
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          l
                          ⁢
                          
                            
                          
                          ⁢
                          δ
                        
                      
                      }
                    
                  
                  _
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          
                            (
                            
                              l
                              -
                              j
                            
                            )
                          
                          ⁢
                          θ
                        
                        -
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          l
                          ⁢
                          
                            
                          
                          ⁢
                          δ
                        
                      
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  --
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              }
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  F
                  k
                
                ⁢
                
                  
                    F
                    l
                  
                  [
                  
                    
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                l
                                +
                                k
                              
                              )
                            
                            ⁢
                            θ
                          
                          +
                          
                            k
                            ⁢
                            
                              
                            
                            ⁢
                            γ
                          
                          +
                          
                            l
                            ⁢
                            
                              
                            
                            ⁢
                            δ
                          
                        
                      
                    
                    _
                  
                  }
                
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            l
                            -
                            k
                          
                          )
                        
                        ⁢
                        θ
                      
                      -
                      
                        k
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      +
                      
                        l
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                --
                              
                            
                          
                        
                      
                    
                  
                
              
            
            }
          
          ]
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            -
                            s
                          
                          )
                        
                      
                    
                    +
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            m
                          
                        
                        2
                      
                      ⁢
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    m
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              -
                              α
                              +
                              mt
                            
                            }
                          
                        
                        _
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      ⁢
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    n
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              -
                              α
                              +
                              nu
                            
                            }
                          
                        
                        _
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            j
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      j
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                +
                                s
                              
                              }
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      j
                                      -
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                -
                                s
                              
                              }
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            j
                          
                          ⁢
                          
                            I
                            m
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      j
                                      +
                                      m
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                +
                                mt
                              
                              }
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      j
                                      -
                                      m
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                -
                                mt
                              
                              }
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            j
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      j
                                      +
                                      n
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                +
                                nu
                              
                              }
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      j
                                      -
                                      n
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                -
                                nu
                              
                              }
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            k
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      k
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  k
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                +
                                s
                              
                              }
                            
                          
                          _
                        
                        -
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      k
                                      -
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  k
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                -
                                s
                              
                              }
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          k
                        
                        ⁢
                        
                          I
                          m
                        
                      
                      2
                    
                    [
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    +
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              +
                              mt
                            
                            }
                          
                        
                        _
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    -
                                    m
                                  
                                  )
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              -
                              mt
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            k
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    +
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              +
                              nu
                            
                          
                        
                        _
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    -
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              -
                              nu
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            l
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      l
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  l
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  δ
                                
                                +
                                s
                              
                              }
                            
                          
                          _
                        
                        -
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      l
                                      -
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  l
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  δ
                                
                                -
                                s
                              
                              }
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            l
                          
                          ⁢
                          
                            I
                            m
                          
                        
                        2
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    +
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              mt
                            
                          
                        
                        _
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    -
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              mt
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          l
                        
                        ⁢
                        
                          I
                          n
                        
                      
                      2
                    
                    [
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    +
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              nu
                            
                            }
                          
                        
                        _
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  l
                                  -
                                  n
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              l
                              ⁢
                              
                                
                              
                              ⁢
                              δ
                            
                            -
                            nu
                          
                          }
                        
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          I
                          1
                          2
                        
                        +
                        
                          I
                          m
                          2
                        
                        +
                        
                          I
                          n
                          2
                        
                      
                      )
                    
                  
                  +
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      I
                      m
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  m
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            -
                            s
                            +
                            mt
                          
                          }
                        
                      
                      _
                    
                  
                  +
                
              
            
            
              
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      I
                      n
                    
                    ⁢
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  n
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            -
                            s
                            +
                            nu
                          
                          }
                        
                      
                      _
                    
                  
                  +
                  
                    
                      I
                      m
                    
                    ⁢
                    
                      I
                      n
                    
                    ⁢
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  n
                                  -
                                  m
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            -
                            mt
                            +
                            nu
                          
                          }
                        
                      
                      _
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  
  
    
      
        
          V
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          fv
        
        ∝
      
    
    
      
        (
        14
        )
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    F
                    1
                    2
                  
                  +
                  
                    F
                    j
                    2
                  
                  +
                  
                    F
                    k
                    2
                  
                  +
                  
                    F
                    l
                    2
                  
                
                )
              
            
            -
            
              
                
                  F
                  j
                  2
                
                2
              
              ⁢
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    j
                    (
                    
                      θ
                      +
                      β
                      -
                      120
                    
                  
                
                
                  -
                  
                    
                  
                  ⁢
                  
                    ·
                    
                      
                    
                    ⁢
                    
                      -
                      
                        
                      
                      ⁢
                      
                        ·
                        
                          
                        
                        ⁢
                        
                          -
                          
                            
                          
                          ⁢
                          
                            ·
                            
                              
                            
                            ⁢
                            
                              -
                              
                                
                              
                              ⁢
                              
                                ·
                                
                                  
                                
                                ⁢
                                
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    ·
                                    
                                      
                                    
                                    ⁢
                                    
                                      -
                                      
                                        
                                          
                                        
                                        ⁢
                                        
                                          
                                        
                                      
                                      ⁢
                                      
                                        ·
                                        
                                          
                                        
                                        ⁢
                                        
                                          -
                                          
                                            
                                          
                                          ⁢
                                          
                                            ·
                                            
                                              
                                            
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          )
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              F
              k
              2
            
            2
          
          ⁢
          
            
              cos
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                
              
              ⁢
              
                k
                ⁡
                
                  (
                  
                    θ
                    +
                    γ
                    -
                    120
                  
                  )
                
              
            
            
              
                -
                
                  
                    
                  
                  ⁢
                  
                    
                  
                
                ⁢
                
                  ·
                  
                    
                  
                  ⁢
                  
                    -
                    
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ⁢
              
                
              
            
          
        
        -
        
          
            
              F
              l
              2
            
            2
          
          ⁢
          
            
              cos
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                
              
              ⁢
              
                l
                ⁡
                
                  (
                  
                    θ
                    +
                    δ
                    -
                    120
                  
                  )
                
              
            
            
              
                -
                
                  
                
                ⁢
                
                  ·
                  
                    
                  
                  ⁢
                  
                    -
                    
                      
                        
                      
                      ⁢
                      
                        
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                            
                          
                          ⁢
                          
                            
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                                
                              
                              ⁢
                              
                                
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                    
                                  
                                  ⁢
                                  
                                    
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ⁢
              
                
              
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          1
        
        ⁢
        
          
            F
            j
          
          [
          
            
              cos
              ⁢
              
                {
                
                  
                    
                      (
                      
                        j
                        +
                        1
                      
                      )
                    
                    ⁢
                    θ
                  
                  +
                  α
                  +
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    β
                  
                  -
                  
                    120
                    ⁢
                    
                      (
                      
                        j
                        +
                        1
                      
                      )
                    
                  
                
                }
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            cos
            ⁢
            
              {
              
                
                  
                    (
                    
                      j
                      -
                      1
                    
                    )
                  
                  ⁢
                  θ
                
                -
                α
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  β
                
                -
                
                  120
                  ⁢
                  
                    (
                    
                      j
                      -
                      1
                    
                    )
                  
                
              
              }
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              k
            
            [
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          k
                          +
                          1
                        
                        )
                      
                      ⁢
                      θ
                    
                    +
                    α
                    +
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                    -
                    
                      120
                      ⁢
                      
                        (
                        
                          k
                          +
                          1
                        
                        )
                      
                    
                  
                
              
              _
            
            }
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              cos
              ⁢
              
                {
                
                  
                    
                      (
                      
                        k
                        -
                        1
                      
                      )
                    
                    ⁢
                    θ
                  
                  -
                  α
                  +
                  
                    k
                    ⁢
                    
                      
                    
                    ⁢
                    γ
                  
                  -
                  
                    120
                    ⁢
                    
                      (
                      
                        k
                        -
                        1
                      
                      )
                    
                  
                
                }
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        --
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          1
        
        ⁢
        
          
            F
            l
          
          [
          
            
              c
              ⁢
              
                
                  os
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            l
                            +
                            1
                          
                          )
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        l
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      
                        120
                        ⁢
                        
                          (
                          
                            l
                            +
                            1
                          
                          )
                        
                      
                    
                    }
                  
                
                _
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          l
                          -
                          1
                        
                        )
                      
                      ⁢
                      θ
                    
                    -
                    α
                    +
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    -
                    
                      120
                      ⁢
                      
                        (
                        
                          l
                          -
                          1
                        
                        )
                      
                    
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        --
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            }
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          j
        
        ⁢
        
          
            F
            k
          
          [
          
            
              c
              ⁢
              
                
                  os
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            k
                            +
                            j
                          
                          )
                        
                        ⁢
                        θ
                      
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        k
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      
                        120
                        ⁢
                        
                          (
                          
                            k
                            +
                            j
                          
                          )
                        
                      
                    
                    }
                  
                
                _
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              cos
              ⁢
              
                {
                
                  
                    
                      (
                      
                        k
                        -
                        j
                      
                      )
                    
                    ⁢
                    θ
                  
                  -
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    β
                  
                  +
                  
                    k
                    ⁢
                    
                      
                    
                    ⁢
                    γ
                  
                  -
                  
                    120
                    ⁢
                    
                      (
                      
                        k
                        -
                        j
                      
                      )
                    
                  
                
                }
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            j
          
          ⁢
          
            
              F
              l
            
            [
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          l
                          +
                          j
                        
                        )
                      
                      ⁢
                      θ
                    
                    +
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      β
                    
                    +
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    -
                    
                      120
                      ⁢
                      
                        (
                        
                          l
                          +
                          j
                        
                        )
                      
                    
                  
                
              
              _
            
            }
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          l
                          -
                          j
                        
                        )
                      
                      ⁢
                      θ
                    
                    -
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      β
                    
                    +
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    -
                    
                      120
                      ⁢
                      
                        (
                        
                          l
                          -
                          j
                        
                        )
                      
                    
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            }
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          k
        
        ⁢
        
          
            F
            l
          
          [
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          l
                          +
                          k
                        
                        )
                      
                      ⁢
                      θ
                    
                    +
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                    +
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    -
                    
                      120
                      ⁢
                      
                        (
                        
                          l
                          +
                          k
                        
                        )
                      
                    
                  
                  }
                
              
              _
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          l
                          -
                          k
                        
                        )
                      
                      ⁢
                      θ
                    
                    -
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                    +
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    -
                    
                      120
                      ⁢
                      
                        (
                        
                          l
                          -
                          k
                        
                        )
                      
                    
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            }
          
          ]
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            -
                            s
                          
                          )
                        
                      
                    
                    +
                    
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              m
                            
                          
                          2
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  m
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            -
                            α
                            +
                            mt
                          
                          }
                        
                      
                      _
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      ⁢
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    n
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              -
                              α
                              +
                              nu
                            
                            }
                          
                        
                        _
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          j
                        
                        ⁢
                        
                          I
                          1
                        
                      
                      2
                    
                    [
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            +
                            s
                            -
                            
                              120
                              ⁢
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            -
                            s
                            -
                            
                              120
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          j
                        
                        ⁢
                        
                          I
                          m
                        
                      
                      2
                    
                    [
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  m
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            +
                            mt
                            -
                            
                              120
                              ⁢
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  -
                                  m
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            -
                            mt
                            -
                            
                              120
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          j
                        
                        ⁢
                        
                          I
                          n
                        
                      
                      2
                    
                    [
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  n
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            +
                            nu
                            -
                            
                              120
                              ⁢
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  -
                                  n
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            -
                            nu
                            -
                            
                              120
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              F
                              k
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          
                            [
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        k
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    k
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    γ
                                  
                                  +
                                  s
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        k
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                              
                            
                          
                          _
                        
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              -
                              s
                              -
                              
                                120
                                ⁢
                                
                                  (
                                  
                                    k
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    -
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              F
                              k
                            
                            ⁢
                            
                              I
                              m
                            
                          
                          2
                        
                        [
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      k
                                      +
                                      m
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  k
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                +
                                mt
                                -
                                
                                  120
                                  ⁢
                                  
                                    (
                                    
                                      k
                                      +
                                      1
                                    
                                  
                                
                              
                            
                          
                          _
                        
                        )
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      k
                                      -
                                      m
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  k
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                -
                                mt
                                -
                                
                                  120
                                  ⁢
                                  
                                    (
                                    
                                      k
                                      -
                                      1
                                    
                                    )
                                  
                                
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        -
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        }
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            k
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    +
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              +
                              nu
                              -
                              
                                120
                                ⁢
                                
                                  (
                                  
                                    k
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        _
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    -
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              -
                              nu
                              -
                              
                                120
                                ⁢
                                
                                  (
                                  
                                    k
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      -
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              F
                              l
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          
                            [
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    
                                      (
                                      
                                        l
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    l
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    δ
                                  
                                  +
                                  s
                                  -
                                  
                                    120
                                    ⁢
                                    
                                      (
                                      
                                        l
                                        +
                                        1
                                      
                                      )
                                    
                                  
                                
                              
                            
                          
                          _
                        
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              s
                              -
                              
                                120
                                ⁢
                                
                                  (
                                  
                                    l
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  --
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            l
                          
                          ⁢
                          
                            I
                            m
                          
                        
                        2
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    +
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              mt
                              -
                              
                                120
                                ⁢
                                
                                  (
                                  
                                    l
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        _
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    -
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              mt
                              -
                              
                                120
                                ⁢
                                
                                  (
                                  
                                    l
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        ]
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    -
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            l
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    +
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              nu
                              -
                              
                                120
                                ⁢
                                
                                  (
                                  
                                    l
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        _
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  l
                                  -
                                  n
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              l
                              ⁢
                              
                                
                              
                              ⁢
                              δ
                            
                            -
                            nu
                            -
                            
                              120
                              ⁢
                              
                                (
                                
                                  l
                                  -
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          I
                          1
                          2
                        
                        +
                        
                          I
                          m
                          2
                        
                        +
                        
                          I
                          n
                          2
                        
                      
                      )
                    
                  
                  +
                  
                    
                      
                        I
                        1
                      
                      ⁢
                      
                        I
                        m
                      
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                m
                                -
                                1
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                          -
                          s
                          +
                          mt
                        
                        }
                      
                    
                    _
                  
                  +
                
              
            
            
              
                
                  
                    
                      
                        I
                        1
                      
                      ⁢
                      
                        I
                        n
                      
                      ⁢
                      cos
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                n
                                -
                                1
                              
                              )
                            
                            ⁢
                            θ
                          
                          -
                          s
                          +
                          nu
                        
                        }
                      
                    
                    _
                  
                  +
                  
                    
                      
                        I
                        m
                      
                      ⁢
                      
                        I
                        n
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                n
                                -
                                m
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                          -
                          mt
                          +
                          nu
                        
                        }
                      
                    
                    _
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  
  
    
      
        
          W
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          fw
        
        ∝
      
    
    
      
        (
        15
        )
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    F
                    1
                    2
                  
                  +
                  
                    F
                    j
                    2
                  
                  +
                  
                    F
                    k
                    2
                  
                  +
                  
                    F
                    l
                    2
                  
                
                )
              
            
            -
            
              
                
                  F
                  j
                  2
                
                2
              
              ⁢
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    j
                    (
                    
                      θ
                      +
                      β
                      -
                      240
                    
                  
                
                
                  
                    -
                    
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          ⁢
                                          
                                            -
                                            
                                              
                                            
                                            ⁢
                                            
                                              ·
                                              
                                                
                                              
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                  
                
              
            
          
          )
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                F
                k
                2
              
              2
            
            ⁢
            
              
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  
                
                ⁢
                
                  k
                  (
                  
                    θ
                    +
                    γ
                    -
                    240
                  
                
              
              
                -
                
                  
                
                ⁢
                
                  ·
                  
                    
                  
                  ⁢
                  
                    -
                    
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          )
        
        -
        
          
            
              F
              l
              2
            
            2
          
          ⁢
          
            
              cos
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                
              
              ⁢
              
                l
                ⁡
                
                  (
                  
                    θ
                    +
                    δ
                    -
                    240
                  
                  )
                
              
            
            
              
                -
                
                  
                
                ⁢
                
                  ·
                  
                    
                  
                  ⁢
                  
                    -
                    
                      
                        
                      
                      ⁢
                      
                        
                      
                    
                    ⁢
                    
                      ·
                      
                        
                      
                      ⁢
                      
                        -
                        
                          
                            
                          
                          ⁢
                          
                            
                          
                        
                        ⁢
                        
                          ·
                          
                            
                          
                          ⁢
                          
                            -
                            
                              
                            
                            ⁢
                            
                              ·
                              
                                
                              
                              ⁢
                              
                                -
                                
                                  
                                
                                ⁢
                                
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  
                                    -
                                    
                                      
                                    
                                    ⁢
                                    
                                      ·
                                      
                                        
                                      
                                      ⁢
                                      
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          ·
                                          
                                            
                                          
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ⁢
              
                
              
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          1
        
        ⁢
        
          
            F
            j
          
          [
          
            
              cos
              ⁢
              
                {
                
                  
                    
                      (
                      
                        j
                        +
                        1
                      
                      )
                    
                    ⁢
                    θ
                  
                  +
                  α
                  +
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    β
                  
                  -
                  
                    240
                    ⁢
                    
                      (
                      
                        j
                        +
                        1
                      
                      )
                    
                  
                
                }
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            cos
            ⁢
            
              {
              
                
                  
                    (
                    
                      j
                      -
                      1
                    
                    )
                  
                  ⁢
                  θ
                
                -
                α
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  β
                
                -
                
                  240
                  ⁢
                  
                    (
                    
                      j
                      -
                      1
                    
                    )
                  
                
              
              }
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          1
        
        ⁢
        
          
            F
            k
          
          [
          
            
              c
              ⁢
              
                
                  os
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            k
                            +
                            1
                          
                          )
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        k
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      
                        240
                        ⁢
                        
                          (
                          
                            k
                            +
                            1
                          
                          )
                        
                      
                    
                    }
                  
                
                _
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              cos
              ⁢
              
                {
                
                  
                    
                      (
                      
                        k
                        -
                        1
                      
                      )
                    
                    ⁢
                    θ
                  
                  -
                  α
                  +
                  
                    k
                    ⁢
                    
                      
                    
                    ⁢
                    γ
                  
                  -
                  
                    240
                    ⁢
                    
                      (
                      
                        k
                        -
                        1
                      
                      )
                    
                  
                
                }
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          1
        
        ⁢
        
          
            F
            l
          
          [
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          l
                          +
                          1
                        
                        )
                      
                      ⁢
                      θ
                    
                    +
                    α
                    +
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    -
                    
                      240
                      ⁢
                      
                        (
                        
                          l
                          +
                          1
                        
                        )
                      
                    
                  
                  }
                
              
              _
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              cos
              ⁢
              
                {
                
                  
                    
                      (
                      
                        l
                        -
                        1
                      
                      )
                    
                    ⁢
                    θ
                  
                  -
                  α
                  +
                  
                    l
                    ⁢
                    
                      
                    
                    ⁢
                    δ
                  
                  -
                  
                    240
                    ⁢
                    
                      (
                      
                        l
                        -
                        1
                      
                      )
                    
                  
                
                }
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          j
        
        ⁢
        
          
            F
            k
          
          [
          
            
              c
              ⁢
              
                
                  os
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            k
                            +
                            j
                          
                          )
                        
                        ⁢
                        θ
                      
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        k
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      
                        240
                        ⁢
                        
                          (
                          
                            k
                            +
                            j
                          
                          )
                        
                      
                    
                    }
                  
                
                _
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          k
                          -
                          j
                        
                        )
                      
                      ⁢
                      θ
                    
                    -
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      β
                    
                    +
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                    -
                    
                      240
                      ⁢
                      
                        (
                        
                          k
                          -
                          j
                        
                        )
                      
                    
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            }
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            j
          
          ⁢
          
            
              F
              l
            
            [
            
              c
              ⁢
              
                
                  os
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            l
                            +
                            j
                          
                          )
                        
                        ⁢
                        θ
                      
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        l
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      
                        240
                        ⁢
                        
                          (
                          
                            l
                            +
                            j
                          
                          )
                        
                      
                    
                  
                
                _
              
            
            }
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        
                          (
                          
                            l
                            -
                            j
                          
                          )
                        
                        ⁢
                        θ
                      
                      -
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        l
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      
                        240
                        ⁢
                        
                          (
                          
                            l
                            -
                            j
                          
                        
                      
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              )
            
            }
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          k
        
        ⁢
        
          
            F
            l
          
          [
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      
                        (
                        
                          l
                          +
                          k
                        
                        )
                      
                      ⁢
                      θ
                    
                    +
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                    +
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    -
                    
                      240
                      ⁢
                      
                        (
                        
                          l
                          +
                          k
                        
                        )
                      
                    
                  
                  }
                
              
              _
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              cos
              ⁢
              
                {
                
                  
                    
                      (
                      
                        l
                        -
                        k
                      
                      )
                    
                    ⁢
                    θ
                  
                  -
                  
                    k
                    ⁢
                    
                      
                    
                    ⁢
                    γ
                  
                  +
                  
                    l
                    ⁢
                    
                      
                    
                    ⁢
                    δ
                  
                  -
                  
                    240
                    ⁢
                    
                      (
                      
                        l
                        -
                        k
                      
                      )
                    
                  
                
                }
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          ]
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            -
                            s
                          
                          )
                        
                      
                    
                    +
                    
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              m
                            
                          
                          2
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  m
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            -
                            α
                            +
                            mt
                          
                          }
                        
                      
                      _
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      ⁢
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    n
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              -
                              α
                              +
                              nu
                            
                            }
                          
                        
                        _
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          j
                        
                        ⁢
                        
                          I
                          1
                        
                      
                      2
                    
                    [
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            +
                            s
                            -
                            
                              240
                              ⁢
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            -
                            s
                            -
                            
                              240
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          j
                        
                        ⁢
                        
                          I
                          m
                        
                      
                      2
                    
                    [
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  m
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            +
                            mt
                            -
                            
                              240
                              ⁢
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  -
                                  m
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            -
                            mt
                            -
                            
                              240
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          j
                        
                        ⁢
                        
                          I
                          n
                        
                      
                      2
                    
                    [
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  n
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            +
                            nu
                            -
                            
                              240
                              ⁢
                              
                                (
                                
                                  j
                                  +
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  j
                                  -
                                  n
                                
                                )
                              
                              ⁢
                              θ
                            
                            +
                            
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              β
                            
                            -
                            nu
                            -
                            
                              240
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  1
                                
                                )
                              
                            
                          
                          }
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            k
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁢
                      
                        
                          [
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      k
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  k
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                +
                                s
                                -
                                
                                  240
                                  ⁢
                                  
                                    (
                                    
                                      k
                                      +
                                      1
                                    
                                    )
                                  
                                
                              
                              }
                            
                          
                        
                        _
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              -
                              s
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    k
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    --
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            k
                          
                          ⁢
                          
                            I
                            m
                          
                        
                        2
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    +
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              +
                              mt
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    k
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        _
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      k
                                      -
                                      m
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  k
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                -
                                mt
                                -
                                
                                  240
                                  ⁢
                                  
                                    (
                                    
                                      k
                                      -
                                      1
                                    
                                    )
                                  
                                
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        -
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        }
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            k
                          
                          ⁢
                          
                            I
                            n
                          
                        
                        2
                      
                      [
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    +
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              +
                              nu
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    k
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        _
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    k
                                    -
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                k
                                ⁢
                                
                                  
                                
                                ⁢
                                γ
                              
                              -
                              nu
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    k
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      -
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            l
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁢
                      
                        
                          [
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  
                                    (
                                    
                                      l
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  l
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  δ
                                
                                +
                                s
                                -
                                
                                  240
                                  ⁢
                                  
                                    (
                                    
                                      l
                                      +
                                      1
                                    
                                    )
                                  
                                
                              
                              }
                            
                          
                        
                        _
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              s
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    l
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  --
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          l
                        
                        ⁢
                        
                          I
                          m
                        
                      
                      2
                    
                    [
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    +
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              mt
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    l
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        _
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    -
                                    m
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              mt
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    l
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    --
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          l
                        
                        ⁢
                        
                          I
                          n
                        
                      
                      2
                    
                    [
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    +
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              nu
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    l
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                            }
                          
                        
                        _
                      
                      -
                    
                  
                
              
              
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                
                                  (
                                  
                                    l
                                    -
                                    n
                                  
                                  )
                                
                                ⁢
                                θ
                              
                              +
                              
                                l
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              nu
                              -
                              
                                240
                                ⁢
                                
                                  (
                                  
                                    l
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    -
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      }
                    
                    ]
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          I
                          1
                          2
                        
                        +
                        
                          I
                          m
                          2
                        
                        +
                        
                          I
                          n
                          2
                        
                      
                      )
                    
                  
                  +
                  
                    
                      
                        I
                        1
                      
                      ⁢
                      
                        I
                        m
                      
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              (
                              
                                m
                                -
                                1
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                          -
                          s
                          +
                          mt
                        
                        }
                      
                    
                    _
                  
                  +
                
              
            
            
              
                
                  
                    
                      
                        
                          I
                          1
                        
                        ⁢
                        
                          I
                          n
                        
                        ⁢
                        cos
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  n
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              θ
                            
                            -
                            s
                            +
                            nu
                          
                          }
                        
                      
                      _
                    
                    +
                    
                      
                        
                          I
                          m
                        
                        ⁢
                        
                          I
                          n
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              
                                (
                                
                                  n
                                  -
                                  m
                                
                                )
                              
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            -
                            mt
                            +
                            nu
                          
                        
                      
                      _
                    
                  
                  }
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  




Of the harmonics of the rotor magnetomotive force Fmag of a three-phase AC rotary electric machine, generally the third harmonic, the fifth harmonic, and the seventh harmonic are much more dominant than the other harmonics, though it depends on the number of poles and the number of stator slots; therefore, a description will be given below of the magnetic noise (sound) that has the fundamental frequency of the rotor magnetomotive force Fmag and contains the third, fifth, and seventh harmonics, for the case where no harmonic currents for altering the magnetic sound are superimposed.
The integer values (order values) are j=3, k=5, and L=7, which are substituted into the equations (7) to (9) to yield the following equations.




  
    
      
        U
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      3
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    3
                    ⁢
                    
                      (
                      
                        θ
                        +
                        β
                      
                      )
                    
                  
                  +
                  
                    
                      F
                      5
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    5
                    ⁢
                    
                      (
                      
                        θ
                        +
                        γ
                      
                      )
                    
                  
                  +
                  
                    
                      F
                      7
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    7
                    ⁢
                    
                      (
                      
                        θ
                        +
                        δ
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  Fundamental
                  ⁢
                  
                    
                  
                  ⁢
                  3
                  ⁢
                  rd
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  5
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                  ⁢
                  
                    
                  
                  ⁢
                  7
                  ⁢
                  
                    
                  
                  ⁢
                  th
                  ⁢
                  
                    
                  
                  ⁢
                  harmonic
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  (
                  
                    first
                    ⁢
                    
                      
                    
                    ⁢
                    harmonic
                  
                  )
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    I
                    1
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        θ
                        +
                        s
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                
                ⁢
                Fundamental
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  (
                  
                    first
                    ⁢
                    
                      
                    
                    ⁢
                    harmonic
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  
  
    
      
        V
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          120
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    3
                    ⁢
                    
                      (
                      
                        θ
                        +
                        β
                        -
                        120
                      
                      )
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      5
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    5
                    ⁢
                    
                      (
                      
                        θ
                        +
                        γ
                        -
                        120
                      
                      )
                    
                  
                  +
                  
                    
                      F
                      7
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    7
                    ⁢
                    
                      (
                      
                        θ
                        +
                        δ
                        -
                        120
                      
                      )
                    
                  
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    I
                    1
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        θ
                        +
                        s
                        -
                        120
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        W
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          240
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    3
                    ⁢
                    
                      (
                      
                        θ
                        +
                        β
                        -
                        240
                      
                      )
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      5
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    5
                    ⁢
                    
                      (
                      
                        θ
                        +
                        γ
                        -
                        240
                      
                      )
                    
                  
                  +
                  
                    
                      F
                      7
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    7
                    ⁢
                    
                      (
                      
                        θ
                        +
                        δ
                        -
                        240
                      
                      )
                    
                  
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    I
                    1
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        θ
                        +
                        s
                        -
                        240
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




Substituting these equations (16), (17), and (18) into the equation (3), and setting m=0 (Im=0) and n=0 (In=0) since no harmonic currents are superimposed, the following equations are obtained.




  
    
      
        PRIOR
        ⁢
        
          
        
        ⁢
        ART
      
    
    
      
        
      
    
  
  
    
      
        
          U
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          fu
        
        ∝
      
    
    
      
        (
        19
        )
      
    
  
  
    
      
        
          
            1
            2
          
          ⁢
          
            (
            
              
                F
                1
                2
              
              +
              
                F
                3
                2
              
              +
              
                F
                5
                2
              
              +
              
                F
                7
                2
              
            
            )
          
        
        -
        
          
            
              
                F
                1
                2
              
              2
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            2
            ⁢
            
              (
              
                θ
                +
                α
              
              )
            
          
          
            --
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      -
                    
                  
                
              
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                F
                3
                2
              
              2
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            6
            ⁢
            
              (
              
                θ
                +
                β
              
              )
            
          
          _
        
        -
        
          
            
              
                
                  F
                  5
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              10
              ⁢
              
                (
                
                  θ
                  +
                  γ
                
                )
              
            
            -
            
              
                
                  F
                  7
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              14
              ⁢
              
                (
                
                  θ
                  +
                  δ
                
                )
              
            
          
          
            --
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              F
              1
            
            ⁢
            
              
                F
                3
              
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          3
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          3
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                      
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            --
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              }
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                F
                1
              
              ⁢
              
                
                  F
                  5
                
                [
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          6
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          5
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                      
                    
                  
                  _
                
                }
              
            
            -
            
              
                cos
                ⁢
                
                  {
                  
                    
                      4
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    -
                    α
                    +
                    
                      5
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                  
                  }
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          --
                        
                      
                    
                  
                
              
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        8
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                    
                    }
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            --
                          
                        
                      
                    
                  
                
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        6
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                    
                    }
                  
                
                _
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              F
              3
            
            ⁢
            
              
                F
                5
              
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          8
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                      
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              }
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            3
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        10
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                    
                    }
                  
                
                -
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        4
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                    
                    }
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          F
          5
        
        ⁢
        
          
            F
            7
          
          [
          
            
              
                cos
                ⁢
                
                  {
                  
                    
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    +
                    
                      5
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                    +
                    
                      7
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                  
                  }
                
              
              _
            
            -
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                    
                    }
                  
                
                ]
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            --
                          
                        
                      
                    
                  
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        
                          -
                          
                            
                              
                                F
                                1
                              
                              ⁢
                              
                                I
                                1
                              
                            
                            2
                          
                        
                        ⁢
                        
                          
                            {
                            
                              
                                cos
                                ⁢
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    α
                                    +
                                    s
                                  
                                  )
                                
                              
                              -
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              -
                              s
                            
                            )
                          
                        
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              F
                              3
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        [
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  4
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                +
                                s
                              
                              }
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                -
                                s
                              
                              }
                            
                          
                        
                        ]
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        -
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    F
                                    5
                                  
                                  ⁢
                                  
                                    I
                                    1
                                  
                                
                                2
                              
                              [
                              
                                cos
                                ⁢
                                
                                  {
                                  
                                    
                                      6
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      5
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      γ
                                    
                                    +
                                    s
                                  
                                
                              
                            
                            _
                          
                          }
                        
                        -
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  4
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  5
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                -
                                s
                              
                              }
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      --
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          
                            [
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    δ
                                  
                                  +
                                  s
                                
                                }
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      --
                                    
                                  
                                
                              
                            
                          
                        
                      
                      -
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            {
                            
                              
                                6
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              s
                            
                            }
                          
                        
                        _
                      
                    
                    ]
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                1
                2
              
              ⁢
              
                I
                1
                2
              
            
            -
            
              
                
                  
                    I
                    1
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    s
                  
                  )
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        --
                      
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  
  
    
      
        
          V
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          fv
        
        ∝
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    F
                    1
                    2
                  
                  +
                  
                    F
                    3
                    2
                  
                  +
                  
                    F
                    5
                    2
                  
                  +
                  
                    F
                    7
                    2
                  
                
                )
              
            
            -
            
              
                
                  
                    F
                    1
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    θ
                    +
                    α
                    -
                    120
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              --
                            
                          
                        
                      
                    
                  
                
              
            
          
          )
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  F
                  3
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              6
              ⁢
              
                (
                
                  θ
                  +
                  β
                  -
                  120
                
              
            
            _
          
          )
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  F
                  5
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              10
              ⁢
              
                (
                
                  θ
                  +
                  γ
                  -
                  120
                
                )
              
            
            -
            
              
                
                  F
                  7
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              14
              ⁢
              
                (
                
                  θ
                  +
                  δ
                  -
                  120
                
                )
              
            
          
          
            --
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              3
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        4
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      -
                      480
                    
                    }
                  
                
                -
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      -
                      240
                    
                    }
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      --
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              5
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        6
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      720
                    
                    }
                  
                
                _
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        4
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      480
                    
                    }
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  -
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        8
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      960
                    
                    }
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  -
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        6
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      720
                    
                    }
                  
                
                _
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            3
          
          ⁢
          
            
              F
              5
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        8
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      960
                    
                    }
                  
                
                -
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      240
                    
                    }
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        -
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            3
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        10
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      1200
                    
                    }
                  
                
                -
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        4
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      480
                    
                    }
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          -
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            5
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        12
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      1440
                    
                    }
                  
                
                _
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      240
                    
                    }
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  --
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        -
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  s
                                  -
                                  240
                                
                                )
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            --
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                -
                                s
                              
                              )
                            
                          
                        
                        }
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  4
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                +
                                s
                                -
                                480
                              
                              }
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                -
                                s
                                -
                                240
                              
                              }
                            
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                -
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  6
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  5
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                +
                                s
                                -
                                720
                              
                              }
                            
                          
                          _
                        
                        -
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  4
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  5
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                -
                                s
                                -
                                480
                              
                              }
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          7
                        
                        ⁢
                        
                          I
                          1
                        
                      
                      2
                    
                    [
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                8
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              s
                              -
                              960
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      -
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            {
                            
                              
                                6
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              s
                              -
                              720
                            
                            }
                          
                        
                        _
                      
                    
                    ]
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                1
                2
              
              ⁢
              
                I
                1
                2
              
            
            -
            
              
                
                  
                    I
                    1
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    s
                    -
                    120
                  
                  )
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              -
                            
                          
                        
                      
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  
  
    
      
        
          W
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          fw
        
        ∝
      
    
    
      
        (
        21
        )
      
    
  
  
    
      
        
          
            1
            2
          
          ⁢
          
            (
            
              
                F
                1
                2
              
              +
              
                F
                3
                2
              
              +
              
                F
                5
                2
              
              +
              
                F
                7
                2
              
            
            )
          
        
        -
        
          
            
              
                F
                1
                2
              
              2
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            2
            ⁢
            
              
            
            ⁢
            
              (
              
                θ
                +
                α
                -
                240
              
              )
            
          
          
            --
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          --
                        
                      
                    
                  
                
              
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                F
                3
                2
              
              2
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            6
            ⁢
            
              (
              
                θ
                +
                β
                -
                240
              
              )
            
          
          _
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  F
                  5
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              10
              ⁢
              
                (
                
                  θ
                  +
                  γ
                  -
                  240
                
                )
              
            
            -
            
              
                
                  F
                  7
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              14
              ⁢
              
                (
                
                  θ
                  +
                  δ
                  -
                  240
                
                )
              
            
          
          
            --
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              3
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        4
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      -
                      960
                    
                    }
                  
                
                -
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      -
                      480
                    
                    }
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      --
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              5
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        6
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      1440
                    
                    }
                  
                
                _
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        4
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      -
                      960
                    
                    }
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  -
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        8
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      1920
                    
                    }
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  --
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        6
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      1440
                    
                    }
                  
                
                _
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              F
              3
            
            ⁢
            
              
                F
                5
              
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          8
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                        -
                        1920
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          
                            
                          
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          
                            
                          
                          ⁢
                          γ
                        
                        -
                        480
                      
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            -
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              }
            
          
          ]
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            3
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        10
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      2400
                    
                    }
                  
                
                -
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        4
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      960
                    
                    }
                  
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          -
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            5
          
          ⁢
          
            
              F
              7
            
            [
            
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        12
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      2880
                    
                    }
                  
                
                _
              
              -
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                      -
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        γ
                      
                      +
                      
                        7
                        ⁢
                        
                          
                        
                        ⁢
                        δ
                      
                      -
                      480
                    
                    }
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  --
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        
                          
                            -
                            
                              
                                
                                  F
                                  1
                                
                                ⁢
                                
                                  I
                                  1
                                
                              
                              2
                            
                          
                          ⁢
                          
                            {
                            
                              
                                cos
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  s
                                  -
                                  480
                                
                              
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            )
                          
                        
                        -
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              -
                              s
                            
                            )
                          
                        
                      
                      }
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  4
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                +
                                s
                                -
                                960
                              
                              }
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  3
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  β
                                
                                -
                                s
                                -
                                480
                              
                              }
                            
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                -
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  6
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  5
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                +
                                s
                                -
                                1440
                              
                              }
                            
                          
                          _
                        
                        -
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  4
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                +
                                
                                  5
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  γ
                                
                                -
                                s
                                -
                                960
                              
                              }
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          F
                          7
                        
                        ⁢
                        
                          I
                          1
                        
                      
                      2
                    
                    [
                    
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                8
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              +
                              s
                              -
                              1920
                            
                            }
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          -
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      -
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            {
                            
                              
                                6
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                
                                  
                                
                                ⁢
                                δ
                              
                              -
                              s
                              -
                              1440
                            
                            }
                          
                        
                        _
                      
                    
                    ]
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                1
                2
              
              ⁢
              
                I
                1
                2
              
            
            -
            
              
                
                  
                    I
                    1
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    s
                    -
                    240
                  
                  )
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              -
                            
                          
                        
                      
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  




Those terms for which the vector sum is zero are eliminated from the equations (19), (20), and (21), and the DC component terms and the terms where the constituent terms are in phase and therefore reinforcing each other are extracted to obtain the following equations.




  
    
      
        
          
            U­
            ⁢
            phase
          
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
        
        ⁢
        
          

        
        ⁢
        
          fu
          ∝
          
          ⁢
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    F
                    1
                    2
                  
                  +
                  
                    F
                    3
                    2
                  
                  +
                  
                    F
                    5
                    2
                  
                  +
                  
                    F
                    7
                    2
                  
                
                )
              
            
            -
            
              
                
                  F
                  3
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              6
              ⁢
              
                (
                
                  θ
                  +
                  β
                
                )
              
            
            -
            
              

            
            ⁢
            
              
                F
                1
              
              ⁢
              
                F
                5
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      6
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    +
                    α
                    +
                    
                      5
                      ⁢
                      γ
                    
                  
                  )
                
              
            
            +
            
              
                F
                1
              
              ⁢
              
                F
                7
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      6
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    -
                    α
                    +
                    
                      7
                      ⁢
                      δ
                    
                  
                  )
                
              
            
            -
            
              

            
            ⁢
            
              
                F
                5
              
              ⁢
              
                F
                7
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    +
                    
                      5
                      ⁢
                      γ
                    
                    +
                    
                      7
                      ⁢
                      δ
                    
                  
                  )
                
              
            
            +
            
              N
              ⁢
              
                〈
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      I
                      1
                    
                    ⁢
                    cos
                    ⁢
                    
                      (
                      
                        α
                        -
                        s
                      
                      )
                    
                  
                  -
                  
                    

                  
                  ⁢
                  
                    
                      F
                      5
                    
                    ⁢
                    
                      I
                      1
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            6
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                          +
                          
                            5
                            ⁢
                            γ
                          
                          +
                          s
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      7
                    
                    ⁢
                    
                      I
                      1
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            6
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                          +
                          
                            7
                            ⁢
                            δ
                          
                          -
                          s
                        
                        )
                      
                    
                  
                
                〉
              
            
            +
            
              
                
                  N
                  2
                
                2
              
              ⁢
              
                I
                1
                2
              
            
          
        
      
    
    
      
        (
        22
        )
      
    
  












Hence, the sum of the excitation forces, obtained by summing together the equations (22), (23), and (24), is given by the following equation (25).




  
    
      
        
          fu
          +
          fv
          +
          fw
        
        ∝
        
          
            
              
                
                  3
                  2
                
                ⁢
                
                  (
                  
                    
                      F
                      1
                      2
                    
                    +
                    
                      F
                      3
                      2
                    
                    +
                    
                      F
                      5
                      2
                    
                    +
                    
                      F
                      7
                      2
                    
                  
                  )
                
              
              +
              
                3
                ⁢
                
                  NF
                  1
                
                ⁢
                
                  I
                  1
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      α
                      -
                      s
                    
                    )
                  
                
              
              +
              
                
                  
                    3
                    ⁢
                    
                      N
                      2
                    
                  
                  2
                
                ⁢
                
                  I
                  1
                  2
                
              
            
            
              _
              
                (
                1
                )
              
            
          
          -
          
            
              
                
                  
                    3
                    ⁢
                    
                      F
                      3
                      2
                    
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                6
                ⁢
                
                  (
                  
                    θ
                    +
                    β
                  
                  )
                
              
              _
            
            
              (
              2
              )
            
          
          -
          
            
              
                3
                ⁢
                
                  F
                  1
                
                ⁢
                
                  F
                  5
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        5
                        ⁢
                        γ
                      
                    
                    )
                  
                
              
              _
            
            
              (
              3
              )
            
          
          -
          
            
              
                3
                ⁢
                
                  F
                  1
                
                ⁢
                
                  F
                  7
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        7
                        ⁢
                        δ
                      
                    
                    )
                  
                
              
              _
            
            
              (
              4
              )
            
          
          -
          
            
              
                
                  3
                  ⁢
                  
                    F
                    5
                  
                  ⁢
                  
                    F
                    7
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          12
                          ⁢
                          θ
                        
                        -
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      )
                    
                  
                
                )
              
              _
            
            
              (
              5
              )
            
          
          +
          
            
              
                3
                ⁢
                N
                ⁢
                
                  〈
                  
                    
                      -
                      
                        F
                        5
                      
                    
                    ⁢
                    
                      I
                      1
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            6
                            ⁢
                            θ
                          
                          +
                          
                            5
                            ⁢
                            γ
                          
                          +
                          s
                        
                        )
                      
                    
                  
                
              
              _
            
            
              (
              6
              )
            
          
          +
          
            
              
                
                  F
                  7
                
                ⁢
                
                  I
                  1
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      +
                      
                        7
                        ⁢
                        δ
                      
                      -
                      s
                    
                    )
                  
                
              
              〉
            
            
              _
              
                (
                7
                )
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




Term (1): DC component . . . No pulsation, hence no magnetic sound
Term (2): Sixth component occurring due to the third harmonic of the rotor magnetomotive force
Term (3): Sixth component occurring due to the first and fifth harmonics of the rotor magnetomotive force
Term (4): Sixth component occurring due to the first and seventh harmonics of the rotor magnetomotive force
Term (5): 12th component occurring due to the fifth and seventh harmonics of the rotor magnetomotive force
Term (6): Sixth component occurring due to the fifth harmonic of the rotor magnetomotive force and the first harmonic of the stator current.
Term (7): Sixth component occurring due to the seventh harmonic of the rotor magnetomotive force and the first harmonic of the stator current
These sixth and 12th excitation force pulsation components (term (2)–(7)) cause magnetic sound.
In the equation (25), the term (1) is the DC component term in the sum of the excitation forces; this term does not pulsate, and therefore does not cause magnetic sound.
On the other hand, the term (2) is the sixth harmonic component occurring due to the third harmonic of the rotor magnetomotive force Fmag, the term (3) is the sixth harmonic component occurring due to the first and fifth harmonics of the rotor magnetomotive force Fmag, the term (4) is the sixth harmonic component occurring due to the first and seventh harmonics of the rotor magnetomotive force Fmag, the term (5) is the 12th harmonic component occurring due to the fifth and seventh harmonics of the rotor magnetomotive force Fmag, the term (6) is the sixth harmonic component occurring due to the fifth harmonic of the rotor magnetomotive force Fmag and the first current component (fundamental) of the stator current, and the term (7) is the sixth harmonic component occurring due to the seventh harmonic of the rotor magnetomotive force Fmag and the first current component (fundamental) of the stator current. These sixth and 12th excitation force pulsation components cause magnetic noise.
That is, it can be seen from the equation (25) that the magnetic noise of the three-phase AC rotary electric machine is caused by the sixth and 12th magnetic noise components occurring due to the fundamental component and the third, fifth, and seventh harmonic components of the rotor magnetomotive force Fmag.
With this in mind, a description will be given below of the case where the seventh and 13th harmonic components are superimposed on the stator current Icoil in order to alter (reduce or increase) the sixth and 12th magnetic sound components.
In this case, setting j=3, k=5, L=7, m=7, and n=13 in the equations (7), (8), and (9), the following equations are obtained.




  
    
      
        U
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        (
        26
        )
      
    
  
  
    
      
        {
        
          
            
              
                
                  Rotor
                  ⁢
                  
                    
                  
                  ⁢
                  magnetomotive
                  ⁢
                  
                    
                  
                  ⁢
                  force
                
                ⁢
                
                  
                
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      
                        F
                        1
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            θ
                            +
                            α
                          
                          )
                        
                      
                    
                    
                      
                        
                          Fundamental
                        
                      
                      
                        
                          
                            (
                            
                              first
                              ⁢
                              
                                
                              
                              ⁢
                              harmonic
                            
                            )
                          
                        
                      
                    
                  
                  +
                  
                    
                      
                        F
                        3
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      3
                      ⁢
                      
                        (
                        
                          θ
                          +
                          β
                        
                        )
                      
                    
                    
                      3
                      ⁢
                      
                        
                      
                      ⁢
                      r
                      ⁢
                      
                        
                      
                      ⁢
                      d
                      ⁢
                      
                        
                      
                      ⁢
                      harmonic
                    
                  
                  +
                  
                    
                      
                        F
                        5
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      5
                      ⁢
                      
                        (
                        
                          θ
                          +
                          γ
                        
                        )
                      
                    
                    
                      5
                      ⁢
                      
                        
                      
                      ⁢
                      th
                      ⁢
                      
                        
                      
                      ⁢
                      harmonic
                    
                  
                  +
                  
                    
                      
                        F
                        7
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      7
                      ⁢
                      
                        (
                        
                          θ
                          +
                          δ
                        
                        )
                      
                    
                    
                      7
                      ⁢
                      th
                      ⁢
                      
                        
                      
                      ⁢
                      harmonic
                    
                  
                
              
            
          
          
            
              
                Stator
                ⁢
                
                  
                
                ⁢
                current
              
            
          
          
            
              
                
                  I
                  coil
                
                =
                
                  
                    
                      
                        I
                        1
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            θ
                            +
                            s
                          
                          )
                        
                      
                    
                    
                      
                        
                          Fundamental
                        
                      
                      
                        
                          
                            (
                            
                              first
                              ⁢
                              
                                
                              
                              ⁢
                              harmonic
                            
                            )
                          
                        
                      
                    
                  
                  +
                  
                    
                      
                        I
                        7
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      7
                      ⁢
                      
                        (
                        
                          θ
                          +
                          t
                        
                        )
                      
                    
                    
                      7
                      ⁢
                      th
                      ⁢
                      
                        
                      
                      ⁢
                      harmonic
                    
                  
                  +
                  
                    
                      
                        I
                        13
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      13
                      ⁢
                      
                        (
                        
                          θ
                          +
                          u
                        
                        )
                      
                    
                    
                      13
                      ⁢
                      
                        
                      
                      ⁢
                      th
                      ⁢
                      
                        
                      
                      ⁢
                      harmonic
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        V
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        (
        27
        )
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          120
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      3
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    3
                    ⁢
                    
                      (
                      
                        θ
                        +
                        β
                        -
                        120
                      
                      )
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      5
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    5
                    ⁢
                    
                      (
                      
                        θ
                        +
                        γ
                        -
                        120
                      
                      )
                    
                  
                  +
                  
                    
                      F
                      7
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    7
                    ⁢
                    
                      (
                      
                        θ
                        +
                        δ
                        -
                        120
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      Stator
                      ⁢
                      
                        
                      
                      ⁢
                      current
                    
                  
                
                
                  
                    
                      
                        I
                        coil
                      
                      =
                      
                        
                          
                            I
                            1
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                θ
                                +
                                s
                                -
                                120
                              
                              )
                            
                          
                        
                        +
                        
                          
                            I
                            7
                          
                          ⁢
                          sin
                          ⁢
                          
                            {
                            
                              
                                7
                                ⁢
                                
                                  (
                                  
                                    θ
                                    +
                                    t
                                  
                                  )
                                
                              
                              -
                              120
                            
                            }
                          
                        
                        +
                      
                    
                  
                
                
                  
                    
                      
                        
                      
                      ⁢
                      
                        
                          I
                          13
                        
                        ⁢
                        sin
                        ⁢
                        
                          {
                          
                            
                              13
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  u
                                
                                )
                              
                            
                            -
                            120
                          
                          }
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        W
        ⁢
        
          
        
        ⁢
        phase
      
    
    
      
        (
        28
        )
      
    
  
  
    
      
        {
        
          
            
              
                Rotor
                ⁢
                
                  
                
                ⁢
                magnetomotive
                ⁢
                
                  
                
                ⁢
                force
              
            
          
          
            
              
                
                  F
                  mag
                
                =
                
                  
                    
                      F
                      1
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          θ
                          +
                          α
                          -
                          240
                        
                        )
                      
                    
                  
                  +
                  
                    
                      F
                      3
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    3
                    ⁢
                    
                      (
                      
                        θ
                        +
                        β
                        -
                        240
                      
                      )
                    
                  
                  +
                
              
            
          
          
            
              
                
                  
                
                ⁢
                
                  
                    
                      F
                      5
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    5
                    ⁢
                    
                      (
                      
                        θ
                        +
                        γ
                        -
                        240
                      
                      )
                    
                  
                  +
                  
                    
                      F
                      7
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    7
                    ⁢
                    
                      (
                      
                        θ
                        +
                        δ
                        -
                        240
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      Stator
                      ⁢
                      
                        
                      
                      ⁢
                      current
                    
                  
                
                
                  
                    
                      
                        I
                        coil
                      
                      =
                      
                        
                          
                            I
                            1
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                θ
                                +
                                s
                                -
                                240
                              
                              )
                            
                          
                        
                        +
                        
                          
                            I
                            7
                          
                          ⁢
                          sin
                          ⁢
                          
                            {
                            
                              
                                7
                                ⁢
                                
                                  (
                                  
                                    θ
                                    +
                                    t
                                  
                                  )
                                
                              
                              -
                              240
                            
                            }
                          
                        
                        +
                      
                    
                  
                
                
                  
                    
                      
                        
                      
                      ⁢
                      
                        
                          I
                          13
                        
                        ⁢
                        sin
                        ⁢
                        
                          {
                          
                            
                              13
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  u
                                
                                )
                              
                            
                            -
                            240
                          
                          }
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  




Calculating the equations (10), (11), and (12) using the equations (26), (27), and (28), the following equations are obtained.




  
    
      
        
          U
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          
            
              
            
            _
          
          ⁢
          
            
          
          ⁢
          
            
              
                
                  
                    
                      
                        Inphase
                        ⁢
                        
                          
                        
                        ⁢
                        term
                        ⁢
                        
                          
                        
                        ⁢
                        which
                      
                      ,
                      when
                    
                  
                
                
                  
                    
                      vector
                      ⁢
                      
                        
                      
                      ⁢
                      summed
                      ⁢
                      
                        
                      
                      ⁢
                      between
                    
                  
                
                
                  
                    
                      
                        the
                        ⁢
                        
                          
                        
                        ⁢
                        three
                        ⁢
                        
                          
                        
                        ⁢
                        phases
                      
                      ,
                    
                  
                
                
                  
                    
                      contributes
                      ⁢
                      
                        
                      
                      ⁢
                      to
                      ⁢
                      
                        
                      
                      ⁢
                      producing
                    
                  
                
                
                  
                    reinforcement
                  
                
              
              ⁢
              
                
              
              --
            
            --
          
        
        -
        
          
            
              
                Term
                ⁢
                
                  
                
                ⁢
                for
                ⁢
                
                  
                
                ⁢
                which
                ⁢
                
                  
                
                ⁢
                the
              
            
          
          
            
              
                vector
                ⁢
                
                  
                
                ⁢
                sum
                ⁢
                
                  
                
                ⁢
                between
                ⁢
                
                  
                
                ⁢
                the
              
            
          
          
            
              
                three
                ⁢
                
                  
                
                ⁢
                phases
                ⁢
                
                  
                
                ⁢
                is
                ⁢
                
                  
                
                ⁢
                
                  zero
                  .
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        fu
        ∝
        
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  F
                  1
                  2
                
                +
                
                  F
                  3
                  2
                
                +
                
                  F
                  5
                  2
                
                +
                
                  F
                  7
                  2
                
              
              )
            
          
          -
          
            
              
                
                  F
                  1
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                (
                
                  θ
                  +
                  α
                
                )
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        -
                      
                    
                  
                
              
            
          
          -
          
            
              
                
                  F
                  3
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              6
              ⁢
              
                (
                
                  θ
                  +
                  β
                
                )
              
            
            _
          
          -
          
            
              
                
                  
                    F
                    5
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                10
                ⁢
                
                  (
                  
                    θ
                    +
                    γ
                  
                  )
                
              
              -
              
                
                  
                    F
                    7
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                14
                ⁢
                
                  (
                  
                    θ
                    +
                    δ
                  
                  )
                
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        --
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          -
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              3
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          3
                          ⁢
                          β
                        
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          3
                          ⁢
                          β
                        
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
        
          
            F
            1
          
          ⁢
          
            
              F
              5
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          6
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          5
                          ⁢
                          γ
                        
                      
                      }
                    
                  
                  _
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          5
                          ⁢
                          γ
                        
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              --
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  cos
                  ⁢
                  
                    {
                    
                      
                        8
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        7
                        ⁢
                        δ
                      
                    
                    }
                  
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          6
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      }
                    
                  
                  _
                
              
              ]
            
          
        
        -
        
          
            F
            3
          
          ⁢
          
            
              F
              5
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          8
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            3
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          10
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                --
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
        
          
            F
            5
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          12
                          ⁢
                          θ
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      }
                    
                  
                  _
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              --
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        -
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  s
                                
                                )
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      --
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                -
                                s
                              
                              )
                            
                          
                        
                        }
                      
                    
                    -
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  α
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  -
                                  α
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  s
                                
                                }
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  -
                                  s
                                
                                }
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      --
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    10
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                )
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                )
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          -
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    16
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    10
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              -
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  s
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  -
                                  s
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        --
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        --
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    18
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    -
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            --
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  s
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  -
                                  s
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                )
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  7
                                  ⁢
                                  δ
                                
                                -
                                
                                  7
                                  ⁢
                                  t
                                
                              
                              }
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    20
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        (
        29
        )
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              I
                              1
                              2
                            
                            +
                            
                              I
                              7
                              2
                            
                            +
                            
                              I
                              13
                              2
                            
                          
                          )
                        
                      
                      -
                      
                        
                          
                            
                              I
                              1
                              2
                            
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          2
                          ⁢
                          
                            (
                            
                              θ
                              +
                              s
                            
                            )
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  --
                                
                              
                            
                          
                        
                      
                      -
                      
                        
                          
                            
                              I
                              7
                              2
                            
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          14
                          ⁢
                          
                            (
                            
                              θ
                              +
                              t
                            
                            )
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  --
                                
                              
                            
                          
                        
                      
                      -
                      
                        
                          
                            
                              I
                              13
                              2
                            
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          26
                          ⁢
                          
                            (
                            
                              θ
                              +
                              u
                            
                            )
                          
                        
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    --
                                  
                                
                              
                            
                          
                        
                      
                      -
                      
                        
                          I
                          1
                        
                        ⁢
                        
                          
                            I
                            7
                          
                          [
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  +
                                  s
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        -
                                      
                                    
                                  
                                
                              
                            
                          
                          }
                        
                      
                      -
                      
                        
                          cos
                          ⁢
                          
                            {
                            
                              
                                6
                                ⁢
                                θ
                              
                              -
                              s
                              +
                              
                                7
                                ⁢
                                t
                              
                            
                            }
                          
                        
                        _
                      
                    
                    ]
                  
                  -
                
              
            
            
              
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      
                        I
                        13
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  s
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  -
                                  s
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    
                      I
                      7
                    
                    ⁢
                    
                      
                        I
                        13
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    20
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          --
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  
  
    
      
        
          V
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          
            
              
            
            _
          
          ⁢
          
            
          
          ⁢
          
            
              
                
                  
                    
                      
                        Inphase
                        ⁢
                        
                          
                        
                        ⁢
                        term
                        ⁢
                        
                          
                        
                        ⁢
                        which
                      
                      ,
                      when
                    
                  
                
                
                  
                    
                      vector
                      ⁢
                      
                        
                      
                      ⁢
                      summed
                      ⁢
                      
                        
                      
                      ⁢
                      between
                    
                  
                
                
                  
                    
                      
                        the
                        ⁢
                        
                          
                        
                        ⁢
                        three
                        ⁢
                        
                          
                        
                        ⁢
                        phases
                      
                      ,
                    
                  
                
                
                  
                    
                      contributes
                      ⁢
                      
                        
                      
                      ⁢
                      to
                      ⁢
                      
                        
                      
                      ⁢
                      producing
                    
                  
                
                
                  
                    reinforcement
                  
                
              
              ⁢
              
                
              
              --
            
            --
          
        
        -
        
          
            
              
                Term
                ⁢
                
                  
                
                ⁢
                for
                ⁢
                
                  
                
                ⁢
                which
                ⁢
                
                  
                
                ⁢
                the
              
            
          
          
            
              
                vector
                ⁢
                
                  
                
                ⁢
                sum
                ⁢
                
                  
                
                ⁢
                between
                ⁢
                
                  
                
                ⁢
                the
              
            
          
          
            
              
                three
                ⁢
                
                  
                
                ⁢
                phases
                ⁢
                
                  
                
                ⁢
                is
                ⁢
                
                  
                
                ⁢
                
                  zero
                  .
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        fu
        ∝
        
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  F
                  1
                  2
                
                +
                
                  F
                  3
                  2
                
                +
                
                  F
                  5
                  2
                
                +
                
                  F
                  7
                  2
                
              
              )
            
          
          -
          
            
              
                F
                1
                2
              
              2
            
            ⁢
            
              
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    α
                    -
                    120
                  
                  )
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            -
                          
                        
                      
                    
                  
                
              
            
          
          -
          
            
              
                F
                3
                2
              
              2
            
            ⁢
            
              
                cos
                ⁢
                
                  
                
                ⁢
                6
                ⁢
                
                  (
                  
                    θ
                    +
                    β
                    -
                    120
                  
                  )
                
              
              _
            
          
          -
          
            
              
                
                  
                    F
                    5
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                10
                ⁢
                
                  (
                  
                    θ
                    +
                    γ
                    -
                    120
                  
                  )
                
              
              -
              
                
                  
                    F
                    7
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                14
                ⁢
                
                  (
                  
                    θ
                    +
                    δ
                    -
                    120
                  
                  )
                
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      -
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    -
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          -
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              3
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          3
                          ⁢
                          β
                        
                        -
                        480
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          3
                          ⁢
                          β
                        
                        -
                        240
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        --
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
        
          
            F
            1
          
          ⁢
          
            
              F
              5
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          6
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        720
                      
                      }
                    
                  
                  _
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        480
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    -
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          8
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        960
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    -
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          6
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        720
                      
                      }
                    
                  
                  _
                
              
              ]
            
          
        
        -
        
          
            F
            3
          
          ⁢
          
            
              F
              5
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          8
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        960
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        240
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          --
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            3
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          10
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        1200
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        480
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              -
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
        
          
            F
            5
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          12
                          ⁢
                          θ
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        1440
                      
                      }
                    
                  
                  _
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        240
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    --
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        -
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  s
                                  -
                                  240
                                
                                )
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            --
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                -
                                s
                              
                              )
                            
                          
                        
                        }
                      
                    
                    -
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  240
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  α
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  240
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                --
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  -
                                  α
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  s
                                  -
                                  480
                                
                                }
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  -
                                  s
                                  -
                                  240
                                
                                }
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                
                                                                  --
                                                                  -
                                                                
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    10
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  480
                                
                                )
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  240
                                
                                )
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          -
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    16
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  480
                                
                                }
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    10
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  +
                                  240
                                
                                }
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                
                                                                  --
                                                                  
                                                                    --
                                                                    
                                                                      --
                                                                      
                                                                        --
                                                                        --
                                                                      
                                                                    
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  s
                                  -
                                  720
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  -
                                  s
                                  -
                                  480
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  720
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  480
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    18
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  720
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  +
                                  480
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    -
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                --
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  s
                                  -
                                  960
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  -
                                  s
                                  -
                                  720
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  960
                                
                                )
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  7
                                  ⁢
                                  δ
                                
                                -
                                
                                  7
                                  ⁢
                                  t
                                
                                -
                                720
                              
                              }
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    20
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  960
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  +
                                  720
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        (
        30
        )
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          I
                          1
                          2
                        
                        +
                        
                          I
                          7
                          2
                        
                        +
                        
                          I
                          13
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    
                      
                        I
                        1
                        2
                      
                      2
                    
                    ⁢
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            θ
                            +
                            s
                            -
                            120
                          
                          )
                        
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    -
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  -
                  
                    
                      
                        I
                        7
                        2
                      
                      2
                    
                    ⁢
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              14
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  t
                                
                                )
                              
                            
                            -
                            240
                          
                          }
                        
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  -
                  
                    
                      
                        I
                        13
                        2
                      
                      2
                    
                    ⁢
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          {
                          
                            
                              26
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  u
                                
                                )
                              
                            
                            -
                            240
                          
                          }
                        
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  -
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      
                        I
                        7
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  8
                                  ⁢
                                  θ
                                
                                +
                                s
                                +
                                
                                  7
                                  ⁢
                                  t
                                
                                -
                                240
                              
                              }
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        -
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  6
                                  ⁢
                                  θ
                                
                                -
                                s
                                +
                                
                                  7
                                  ⁢
                                  t
                                
                              
                              }
                            
                          
                          _
                        
                      
                      ]
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      
                        I
                        13
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  s
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  240
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  -
                                  s
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    
                      I
                      7
                    
                    ⁢
                    
                      
                        I
                        13
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    20
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  240
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  
  
    
      
        
          W
          ⁢
          
            -
          
          ⁢
          phase
          ⁢
          
            
          
          ⁢
          excitation
          ⁢
          
            
          
          ⁢
          force
          ⁢
          
            
          
          ⁢
          
            
              
            
            _
          
          ⁢
          
            
          
          ⁢
          
            
              
                
                  
                    
                      
                        Inphase
                        ⁢
                        
                          
                        
                        ⁢
                        term
                        ⁢
                        
                          
                        
                        ⁢
                        which
                      
                      ,
                      when
                    
                  
                
                
                  
                    
                      vector
                      ⁢
                      
                        
                      
                      ⁢
                      summed
                      ⁢
                      
                        
                      
                      ⁢
                      between
                    
                  
                
                
                  
                    
                      contributes
                      ⁢
                      
                        
                      
                      ⁢
                      to
                      ⁢
                      
                        
                      
                      ⁢
                      producing
                    
                  
                
                
                  
                    reinforcement
                  
                
              
              ⁢
              
                
              
              --
            
            --
          
        
        -
        
          
            
              
                Term
                ⁢
                
                  
                
                ⁢
                for
                ⁢
                
                  
                
                ⁢
                which
                ⁢
                
                  
                
                ⁢
                the
              
            
          
          
            
              
                vector
                ⁢
                
                  
                
                ⁢
                sum
                ⁢
                
                  
                
                ⁢
                between
                ⁢
                
                  
                
                ⁢
                the
              
            
          
          
            
              
                three
                ⁢
                
                  
                
                ⁢
                phases
                ⁢
                
                  
                
                ⁢
                is
                ⁢
                
                  
                
                ⁢
                
                  zero
                  .
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        fu
        ∝
        
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  F
                  1
                  2
                
                +
                
                  F
                  3
                  2
                
                +
                
                  F
                  5
                  2
                
                +
                
                  F
                  7
                  2
                
              
              )
            
          
          -
          
            
              
                F
                1
                2
              
              2
            
            ⁢
            
              
                cos
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  (
                  
                    θ
                    +
                    α
                    -
                    240
                  
                  )
                
              
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            -
                          
                        
                      
                    
                  
                
              
            
          
          -
          
            
              
                
                  F
                  3
                  2
                
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              6
              ⁢
              
                (
                
                  θ
                  +
                  β
                  -
                  240
                
                )
              
            
            _
          
          -
          
            
              
                
                  
                    F
                    5
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                10
                ⁢
                
                  (
                  
                    θ
                    +
                    γ
                    -
                    240
                  
                  )
                
              
              -
              
                
                  
                    F
                    7
                    2
                  
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                14
                ⁢
                
                  (
                  
                    θ
                    +
                    δ
                    -
                    240
                  
                  )
                
              
            
            
              --
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  --
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
          -
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              3
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          3
                          ⁢
                          β
                        
                        -
                        960
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          3
                          ⁢
                          β
                        
                        -
                        480
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        --
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
        
          
            F
            1
          
          ⁢
          
            
              F
              5
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          6
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        1440
                      
                      }
                    
                  
                  _
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        960
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    --
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            1
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          8
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        1920
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    --
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          6
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        1440
                      
                      }
                    
                  
                  _
                
              
              ]
            
          
        
        -
        
          
            F
            3
          
          ⁢
          
            
              F
              5
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          8
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        1920
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        -
                        480
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            -
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
      
    
    
      
        
      
    
  
  
    
      
        
          
            F
            3
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          10
                          ⁢
                          θ
                        
                        +
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        2400
                      
                      }
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          4
                          ⁢
                          θ
                        
                        -
                        
                          3
                          ⁢
                          β
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        960
                      
                      }
                    
                  
                
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            --
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        -
        
          
            F
            5
          
          ⁢
          
            
              F
              7
            
            ⁡
            
              [
              
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          12
                          ⁢
                          θ
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        2880
                      
                      }
                    
                  
                  _
                
                -
                
                  
                    cos
                    ⁢
                    
                      {
                      
                        
                          2
                          ⁢
                          θ
                        
                        -
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                        -
                        480
                      
                      }
                    
                  
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    --
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        +
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          N
          ⁢
          
            〈
            
              
                
                  
                    
                      
                        -
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  s
                                  -
                                  480
                                
                                )
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            --
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                -
                                s
                              
                              )
                            
                          
                        
                        }
                      
                    
                    -
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  480
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  α
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            1
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  α
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  480
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                --
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  -
                                  α
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  s
                                  -
                                  960
                                
                                }
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  -
                                  s
                                  -
                                  480
                                
                                }
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                
                                                                  --
                                                                  -
                                                                
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    10
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  960
                                
                                )
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  480
                                
                                )
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                
                                                                  --
                                                                  
                                                                    --
                                                                    --
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            3
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    16
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  960
                                
                                }
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    10
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    3
                                    ⁢
                                    β
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  +
                                  480
                                
                                }
                              
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                
                                                                  --
                                                                  
                                                                    --
                                                                    
                                                                      --
                                                                      
                                                                        --
                                                                        --
                                                                      
                                                                    
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  s
                                  -
                                  1440
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    4
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  -
                                  s
                                  -
                                  960
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              -
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  1440
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  960
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            5
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    18
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  1440
                                
                                }
                              
                            
                            _
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    5
                                    ⁢
                                    γ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  +
                                  960
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    -
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  -
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            1
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    8
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  s
                                  -
                                  1920
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  -
                                  s
                                  -
                                  1440
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            7
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  -
                                  1920
                                
                                )
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                --
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  7
                                  ⁢
                                  δ
                                
                                -
                                
                                  7
                                  ⁢
                                  t
                                
                                -
                                1440
                              
                              }
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          
                            F
                            7
                          
                          ⁢
                          
                            I
                            13
                          
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    20
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  1920
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  --
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    7
                                    ⁢
                                    δ
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  +
                                  1440
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                
              
            
            〉
          
        
        +
      
    
    
      
        
          (
          31
          )
        
        ⁢
        
          
        
      
    
  
  
    
      
        
          N
          2
        
        ⁢
        
          〈
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          I
                          1
                          2
                        
                        +
                        
                          I
                          7
                          2
                        
                        +
                        
                          I
                          13
                          2
                        
                      
                      )
                    
                  
                  -
                  
                    
                      
                        
                          I
                          1
                          2
                        
                        2
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        (
                        
                          θ
                          +
                          s
                          -
                          240
                        
                        )
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    -
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  -
                  
                    
                      
                        
                          I
                          7
                          2
                        
                        2
                      
                      ⁢
                      cos
                      ⁢
                      
                        {
                        
                          
                            14
                            ⁢
                            
                              (
                              
                                θ
                                +
                                t
                              
                              )
                            
                          
                          -
                          480
                        
                        }
                      
                    
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  -
                  
                    
                      
                        I
                        13
                        2
                      
                      2
                    
                    ⁢
                    
                      
                        cos
                        ⁢
                        
                          {
                          
                            
                              26
                              ⁢
                              
                                (
                                
                                  θ
                                  +
                                  u
                                
                                )
                              
                            
                            -
                            480
                          
                          }
                        
                      
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  -
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      
                        I
                        7
                      
                      [
                      
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  8
                                  ⁢
                                  θ
                                
                                +
                                s
                                +
                                
                                  7
                                  ⁢
                                  t
                                
                                -
                                480
                              
                              }
                            
                          
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        -
                        
                          
                            cos
                            ⁢
                            
                              {
                              
                                
                                  6
                                  ⁢
                                  θ
                                
                                -
                                s
                                +
                                
                                  7
                                  ⁢
                                  t
                                
                              
                              }
                            
                          
                          _
                        
                      
                      ]
                    
                  
                  -
                
              
            
            
              
                
                  
                    
                      I
                      1
                    
                    ⁢
                    
                      
                        I
                        13
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    14
                                    ⁢
                                    θ
                                  
                                  +
                                  s
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  480
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    12
                                    ⁢
                                    θ
                                  
                                  -
                                  s
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    
                      I
                      7
                    
                    ⁢
                    
                      
                        I
                        13
                      
                      ⁡
                      
                        [
                        
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    20
                                    ⁢
                                    θ
                                  
                                  +
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                  -
                                  480
                                
                                }
                              
                            
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                --
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          -
                          
                            
                              cos
                              ⁢
                              
                                {
                                
                                  
                                    6
                                    ⁢
                                    θ
                                  
                                  -
                                  
                                    7
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    13
                                    ⁢
                                    u
                                  
                                
                                }
                              
                            
                            _
                          
                        
                        ]
                      
                    
                  
                
              
            
          
          〉
        
      
    
    
      
        
      
    
  




In the equations (29), (30), and (31), the terms that are in phase and therefore reinforcing each other in the vector sum of the three phase are underlined with a solid line, while the terms for which the vector sum is zero are underlined with a dashed line. When the terms for which the vector sum is zero are eliminated from the equations (29), (30), and (31), and the DC component terms and the terms that are in phase and therefore reinforcing each other are extracted, the following equations are obtained.




  
    
      
        fu
        ∝
        
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  F
                  1
                  2
                
                +
                
                  F
                  3
                  2
                
                +
                
                  F
                  5
                  2
                
                +
                
                  F
                  7
                  2
                
              
              )
            
          
          -
          
            
              
                F
                3
                2
              
              2
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            6
            ⁢
            
              (
              
                θ
                +
                β
              
              )
            
          
          -
          
            
              F
              1
            
            ⁢
            
              F
              5
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    6
                    ⁢
                    θ
                  
                  +
                  α
                  +
                  
                    5
                    ⁢
                    γ
                  
                
                )
              
            
          
          +
          
            
              F
              1
            
            ⁢
            
              F
              7
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    6
                    ⁢
                    θ
                  
                  -
                  α
                  +
                  
                    7
                    ⁢
                    δ
                  
                
                )
              
            
          
          -
          
            
              F
              5
            
            ⁢
            
              F
              7
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    12
                    ⁢
                    θ
                  
                  +
                  
                    5
                    ⁢
                    γ
                  
                  +
                  
                    7
                    ⁢
                    δ
                  
                
                )
              
            
          
          +
        
      
    
    
      
        (
        32
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            2
            ⁢
            N
            ⁢
            
              〈
              
                
                  
                    
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              a
                              -
                              s
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              -
                              a
                              +
                              
                                7
                                ⁢
                                t
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                12
                                ⁢
                                θ
                              
                              -
                              a
                              +
                              
                                13
                                ⁢
                                u
                              
                            
                            )
                          
                        
                      
                      -
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              s
                            
                            )
                          
                        
                      
                      -
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                12
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              
                                7
                                ⁢
                                t
                              
                            
                            )
                          
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                18
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              
                                13
                                ⁢
                                u
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                δ
                              
                              -
                              s
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                7
                                ⁢
                                δ
                              
                              -
                              
                                7
                                ⁢
                                t
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              -
                              
                                7
                                ⁢
                                δ
                              
                              +
                              
                                13
                                ⁢
                                u
                              
                            
                            )
                          
                        
                      
                    
                  
                
              
              〉
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            N
            2
          
          ⁢
          
            〈
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      I
                      1
                      2
                    
                    +
                    
                      I
                      7
                      2
                    
                    +
                    
                      I
                      13
                      2
                    
                  
                  )
                
              
              +
              
                
                  I
                  1
                
                ⁢
                
                  I
                  7
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      -
                      s
                      +
                      
                        7
                        ⁢
                        t
                      
                    
                    )
                  
                
              
              +
              
                
                  I
                  1
                
                ⁢
                
                  I
                  13
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        12
                        ⁢
                        θ
                      
                      -
                      s
                      +
                      
                        13
                        ⁢
                        u
                      
                    
                    )
                  
                
              
              +
              
                
                  I
                  7
                
                ⁢
                
                  I
                  13
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      +
                      
                        7
                        ⁢
                        t
                      
                      -
                      
                        13
                        ⁢
                        u
                      
                    
                    )
                  
                
              
            
            〉
          
        
      
    
    
      
        
      
    
  
  
    
      
        fv
        ∝
        
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  F
                  1
                  2
                
                +
                
                  F
                  3
                  2
                
                +
                
                  F
                  5
                  2
                
                +
                
                  F
                  7
                  2
                
              
              )
            
          
          -
          
            
              
                F
                3
                2
              
              2
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            6
            ⁢
            
              (
              
                θ
                +
                β
                -
                120
              
              )
            
          
          -
          
            
              F
              1
            
            ⁢
            
              F
              5
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    6
                    ⁢
                    θ
                  
                  +
                  α
                  +
                  
                    5
                    ⁢
                    γ
                  
                  -
                  720
                
                )
              
            
          
          +
          
            
              F
              1
            
            ⁢
            
              F
              7
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    6
                    ⁢
                    θ
                  
                  -
                  α
                  +
                  
                    7
                    ⁢
                    δ
                  
                  -
                  720
                
                )
              
            
          
          -
          
            
              F
              5
            
            ⁢
            
              F
              7
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    12
                    ⁢
                    θ
                  
                  +
                  
                    5
                    ⁢
                    γ
                  
                  +
                  
                    7
                    ⁢
                    δ
                  
                  -
                  1440
                
                )
              
            
          
          +
        
      
    
    
      
        (
        33
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            2
            ⁢
            N
            ⁢
            
              〈
              
                
                  
                    
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              a
                              -
                              s
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              -
                              a
                              +
                              
                                7
                                ⁢
                                t
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                12
                                ⁢
                                θ
                              
                              -
                              a
                              +
                              
                                13
                                ⁢
                                u
                              
                            
                            )
                          
                        
                      
                      -
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              s
                              -
                              720
                            
                            )
                          
                        
                      
                      -
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                12
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              
                                7
                                ⁢
                                t
                              
                              -
                              720
                            
                            )
                          
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                18
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              
                                13
                                ⁢
                                u
                              
                              -
                              720
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                δ
                              
                              -
                              s
                              -
                              720
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                7
                                ⁢
                                δ
                              
                              -
                              
                                7
                                ⁢
                                t
                              
                              -
                              720
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              -
                              
                                7
                                ⁢
                                δ
                              
                              +
                              
                                13
                                ⁢
                                u
                              
                              +
                              720
                            
                            )
                          
                        
                      
                    
                  
                
              
              〉
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            N
            2
          
          ⁢
          
            〈
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      I
                      1
                      2
                    
                    +
                    
                      I
                      7
                      2
                    
                    +
                    
                      I
                      13
                      2
                    
                  
                  )
                
              
              +
              
                
                  I
                  1
                
                ⁢
                
                  I
                  7
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      -
                      s
                      +
                      
                        7
                        ⁢
                        t
                      
                    
                    )
                  
                
              
              +
              
                
                  I
                  1
                
                ⁢
                
                  I
                  13
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        12
                        ⁢
                        θ
                      
                      -
                      s
                      +
                      
                        13
                        ⁢
                        u
                      
                    
                    )
                  
                
              
              +
              
                
                  I
                  7
                
                ⁢
                
                  I
                  13
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      -
                      
                        7
                        ⁢
                        t
                      
                      +
                      
                        13
                        ⁢
                        u
                      
                    
                    )
                  
                
              
            
            〉
          
        
      
    
    
      
        
      
    
  
  
    
      
        fw
        ∝
        
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  F
                  1
                  2
                
                +
                
                  F
                  3
                  2
                
                +
                
                  F
                  5
                  2
                
                +
                
                  F
                  7
                  2
                
              
              )
            
          
          -
          
            
              
                F
                3
                2
              
              2
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            6
            ⁢
            
              (
              
                θ
                +
                β
                -
                240
              
              )
            
          
          -
          
            
              F
              1
            
            ⁢
            
              F
              5
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    6
                    ⁢
                    θ
                  
                  +
                  α
                  +
                  
                    5
                    ⁢
                    γ
                  
                  -
                  1440
                
                )
              
            
          
          +
          
            
              F
              1
            
            ⁢
            
              F
              7
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    6
                    ⁢
                    θ
                  
                  -
                  α
                  +
                  
                    7
                    ⁢
                    δ
                  
                  -
                  1440
                
                )
              
            
          
          -
          
            
              F
              5
            
            ⁢
            
              F
              7
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    12
                    ⁢
                    θ
                  
                  +
                  
                    5
                    ⁢
                    γ
                  
                  +
                  
                    7
                    ⁢
                    δ
                  
                  -
                  2880
                
                )
              
            
          
          +
        
      
    
    
      
        
          
        
        ⁢
        
          (
          34
          )
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            2
            ⁢
            N
            ⁢
            
              〈
              
                
                  
                    
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              a
                              -
                              s
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              -
                              a
                              +
                              
                                7
                                ⁢
                                t
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              1
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                12
                                ⁢
                                θ
                              
                              -
                              a
                              +
                              
                                13
                                ⁢
                                u
                              
                            
                            )
                          
                        
                      
                      -
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              s
                              -
                              1440
                            
                            )
                          
                        
                      
                      -
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                12
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              
                                7
                                ⁢
                                t
                              
                              -
                              1440
                            
                            )
                          
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        
                          
                            
                              F
                              5
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                18
                                ⁢
                                θ
                              
                              +
                              
                                5
                                ⁢
                                γ
                              
                              +
                              
                                13
                                ⁢
                                u
                              
                              -
                              1440
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              1
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              +
                              
                                7
                                ⁢
                                δ
                              
                              -
                              s
                              -
                              1440
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              7
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                7
                                ⁢
                                δ
                              
                              -
                              
                                7
                                ⁢
                                t
                              
                              -
                              1440
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            
                              F
                              7
                            
                            ⁢
                            
                              I
                              13
                            
                          
                          2
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                6
                                ⁢
                                θ
                              
                              -
                              
                                7
                                ⁢
                                δ
                              
                              +
                              
                                13
                                ⁢
                                u
                              
                              +
                              1440
                            
                            )
                          
                        
                      
                    
                  
                
              
              〉
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            N
            2
          
          ⁢
          
            〈
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      I
                      1
                      2
                    
                    +
                    
                      I
                      7
                      2
                    
                    +
                    
                      I
                      13
                      2
                    
                  
                  )
                
              
              +
              
                
                  I
                  1
                
                ⁢
                
                  I
                  7
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      -
                      s
                      +
                      
                        7
                        ⁢
                        t
                      
                    
                    )
                  
                
              
              +
              
                
                  I
                  1
                
                ⁢
                
                  I
                  13
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        12
                        ⁢
                        θ
                      
                      -
                      s
                      +
                      
                        13
                        ⁢
                        u
                      
                    
                    )
                  
                
              
              +
              
                
                  I
                  7
                
                ⁢
                
                  I
                  13
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      -
                      
                        7
                        ⁢
                        t
                      
                      +
                      
                        13
                        ⁢
                        u
                      
                    
                    )
                  
                
              
            
            〉
          
        
      
    
    
      
        
      
    
  




Arranging the above equations similarly to the equation (25), the sum of the respective excitation forces is given by the following equation.




  
    
      
        
          
            
              
                fu
                +
                fv
                +
                fw
              
              ∝
            
          
        
        
          
            
              
                
                  
                    3
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        F
                        1
                        2
                      
                      +
                      
                        F
                        3
                        2
                      
                      +
                      
                        F
                        5
                        2
                      
                      +
                      
                        F
                        7
                        2
                      
                    
                    )
                  
                
                +
                
                  
                    3
                    2
                  
                  ⁢
                  
                    
                      N
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          I
                          1
                          2
                        
                        +
                        
                          I
                          7
                          2
                        
                        +
                        
                          I
                          13
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  3
                  ⁢
                  N
                  ⁢
                  
                    
                  
                  ⁢
                  
                    F
                    1
                  
                  ⁢
                  
                    I
                    1
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        α
                        -
                        s
                      
                      )
                    
                  
                
                +
                
                  3
                  ⁢
                  
                    NF
                    7
                  
                  ⁢
                  
                    I
                    7
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          7
                          ⁢
                          δ
                        
                        -
                        
                          7
                          ⁢
                          t
                        
                      
                      )
                    
                  
                
                -
              
              
                _
                
                  (
                  1
                  )
                
              
            
          
        
        
          
            
              
                
                  
                    
                      3
                      ⁢
                      
                        F
                        3
                        2
                      
                    
                    2
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  6
                  ⁢
                  
                    (
                    
                      θ
                      +
                      β
                    
                    )
                  
                
                
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              --
                            
                          
                        
                      
                    
                  
                  
                    (
                    2
                    )
                  
                
              
              -
              
                
                  3
                  ⁢
                  
                    F
                    1
                  
                  ⁢
                  
                    F
                    5
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          6
                          ⁢
                          θ
                        
                        +
                        α
                        +
                        
                          5
                          ⁢
                          γ
                        
                      
                      )
                    
                  
                
                
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    --
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  
                    (
                    3
                    )
                  
                
              
              +
              
                
                  3
                  ⁢
                  
                    F
                    1
                  
                  ⁢
                  
                    F
                    7
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          6
                          ⁢
                          θ
                        
                        -
                        α
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      )
                    
                  
                
                
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    --
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  
                    (
                    4
                    )
                  
                
              
              -
              
                
                  3
                  ⁢
                  
                    F
                    5
                  
                  ⁢
                  
                    F
                    7
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          12
                          ⁢
                          θ
                        
                        +
                        
                          5
                          ⁢
                          γ
                        
                        +
                        
                          7
                          ⁢
                          δ
                        
                      
                      )
                    
                  
                
                
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      --
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  
                    (
                    5
                    )
                  
                
              
              +
            
          
        
        
          
            
              
                N
                ⁢
                
                  〈
                  
                    
                      
                        
                          
                            
                              3
                              ⁢
                              
                                F
                                1
                              
                              ⁢
                              
                                I
                                7
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      6
                                      ⁢
                                      θ
                                    
                                    -
                                    α
                                    +
                                    
                                      7
                                      ⁢
                                      t
                                    
                                  
                                  )
                                
                              
                            
                            
                              _
                              
                                (
                                8
                                )
                              
                            
                          
                          +
                          
                            
                              3
                              ⁢
                              
                                F
                                1
                              
                              ⁢
                              
                                F
                                13
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      12
                                      ⁢
                                      θ
                                    
                                    -
                                    α
                                    +
                                    
                                      13
                                      ⁢
                                      u
                                    
                                  
                                  )
                                
                              
                            
                            
                              _
                              
                                (
                                9
                                )
                              
                            
                          
                          -
                          
                            
                              3
                              ⁢
                              
                                F
                                5
                              
                              ⁢
                              
                                F
                                1
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      6
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      5
                                      ⁢
                                      γ
                                    
                                    +
                                    s
                                  
                                  )
                                
                              
                            
                            
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                -
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                              
                                (
                                6
                                )
                              
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              3
                              ⁢
                              
                                F
                                5
                              
                              ⁢
                              
                                I
                                7
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      12
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      5
                                      ⁢
                                      γ
                                    
                                    +
                                    
                                      7
                                      ⁢
                                      t
                                    
                                  
                                  )
                                
                              
                            
                            
                              _
                              
                                (
                                10
                                )
                              
                            
                          
                          -
                          
                            
                              3
                              ⁢
                              
                                F
                                5
                              
                              ⁢
                              
                                I
                                13
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      18
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      5
                                      ⁢
                                      γ
                                    
                                    +
                                    
                                      13
                                      ⁢
                                      u
                                    
                                  
                                  )
                                
                              
                            
                            
                              (
                              11
                              )
                            
                          
                          +
                          
                            
                              3
                              ⁢
                              
                                F
                                7
                              
                              ⁢
                              
                                I
                                1
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      6
                                      ⁢
                                      θ
                                    
                                    +
                                    
                                      7
                                      ⁢
                                      δ
                                    
                                    -
                                    s
                                  
                                  )
                                
                              
                            
                            
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              --
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                              
                                (
                                7
                                )
                              
                            
                          
                          +
                          
                            
                              3
                              ⁢
                              
                                F
                                7
                              
                              ⁢
                              
                                I
                                13
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      6
                                      ⁢
                                      θ
                                    
                                    -
                                    
                                      7
                                      ⁢
                                      δ
                                    
                                    +
                                    
                                      13
                                      ⁢
                                      u
                                    
                                  
                                  )
                                
                              
                            
                            
                              _
                              
                                (
                                12
                                )
                              
                            
                          
                        
                      
                    
                  
                  〉
                
              
              +
            
          
        
        
          
            
              
                N
                2
              
              ⁢
              
                〈
                
                  
                    
                      3
                      ⁢
                      
                        I
                        1
                      
                      ⁢
                      
                        I
                        7
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              6
                              ⁢
                              θ
                            
                            -
                            s
                            +
                            
                              7
                              ⁢
                              t
                            
                          
                          )
                        
                      
                    
                    
                      _
                      
                        (
                        13
                        )
                      
                    
                  
                  +
                  
                    
                      3
                      ⁢
                      
                        I
                        1
                      
                      ⁢
                      
                        I
                        13
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              12
                              ⁢
                              θ
                            
                            -
                            s
                            +
                            
                              13
                              ⁢
                              u
                            
                          
                          )
                        
                      
                    
                    
                      _
                      
                        (
                        14
                        )
                      
                    
                  
                  +
                  
                    
                      3
                      ⁢
                      
                        I
                        7
                      
                      ⁢
                      
                        I
                        13
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              6
                              ⁢
                              θ
                            
                            +
                            
                              7
                              ⁢
                              t
                            
                            -
                            
                              13
                              ⁢
                              u
                            
                          
                          )
                        
                      
                    
                    
                      _
                      
                        (
                        15
                        )
                      
                    
                  
                
                〉
              
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




Term (1): DC component
Term (2): Sixth component occurring due to the third harmonic of the rotor magnetomotive force
Term (3): Sixth component occurring due to the first and fifth harmonics of the rotor magnetomotive force
Term (4): Sixth component occurring due to the first and seventh harmonics of the rotor magnetomotive force
Term (5): 12th component occurring due to the fifth and seventh harmonics of the rotor magnetomotive force
Term (6): Sixth component occurring due to the fifth harmonic of the rotor magnetomotive force and the first harmonic of the stator current
Term (7): Sixth component occurring due to the seventh harmonic of the rotor magnetomotive force and the first harmonic of the stator current
Terms occurring when the seventh and 13th harmonics of the stator current are superimposed
Term (8): Sixth component occurring due to the first harmonic of the rotor magnetomotive force and the seventh harmonic of the stator current
Term (9): 12th component occurring due to the first harmonic of the rotor magnetomotive force and the 13th harmonic of the stator current
Term (10): 12th component occurring due to the fifth harmonic of the rotor magnetomotive force and the seventh harmonic of the stator current
Term (11): 18th component occurring due to the fifth harmonic of the rotor magnetomotive force and the 13th harmonic of the stator current
Term (12): Sixth component occurring due to the seventh harmonic of the rotor magnetomotive force and the 13th harmonic of the stator current
Term (13): Sixth component occurring due to the first and seventh harmonics of the stator current
Term (14): 12th component occurring due to the first and 13th harmonics of the stator current
Term (15): Sixth component occurring due to the seventh and 13th harmonics of the stator current
In the equation (35), the term (1) is the DC component term, the term (2) is the sixth harmonic component occurring due to the third harmonic of the rotor magnetomotive force Fmag, the term (3) is the sixth harmonic component occurring due to the first and fifth harmonics of the rotor magnetomotive force Fmag, the term (4) is the sixth harmonic component occurring due to the first and seventh harmonics of the rotor magnetomotive force Fmag, the term (5) is the 12th harmonic component occurring due to the fifth and seventh harmonics of the rotor magnetomotive force Fmag, the term (6) is the sixth harmonic component occurring due to the fifth harmonic of the rotor magnetomotive force Fmag and the first current component (fundamental) of the stator current, and the term (7) is the sixth harmonic component occurring due to the seventh harmonic of the rotor magnetomotive force Fmag and the first current component (fundamental) of the stator current.
Further, the term (8) is the sixth harmonic component occurring due to the first harmonic (fundamental) of the rotor magnetomotive force Fmag and the seventh harmonic of the stator current, the term (9) is the 12th harmonic component occurring due to the first harmonic (fundamental) of the rotor magnetomotive force Fmag and the 13th harmonic of the stator current, the term (10) is the 12th harmonic component occurring due to the fifth harmonic of the rotor magnetomotive force Fmag and the seventh harmonic of the stator current, the term (11) is the 18th harmonic component occurring due to the fifth harmonic of the rotor magnetomotive force Fmag and the 13th harmonic of the stator current, the term (12) is the sixth harmonic component occurring due to the seventh harmonic of the rotor magnetomotive force Fmag and the 13th harmonic of the stator current, the term (13) is the sixth harmonic component occurring due to the first (fundamental) and seventh harmonics of the stator current, the term (14) is the 12th harmonic component occurring due to the first (fundamental) and 13th harmonics of the stator current, and the term (15) is the sixth harmonic component occurring due to the seventh and 13th harmonics of the stator current.
After all, the sum of the excitation forces shown by the equation (35) consists of the sixth and 12th harmonics; therefore, by suitably setting the phase angle and amplitude of each of the terms (2), (3), (4), (6), (7), (8), (12), (13), and (15), the vector sum of the terms (2), (3), (4), (6), (7), (8), (12), (13), and (15) can be made equal to zero or reduced or increased, thereby canceling or reducing (or increasing) the sixth harmonic of the magnetic noise. Likewise, by suitably setting the phase angle and amplitude of each of the terms (5), (9), (10), and (14), the vector sum of the terms (5), (9), (10), and (14) can be made equal to zero or reduced or increased, thereby canceling or reducing (or increasing) the 12th harmonic of the magnetic noise.
That is, in the equation (35), by suitably adjusting the phases and amplitudes of the seventh and 13th harmonic current components, the respective sums of the terms shown in the equation (35), excluding the DC component term, can be made equal to zero, to cancel or reduce (or increase) the sixth and 12th harmonics of the magnetic sound which are the most critical sound factors in the three-phase AC rotary electric machine.
The condition for canceling the sixth harmonic of the magnetic sound is shown by the following equation.
(1) Cancellation of the Sixth Harmonic
The equation “Vector sum of the magnetic sound terms ((2)+(3)+(4)+(6)+(7))+Vector sum of the canceling terms ((8)+(12)+(13)+(15))=0” holds.




  
    
      
        
          
            
              
                
                  -
                  
                    
                      3
                      ⁢
                      
                        F
                        3
                        2
                      
                    
                    2
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                6
                ⁢
                
                  (
                  
                    θ
                    +
                    β
                  
                  )
                
              
              -
              
                3
                ⁢
                
                  F
                  1
                
                ⁢
                
                  F
                  5
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      +
                      α
                      +
                      
                        5
                        ⁢
                        γ
                      
                    
                    )
                  
                
              
              +
              
                3
                ⁢
                
                  F
                  1
                
                ⁢
                
                  F
                  7
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      -
                      α
                      +
                      
                        7
                        ⁢
                        δ
                      
                    
                    )
                  
                
              
              -
              
                3
                ⁢
                
                  NF
                  5
                
                ⁢
                
                  I
                  1
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      +
                      
                        5
                        ⁢
                        γ
                      
                      +
                      s
                    
                    )
                  
                
              
              +
              
                3
                ⁢
                
                  NF
                  7
                
                ⁢
                
                  I
                  1
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        6
                        ⁢
                        θ
                      
                      +
                      
                        7
                        ⁢
                        δ
                      
                      -
                      s
                    
                    )
                  
                
              
              +
            
            
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    
                                      --
                                      
                                        --
                                        
                                          --
                                          
                                            --
                                            
                                              --
                                              
                                                --
                                                
                                                  --
                                                  
                                                    --
                                                    
                                                      --
                                                      
                                                        --
                                                        
                                                          --
                                                          
                                                            --
                                                            
                                                              --
                                                              
                                                                --
                                                                
                                                                  --
                                                                  
                                                                    --
                                                                    
                                                                      --
                                                                      
                                                                        --
                                                                        
                                                                          --
                                                                          
                                                                            --
                                                                            
                                                                              --
                                                                              
                                                                                --
                                                                                
                                                                                  --
                                                                                  
                                                                                    --
                                                                                    
                                                                                      --
                                                                                      
                                                                                        --
                                                                                        
                                                                                          --
                                                                                          
                                                                                            --
                                                                                            
                                                                                              --
                                                                                              
                                                                                                --
                                                                                                
                                                                                                  --
                                                                                                  
                                                                                                    --
                                                                                                    
                                                                                                      --
                                                                                                      
                                                                                                        --
                                                                                                        
                                                                                                          --
                                                                                                          
                                                                                                            --
                                                                                                            
                                                                                                              --
                                                                                                              
                                                                                                                --
                                                                                                                
                                                                                                                  --
                                                                                                                  
                                                                                                                    --
                                                                                                                    
                                                                                                                      --
                                                                                                                      -
                                                                                                                    
                                                                                                                  
                                                                                                                
                                                                                                              
                                                                                                            
                                                                                                          
                                                                                                        
                                                                                                      
                                                                                                    
                                                                                                  
                                                                                                
                                                                                              
                                                                                            
                                                                                          
                                                                                        
                                                                                      
                                                                                    
                                                                                  
                                                                                
                                                                              
                                                                            
                                                                          
                                                                        
                                                                      
                                                                    
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              
                Vector
                ⁢
                
                  
                
                ⁢
                sum
                ⁢
                
                  
                
                ⁢
                of
                ⁢
                
                  
                
                ⁢
                magnetic
                ⁢
                
                  
                
                ⁢
                noise
                ⁢
                
                  
                
                ⁢
                terms
              
            
          
          ⁢
          
            
              
                N
                ⁢
                
                  〈
                  
                    
                      3
                      ⁢
                      
                        F
                        1
                      
                      ⁢
                      
                        I
                        7
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              6
                              ⁢
                              θ
                            
                            -
                            α
                            +
                            
                              7
                              ⁢
                              t
                            
                          
                          )
                        
                      
                    
                    +
                    
                      3
                      ⁢
                      
                        F
                        7
                      
                      ⁢
                      
                        I
                        13
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              6
                              ⁢
                              θ
                            
                            -
                            
                              7
                              ⁢
                              δ
                            
                            +
                            
                              13
                              ⁢
                              u
                            
                          
                          )
                        
                      
                    
                  
                  〉
                
              
              +
              
                
                  N
                  2
                
                ⁢
                
                  〈
                  
                    
                      3
                      ⁢
                      
                        I
                        1
                      
                      ⁢
                      
                        I
                        7
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              6
                              ⁢
                              θ
                            
                            -
                            s
                            +
                            
                              7
                              ⁢
                              t
                            
                          
                          )
                        
                      
                    
                    +
                    
                      3
                      ⁢
                      
                        I
                        7
                      
                      ⁢
                      
                        I
                        13
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              6
                              ⁢
                              θ
                            
                            +
                            
                              7
                              ⁢
                              t
                            
                            -
                            
                              13
                              ⁢
                              u
                            
                          
                          )
                        
                      
                    
                  
                  〉
                
              
            
            
              _
              
                Canceling
                ⁢
                
                  
                
                ⁢
                terms
              
            
          
        
        =
        0
      
    
    
      
        (
        36
        )
      
    
  




In the equation (36), the sum or difference of the terms underlined with a dashed line indicates the vector sum of the sixth harmonics of the radially acting magnetic excitation force that contribute to the generation of the magnetic sound, and the sum or difference of the terms underlined with a solid line indicates the vector sum of the sixth harmonics of the excitation force occurring due to the canceling harmonic current. Accordingly, in the equation (36), the phase angle and amplitude should be set so as to make the sum of these vector sums equal to zero. The condition for canceling the 12th harmonic of the magnetic noise is shown by the following equation.
(2) Cancellation of the 12th Harmonic
Vector sum of the magnetic noise terms (5)+Vector sum of the canceling terms ((9)+(10)+(14))=0




  
    
      
        
          
            
              
                
                  -
                  3
                
                ⁢
                
                  F
                  5
                
                ⁢
                
                  F
                  7
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        12
                        ⁢
                        θ
                      
                      +
                      
                        5
                        ⁢
                        γ
                      
                      +
                      
                        7
                        ⁢
                        δ
                      
                    
                    )
                  
                
              
              +
            
            
              
                --
                
                  --
                  
                    --
                    
                      --
                      
                        --
                        
                          --
                          
                            --
                            
                              --
                              
                                --
                                
                                  --
                                  
                                    --
                                    -
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              
                Vector
                ⁢
                
                  
                
                ⁢
                sum
                ⁢
                
                  
                
                ⁢
                of
                ⁢
                
                  
                
                ⁢
                magnetic
                ⁢
                
                  
                
                ⁢
                noise
                ⁢
                
                  
                
                ⁢
                terms
              
            
          
          ⁢
          
            
              
                N
                ⁢
                
                  〈
                  
                    
                      
                        +
                        3
                      
                      ⁢
                      
                        F
                        1
                      
                      ⁢
                      
                        I
                        13
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              12
                              ⁢
                              θ
                            
                            -
                            α
                            +
                            
                              13
                              ⁢
                              u
                            
                          
                          )
                        
                      
                    
                    -
                    
                      3
                      ⁢
                      
                        F
                        5
                      
                      ⁢
                      
                        I
                        7
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              12
                              ⁢
                              θ
                            
                            +
                            
                              5
                              ⁢
                              γ
                            
                            +
                            
                              7
                              ⁢
                              t
                            
                          
                          )
                        
                      
                    
                  
                  〉
                
              
              +
              
                
                  N
                  2
                
                ⁢
                
                  〈
                  
                    3
                    ⁢
                    
                      I
                      1
                    
                    ⁢
                    
                      I
                      13
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            12
                            ⁢
                            θ
                          
                          -
                          s
                          +
                          
                            13
                            ⁢
                            u
                          
                        
                        )
                      
                    
                  
                  〉
                
              
            
            
              _
              
                Canceling
                ⁢
                
                  
                
                ⁢
                terms
              
            
          
        
        =
        0
      
    
    
      
        (
        37
        )
      
    
  




In the equation (37), the sum or difference of the terms underlined with a dashed line indicates the vector sum of the 12th harmonics that contribute to the generation of the magnetic sound, and the sum or difference of the terms underlined with a solid line indicates the vector sum of the 12th harmonics of the excitation force occurring due to the canceling harmonic current. Accordingly, in the equation (37), the phase angle and amplitude should be set so as to make the sum of these vector sums equal to zero.
In the above equations (35), (36), and (37), as θ, as a function of the number of revolutions, changes from moment to moment, the amplitudes and phases of the harmonic currents for controlling the sum of magnetic sound shown by the equation (35) to the prescribed level also change from moment to moment as the number of revolutions and the phase and amplitude of the fundamental wave change with time. Likewise, the amplitudes and phases of the harmonic currents, that satisfy the equations (36) and (37), also change from moment to moment. Accordingly, the amplitudes and phases of the harmonic currents to be superimposed are computed at every predetermined time interval in accordance with the frequency, phase, and amplitude of the fundamental current.
CIRCUIT CONFIGURATION EXAMPLE 1
An example of the circuit for superimposing the above-described harmonic currents is shown in FIG. 3. The motor control circuit shown here is an embodiment that performs the feedback control of the motor current.
Reference numeral 10 is a motor current control means for controlling the motor current of a three-phase synchronous machine 107, and has the following configuration.
Reference numeral 100 is an amplitude/phase commanding circuit block which commands the amplitude and phase of an electric current command value (in three-phase AC coordinate system) corresponding to the fundamental wave. Reference numeral 101 is an amplitude/phase commanding circuit block which commands the amplitude and phase of a harmonic current of a designated order (in three-phase AC coordinate system).
The amplitude/phase commanding circuit block 100 determines the amplitude and phase based on the current command (fundamental wave) received from an external control apparatus such as a vehicle control ECU or the like. This circuit block 100 may be incorporated in the vehicle control ECU. The external control apparatus computes the electric current command value as the fundamental wave, based on a torque command and a rotation angle signal (rotational position signal) related to the three-phase synchronous machine 107.
The circuit block 101 determines the frequency, amplitude, and phase of a harmonic current of a predesignated order by substituting the frequency, amplitude, and phase of the current specified by the current command (fundamental wave) into the equation (13), (14), or (15) and calculating the equation, and outputs an amplitude/phase command that implements them. The other constants in the above equations are preset according to the purpose.
For example, when reducing or canceling the sixth and 12th harmonics of the magnetic sound, the amplitudes and phases of the seventh and/or 13th harmonic currents are determined so that the calculated values of the equations (36) and (37) become lower than a predetermined value or equal to zero. The other constants are preset as numeric values unique to the AC rotary electric machine.
Similarly, when canceling only the sixth harmonic of the magnetic sound, the amplitude and phase of the harmonic current are determined so that the calculated value of the equation (36) becomes lower than a predetermined value or equal to zero. When canceling only the 12th harmonic of the magnetic sound, the amplitude and phase of the harmonic current are determined so that the calculated value of the equation (37) becomes lower than a predetermined value or equal to zero. In any case, by adjusting the phases and/or amplitudes of the seventh and/or 13th harmonic currents to be superimposed in the above equations, the sixth and/or 12th harmonics of the magnetic sound, i.e., most of the magnetic sound, can be amplified or reduced or canceled.
Instead of calculating the above equations, the frequency, phase, and amplitude of the fundamental frequency component may be entered in advance into a map or table corresponding to these equations, to provide the values of the phases and amplitudes of the seventh and/or 13th harmonic currents. Alternatively, based on the computed amplitude and phase of the fundamental frequency component of the stator current, the circuit block 100 may compute its present value for output, and the circuit block 101, based on the computed amplitude and phase of the harmonic current, may compute the present value of the harmonic current for output.
The commands concerning the fundamental current and the harmonic current are supplied to a circuit block 102. The circuit block 102 periodically computes a combined three-phase AC current value by combining on a phase-by-phase basis the fundamental current value and the harmonic current value determined for each phase based on the supplied information.
The thus computed combined three-phase AC current value is converted by a coordinate converting circuit block 103 into the d-q axis coordinate system; then, the converted values are compared in a subtractor 104 with their corresponding detection values (d-q axis) and, after their differences are gain-adjusted by a current amplifier 400, a coordinate converting circuit block 104A converts the values for output as a three-phase AC current value.
The circuit block 104A causes a circuit block 105 to generate a PWM control voltage for each phase for eliminating the above differences and, using the resulting three-phase PWM control voltage, performs on/off control of a switching device in a three-phase inverter 106, thereby performing control to apply the output voltage of the three-phase inverter 106 to the stator coil of the three-phase synchronous machine 107 as a generator-motor so that the three-phase AC current flowing in the three-phase synchronous machine 107 is the sum of the fundamental current and the harmonic current having the frequencies, amplitudes, and phases specified by the respective circuit blocks 100 and 101. This type of PWM feedback control is well known in the art, and will not be described in detail here.
The three-phase synchronous machine 107 includes a rotation angle sensor 108, and a speed/position signal processing circuit block 109 derives a speed signal and a position signal from the rotational position signal output from the rotation angle sensor 108 and supplies them to the circuit block 104A. On the other hand, the stator coil current of the three-phase synchronous machine 107 is detected by a current sensor 110, and converted by a coordinate converting circuit block 111 into a d-axis current detection value and a q-axis current detection value which are input to the subtractor 104.
CIRCUIT CONFIGURATION EXAMPLE 2
An example of the circuit for superimposing the above-described harmonic currents is shown in FIG. 4.
Reference numeral 100 is an amplitude/phase commanding circuit block which determines amplitude and phase as an electric current command value (in three-phase AC coordinate system) corresponding to the fundamental wave. The command value output from the circuit block 100 is fed to a subtractor 104a via a circuit block 300 which performs conversion from the three-phase AC coordinate system into the d-q axis coordinate system as in the first circuit configuration example. An FFT 111 extracts the detected value of the fundamental component (in three-phase coordinate system) from the phase current output from the current detector. The detected value is first converted by a circuit block 403 from the three-phase AC coordinate system into the d-q axis coordinate system, and then compared in the subtractor 104a with the electric current command value, and their difference is supplied to a coordinate converting circuit block 104B via a gain-adjusting current controller 401. The circuit block 104B outputs a three-phase AC current command value for eliminating the above difference to an adder 112.
Reference numeral 101 is an amplitude/phase commanding circuit block which determines amplitude and phase as an electric current command value (in three-phase AC coordinate system) corresponding to a harmonic of a designated order. The command value output from the circuit block 101 is fed to a subtractor 104b via a circuit block 301 which performs conversion from the three-phase AC coordinate system into the d-q axis coordinate system as in the first circuit configuration example. The FFT 111 extracts the detected value of the harmonic component of the designated order (in three-phase coordinate system) from the motor current. The detected value is first converted by a circuit block 404 from the three-phase AC coordinate system into the d-q axis coordinate system, and then compared in the subtractor 104b with the electric current command value, and their difference is supplied to a coordinate converting circuit block 104C via a gain-adjusting current controller 402. The circuit block 104C outputs a three-phase AC current command value, for eliminating the above difference, to the adder 112.
The circuit block 104C outputs the three-phase AC current command value for eliminating the above difference to the adder 112. The circuit block 109 derives a position signal and a speed signal from the detected rotational position signal, and supplies them to the coordinate converting circuit blocks 104B, 104C, 300, and 301.
The circuit block 105 is caused to generate a PWM control voltage for each phase corresponding to the combined three-phase Ac current command value output from the adder 112 and, using the resulting three-phase PWM control voltage, performs on/off control of the switching device in the three-phase inverter 106, thereby performing control to apply the output voltage of the three-phase inverter 106 to the stator coil of the three-phase synchronous machine 107 as a generator-motor so that the three-phase AC current flowing in the three-phase synchronous machine 107 is the sum of the fundamental current and the harmonic current having the frequencies, amplitudes, and phases specified by the respective circuit blocks 100 and 101.
CIRCUIT CONFIGURATION EXAMPLE 3
An example of the circuit for superimposing the above-described harmonic currents is shown in FIG. 5. The circuit shown here employs a filter 113 replacing the FFT 111 shown in FIG. 4, and extracts a fundamental current detection value and a harmonic current detection value.
From the phase current signal detected by the current sensor 110, the detected value of its fundamental component (in three-phase coordinate system) is extracted. The detected value is first converted by the circuit block 403 from the three-phase AC coordinate system into the d-q axis coordinate system, and then compared in the subtractor 104a with the fundamental electric current command value, and their difference is supplied to the coordinate converting circuit block 104B via the gain-adjusting current controller 401. The circuit block 104B outputs a three-phase AC current command value for eliminating the above difference to the adder 112.
A subtractor 117 subtracts the fundamental component (in three-phase coordinate system) from the phase current signal (in three-phase coordinate system) detected by the current sensor 110, and thus extracts its harmonic component. The detected harmonic component is first converted by the circuit block 404 from the three-phase AC coordinate system into the d-q axis coordinate system, and then compared in the subtractor 104b with the harmonic electric current command value, and their difference is supplied to the coordinate converting circuit block 104C via the gain-adjusting current controller 402. The circuit block 104C outputs a three-phase AC current command value for eliminating the above difference to the adder 112.
The circuit can thus operate in the same manner as that described in the second circuit configuration example.
CIRCUIT CONFIGURATION EXAMPLE 4
An example of the circuit for superimposing the above-described harmonic currents is shown in FIG. 6. The motor control circuit shown here is an embodiment that performs the feedback control of the motor current in the three-phase coordinate system alone.
Reference numeral 100 is an amplitude/phase commanding circuit block which determines amplitude and phase as an electric current command value (in three-phase AC coordinate system) corresponding to the fundamental wave. Reference numeral 101 is an amplitude/phase commanding circuit block which determines the amplitude and phase of a harmonic current of a designated order (in three-phase AC coordinate system). The functions of these circuit blocks are the same as those in FIG. 3, and the harmonic block 101 determines the amplitude and phase of the harmonic by substituting the frequency, phase, and amplitude output from the circuit block into the equation (13), (14), or (15), or by performing substantially the same computation using a map or a table.
The amplitude/phase commands output from the circuit blocks 100 and 101 are supplied to the circuit block 102. Based on the amplitude/phase command of the fundamental current command value and the amplitude/phase command of the harmonic current command value thus supplied and on the detected rotational position signal, the circuit block 102 combines the fundamental current command value (in three-phase AC coordinate system) and the harmonic current command value (in three-phase AC coordinate system) for each of the U and V phases, and outputs a U-phase combined current command value (in three-phase AC coordinate system) iu and a V-phase combined current command value (in three-phase AC coordinate system) iv.
A subtractor 300 obtains the difference between the detected U-phase current detection value iu′ and the U-phase combined current command value iu, and supplies the difference to a circuit block 302 which is a current controller. A subtractor 301 obtains the difference between the detected V-phase current detection value iv′ and the V-phase combined current command value iv, and supplies the difference to the circuit block 302, i.e., the current controller. The circuit block 302 creates a U-phase voltage and a V-phase voltage for eliminating the above differences, and a circuit block 105 computes and outputs a U-phase and a V-phase PWM voltage corresponding to the U-phase and the V-phase voltage, respectively. A subtracting inverting circuit 303 computes a W-phase voltage by inverting an analog signal representing the difference between the U-phase voltage and the V-phase voltage, and the circuit block 105 computes and outputs a PWM voltage for the W-phase voltage. The three-phase inverter 106 is controlled on and off in accordance with the duty-cycle corresponding to these three phase PWM voltages.
CIRCUIT CONFIGURATION EXAMPLE 5
An example of the circuit for superimposing the above-described harmonic currents is shown in FIG. 7. In this circuit, the circuit shown in FIG. 3 is changed to an open-loop control type.
The commands concerning the fundamental current and the harmonic current, respectively output from the fundamental circuit block 100 and the harmonic circuit block 101, are supplied to the circuit block 102. The circuit block 102 periodically computes a combined three-phase AC current value by combining, on a phase-by-phase basis, the fundamental current value and the harmonic current value determined for each phase based on the supplied information.
The thus computed combined three-phase AC current value is converted by the coordinate converting circuit block 103 into the d-q axis coordinate system and, after gain adjustment by the current amplifier 400, the coordinate converting circuit block 104A outputs the three-phase AC current value.
The circuit block 104A causes the circuit block 105 to generate a PWM control voltage for each phase and, using the resulting three-phase PWM control voltage, the switching device in the three-phase inverter 106 is controlled on and off, thereby performing control to apply the output voltage of the three-phase inverter 106 to the stator coil of the three-phase synchronous machine 107 as a generator-motor so that the three-phase AC current flowing in the three-phase synchronous machine 107 is the sum of the fundamental current and the harmonic current having the frequencies, amplitudes, and phases specified by the respective circuit blocks 100 and 101.
The three-phase synchronous machine 107 includes a rotation angle sensor 108, and the speed/position signal processing circuit block 109 derives a speed signal and a position signal from the rotational position signal output from the rotation angle sensor 108, and supplies them to the circuit blocks 103 and 104A for coordinate conversion.
EXPERIMENTAL EXAMPLE
An experiment for reduction of the magnetic noise was conducted using the three-phase synchronous machine (8-pole, 24-slot, IPM) shown in FIG. 8. The fundamental current was set to 43A, and the rotor phase angle was controlled to a value that maximizes torque.
FIG. 9 shows the three-phase current waveforms obtained when the motor was driven without superimposing the magnetic noise reducing harmonic currents on the stator current of the synchronous machine. Each phase current contains relatively small harmonic components in addition to its fundamental frequency component. FIG. 10 shows the three-phase current waveforms obtained when the motor was driven by superimposing the magnetic noise reducing harmonic currents on the stator current shown in FIG. 9. In FIGS. 9 and 10, the number of revolutions was 1700 rpm. The magnetic noise reducing harmonic currents were superimposed via a feedback control loop using the circuit of FIG. 3.
The frequency spectrum A of the U-phase current shown in FIG. 9 and the frequency spectrum B of the U-phase current shown in FIG. 10 with the magnetic noise reducing harmonic currents superimposed thereon, each spectrum being obtained by FFT, are shown in FIG. 11. In the pair of bars shown for each order in FIG. 11, the bar on the left (shown by cross hatching) indicates the spectrum A of the U-phase current shown in FIG. 9, and the bar on the right (shown by horizontal hatching) indicates the spectrum B of the U-phase current shown in FIG. 10. In FIG. 9, the seventh current (the seventh harmonic component) has an amplitude equal to 3% of that of the first current (the fundamental frequency component). In FIG. 10, the seventh current (the seventh harmonic component) has an amplitude equal to 12% of that of the first current (the fundamental frequency component).
FIG. 12 shows how the radially acting excitation force acting on three adjacent teeth of the stator varies with the rotation angle of the rotor when the currents shown in FIG. 9 are applied (hereinafter called the magnetic noise non-reduction mode) as compared to when the currents shown in FIG. 10 are applied (hereinafter called the magnetic noise reduction mode). Of the two periodically varying waveforms shown in FIG. 12, the larger periodically varying waveform C shows the variation of the radially acting excitation force in the magnetic noise non-reduction mode shown in FIG. 9, while the smaller periodically varying waveform D shows the variation of the radially acting excitation force in the magnetic noise non-reduction mode shown in FIG. 10. Further, E indicates the average value of the radially acting excitation force in the magnetic noise non-reduction mode. It will be noted that the radially acting excitation force contains a DC component due to the attraction being exerted by the permanent magnet of the rotor.
In FIG. 13, F indicates the frequency spectrum of the radially acting excitation force (the larger periodically varying waveform) C in the magnetic noise non-reduction mode, and G indicates the frequency spectrum of the radially acting excitation force (the smaller periodically varying waveform) D in the magnetic noise reduction mode. In the pair of bars shown for each order, the frequency spectrum F is shown by the bar on the left (indicated by cross hatching). On the other hand, the frequency spectrum G is shown by the bar on the right (indicated by horizontal hatching).
FIG. 14 shows the result of the measurement (in motor mode) taken of the sixth component (frequency 6θ) of the magnetic noise as a function of the number of revolutions (rpm) of the experimental machine described above. From FIG. 14, it can be seen that the sixth component of the magnetic noise can be effectively reduced in the high rpm range (higher than about 1500 rpm) where the magnetic noise is uncomfortable to the ear. FIG. 15 shows the result of the measurement (in generator mode) taken of the sixth component (frequency 6θ) of the magnetic noise as a function of the number of revolutions (rpm) of the experimental machine described above. From FIG. 15, it can be seen that the sixth component of the magnetic noise can be reduced over almost the entire rpm range. It is also shown that, in the rpm range where the magnetic noise is the greatest, the sixth magnetic noise component can be reduced as much as 20 dB when the magnetic noise reducing harmonic current is superimposed. In FIGS. 14 and 15, the sixth magnetic noise component increases in certain rpm ranges due to the superimposition of the magnetic noise reducing seventh harmonic current, but this is because the setting of the control constant is not optimized, and this is not an essential issue. Further, as this reversal of the magnetic noise occurs in the region where the level of the sixth magnetic noise component is small, and as its absolute value is small, this does not present a problem. In the above synchronous machine, it is of course possible to halt the operation of superimposing the magnetic noise reducing seventh harmonic current in the rpm range where a satisfactory reduction in the magnetic noise cannot be expected.
(Modified Mode)
Instead of performing the magnetic noise reduction operation, the operation mode can be switched, as needed, to a torque ripple reduction mode to reduce torque ripple by utilizing the harmonic superimposition technique implemented in the circuit of the embodiment described above, that is, by superimposing on the fundamental frequency component a harmonic current whose order is higher by one than that of the circumferential vibration. More specifically, this can be accomplished by altering the amplitude and phase angle of the harmonic current.
(Modified Mode)
A modified mode of each of the above circuit examples will be described below. The harmonic circuit block in each circuit example includes a temperature compensating circuit block for compensating for the temperature characteristics of the permanent magnet contained in the rotor. The temperature compensating circuit block reads the output of the temperature sensor built into the AC rotary electric machine 107, and changes the corresponding constant in the equation (35), (36), or (37) in order to compensate for the temperature-dependent variation of the residual flux of the permanent magnet.
A temperature compensation control operation routine for the case where the temperature compensating circuit block is implemented in a microcomputer will be described below with reference to the flowchart of FIG. 16. This routine is executed at regular intervals of time.
First, rotor temperature is read from the output signal of the temperature sensor (not shown) built into the AC rotary electric machine 107 (S100); then, a constant is obtained by entering this rotor temperature into a map which prestores the relationship between the rotor temperature and the constant (S102), and the thus obtained constant is output (S104). An existing temperature sensor mounted in a vehicle may be used instead of the temperature sensor built into the AC rotary electric machine 107.
The constant in the equation, which is to be changed as the rotor magnetic flux φ changes as a function of the temperature of the permanent magnet, is thus determined; therefore, using this constant, the equation is calculated to determine the phase and amplitude of the harmonic current.
(Modified Mode)
A modified mode of each of the above circuit examples will be described below. In this modified mode, the magnetic noise level is changed according to the operation mode of the AC rotary electric machine.
The routine for the case where the circuit block for changing the magnetic noise level is implemented in a microcomputer will be described below with reference to the flowchart of FIG. 17. This routine is executed at regular intervals of time.
First, the operation mode of the AC rotary electric machine 107, or the rotational speed of the AC rotary electric machine 107, which is supplied externally, is read (S200); then, a constant is obtained by entering this data (i.e., the operation mode of the AC rotary electric machine 107) into a map which prestores the relationship between the operation mode and the constant (S202), and the thus obtained constant is output (S204). Then, using the thus obtained constant, the equation is calculated to determine the amplitude and phase of the harmonic current of the designated order; in this way, the amplitude of the magnetic noise can be changed according to the operation mode of the AC rotary electric machine 107.
A low rpm operation mode and a high rpm operation mode can each be set as the above operation mode. In this case, the constant may be set so as to reduce primarily the 12th magnetic sound component in the low rpm operation mode and to reduce primarily the sixth magnetic sound component in the high rpm operation mode, or vice versa. That is, the order of the primary magnetic sound component to be reduced may be changed rather than changing the amplitude of the magnetic noise according to the operation mode. In the figures so far given, the circuit blocks, excluding the inverter 106, constitute the inverter control circuit.
(Modified Mode)
The equation (35), (36), or (37) shows an example of canceling or altering the harmonics of the rotor magnetomotive force Fmag by superimposing the harmonic currents on the fundamental stator current, but in actual PWM control of the AC rotary electric machine, harmonics are inevitably superimposed on the fundamental current due to inverter switching, resulting in the generation of magnetic sound. To additionally reduce this magnetic sound, computations should be performed so as to superimpose on the fundamental current the harmonic currents obtained by vector-subtracting the harmonic currents inevitably superimposed on the stator current due to inverter switching from the magnetic sound altering harmonic currents obtained by the above equations.
(Modified Mode)
Control operation for reducing the inherent magnetic noise of the three-phase AC rotary electric machine and producing magnetic sound of a designated frequency will be described with reference to the flowchart of FIG. 18.
First, a command is read from an external automotive electronic control unit (S300), and it is determined whether the command thus read is a command for generating a desired magnetic sound different from the inherent magnetic sound that the three-phase AC rotary electric machine is currently generating (S302); if the answer is yes, the previously described harmonic circuit block is operated. In this embodiment, the harmonic circuit block is a circuit block that superimposes the seventh harmonic current with a prescribed phase and amplitude. The phase and amplitude of the seventh harmonic current are set to such values that, when the harmonic current is superimposed, the sixth harmonic component of the radially acting inherent magnetic excitation force (magnetic sound) of the three-phase AC rotary electric machine is canceled in accordance with the above equation.
(Modified Mode)
Control operation will be described with reference to the flowchart of FIG. 19 for the case where the above-described magnetic noise reducing type rotary electric machine is mounted in an engine-powered vehicle. This type of rotary electric machine is well known and widely used as a generator-motor or an electric air conditioner or the like in a vehicle such as a hybrid vehicle or a torque-assisted vehicle.
First, in step S400, it is checked whether or not the engine is stopped and the electric machine is operating as a motor, or the electric machine is operating as a generator imposing regenerative braking; if the answer is yes, the harmonic circuit block is operated (S402), but if the answer is no, the harmonic circuit block is not operated (S404). The harmonic circuit block is a circuit block that superimposes the seventh harmonic current with a prescribed phase and amplitude. The phase and amplitude of the seventh harmonic current are set to such values that, when the harmonic current is superimposed, the sixth harmonic component of the radially acting inherent magnetic excitation force (magnetic sound) of the three-phase AC rotary electric machine is canceled in accordance with the above equation. In this way, noise inside the vehicle can be reduced by reducing the magnetic noise during motor operation (motor mode) or regenerative braking (generator mode) in an engine stopped condition in which engine noise is not generated and the magnetic noise of the generator-motor tends to become uncomfortable to the ear. In other driving conditions, the operation of the circuit for harmonic current superimposition control can be stopped.
CIRCUIT CONFIGURATION EXAMPLE 6
Another example of the harmonic current superimposition control will be described with reference to FIGS. 20 to 23. FIG. 20 is a block circuit diagram showing the motor control apparatus of this embodiment, FIG. 21 is a block circuit diagram showing one example of a coordinate conversion circuit 2, FIG. 22 is a waveform diagram showing signal waveforms (in a rotating coordinate system) in various circuit parts, and FIG. 23 is a waveform diagram showing signal waveforms (in a stationary coordinate system) in various circuit parts.
The motor control apparatus is an embodiment that performs feedback control of the motor current, wherein reference numeral 1 is a fundamental command value generating circuit, 2 is a harmonic command value generating circuit, 3 and 4 are adders, 5 and 6 are subtractors, 7 and 8 are PI amplifiers (proportional-integral circuits), 9 is a coordinate conversion circuit, 10 is a PWM voltage generating circuit, 11 is a three-phase inverter, 12 is a pair of current sensors (phase current detecting elements), 13 is a three-phase synchronous motor-generator (automotive synchronous AC rotary electric machine), 14 is a resolver (rotation angle detecting element), 15 is a position signal processing circuit, 16 is a delay compensating circuit, and 17 is a coordinate conversion circuit.
Of the above constituent elements 1 to 17, the constituent elements excluding the three-phase synchronous motor-generator 13 constitute the motor control apparatus according to the present invention, and of the above constituent elements 1 to 17, the constituent elements (circuits) excluding the current sensors (phase current detecting elements) 12, three-phase synchronous motor-generator (automotive synchronous AC rotary electric machine) 13, and resolver (rotation angle detecting element) constitute the motor current control element or motor control means according to the present invention.
Further, the constituent element (circuit) 17 constitutes the phase current detection value coordinate system converting element according to the present invention, the constituent element (circuit) 1 constitutes the fundamental command value output element according to the present invention, the constituent element (circuit) 2 constitutes the harmonic command value output element according to the present invention, the constituent elements (circuits) 3 to 6 constitute the current deviation computing element according to the present invention, and the constituent elements (circuits) 7 to 11 constitute the phase voltage control element according to the present invention. Needless to say, the three-phase inverter 11 is fed from a DC power supply and generates a three-phase AC voltage.
The fundamental command value generating circuit (fundamental command value output element) 1 is a known circuit for converting the target value of the fundamental current, corresponding to the input torque command value and rpm command value, into a d-axis fundamental command value Id1* as its d-axis current component and a q-axis fundamental command value Iq1* as its q-axis current component. The torque command value is input from an external control apparatus such as a vehicle control ECU, and the fundamental command value generating circuit 1, based on the input value, determines the d-axis fundamental command value Id1* and the q-axis fundamental command value Iq1*. If necessary for determining the d-axis fundamental command value Id1* and the q-axis fundamental command value Iq1*, values other than the torque command value, such as the voltage of the three-phase inverter 11, the output signal of the resolver 14, etc., are also input to the fundamental command value generating circuit 1.
The harmonic command value generating circuit 2 (harmonic command value output element) is a circuit for converting the predetermined target value of the (6k+1)th harmonic current (k is an integer and is 0 for the fundamental frequency component) into a d-axis harmonic command value Id6k+1* as its d-axis current component and a q-axis harmonic command value Iq6k+1* as its q-axis current component. More specifically, the harmonic command value generating circuit 2 is a circuit that generates a harmonic current command value for reducing the radial vibration of the three-phase synchronous motor-generator 13.
A specific example of the harmonic command value generating circuit 2 will be described with reference to the block diagram of FIG. 21. In FIG. 21, reference numeral 21 is a seventh current command value generating circuit, 22 is a 13th current command value generating circuit, 24 and 25 are coordinate conversion circuits, and 27 and 28 are adders. In this embodiment, the harmonic command value generating circuit 2 generates only the seventh and 13th harmonic command values for reducing the sixth and 12th radial vibration components, but the circuit may be configured to further generate higher harmonic command values, in a like manner, for superimposition in the adders 27 and 28.
The seventh current command value generating circuit 21 is a table that describes the relationships between the d-axis command value Id* and q-axis command value Iq* supplied from the fundamental command value generating circuit 1 and the amplitude I7* and phase angle β7* of the seventh harmonic command value for canceling the sixth radial vibration component. That is, the amplitude I7* and phase angle β7* of the seventh harmonic command value are values that vary as functions of the d-axis command value Id* and q-axis command value Iq* defined in the rotating coordinate system rotating at the fundamental frequency. Here, the amplitude I7* and phase angle β7* of the seventh harmonic command value are values defined in the rotating coordinate system rotating at the fundamental frequency, but the values are the same in the stationary coordinate system.
Likewise, the 13th current command value generating circuit 22 is a table that describes the relationships between the d-axis command value Id* and q-axis command value Iq* supplied from the fundamental command value generating circuit 1 and the amplitude I13* and phase angle β13* of the 13th harmonic command value for canceling the 12th radial vibration component. That is, the amplitude I13* and phase angle β13* of the 13th harmonic command value are values that vary as functions of the d-axis command value Id* and q-axis command value Iq* in the rotating coordinate system rotating at the fundamental frequency. Here, the amplitude I13* and phase angle β13* of the 13th harmonic command value are values defined in the rotating coordinate system rotating at the fundamental frequency, but the values are the same in the stationary coordinate system. These data I7*, β7*, I13*, and β13* are stored in ROMs (not shown) in the seventh and 13th current command value generating circuits 21 and 22. Of the data obtained by entering the d-axis command value Id* and q-axis command value Iq* into the circuits 21 and 22, I7* and β7* are supplied to the coordinate conversion circuit 24, and I13* and β13* are supplied to the coordinate conversion circuit 25.
Based on the amplitude I7* and phase angle β7* (relative to the phase angle θ of the fundamental component) of the seventh harmonic current supplied from the seventh current command value generating circuit 21, the coordinate conversion circuit 24 computes the d-axis harmonic command value Id7* as the d-axis component of the seventh harmonic current command value defined in the rotating coordinate system rotating at the fundamental frequency (also called the d-q axis coordinate system or fundamental dq coordinate system) and the q-axis harmonic command value Iq7* as the q-axis component thereof.
Likewise, based on the amplitude I13* and phase angle β13* (relative to the phase angle θ of the fundamental component) of the 13th harmonic current supplied from the 13th current command value generating circuit 22, the coordinate conversion circuit 25 computes the d-axis harmonic command value Id13* as the d-axis component of the 13th harmonic current command value defined in the rotating coordinate system rotating at the fundamental frequency (also called the d-q axis coordinate system or fundamental dq coordinate system) and the q-axis harmonic command value Iq13* as the q-axis component thereof. This computation is performed using the following equation (38).




  
    
      
        
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                        (
                        
                          
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        (
        38
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In the equation (38), θv is a phase-compensated rotation angle signal which is obtained by phase-compensating the motor rotation angle θ output from the delay compensating circuit (phase compensating circuit) 16 to be described later.
In the above explanation, the circuits 21 and 22 are each described as containing a table that defines the relationships between the d-axis command value Id* and q-axis command value Iq* and the amplitude and phase angle of the harmonic command value to be output, but instead, the detected rotation angle, voltage, and current and the amplitude and phase angle of the harmonic command value to be output may be stored in a table, and the amplitude and phase angle of the harmonic command value to be output may be computed by entering the detected values of the rotation angle, voltage, and current into this table.
Next, the seventh d-axis harmonic command value Id7* and the 13th d-axis harmonic command value Id13* are summed together by the adder 27 and output as the d-axis harmonic command value Id6n+1*, while the seventh q-axis harmonic command value Iq7* and the 13th q-axis harmonic command value Iq13* are summed together by the adder 28 and output as the q-axis harmonic command value Iq6n+1*. Of course, higher harmonic command values such as a 19th harmonic command value may be further generated and summed together in like manner in the adders 27 and 28.
The thus obtained d-axis harmonic command value Id6n+1* is summed with the d-axis fundamental command value Id1* in the adder 3 and output as the d-axis command value Id* and, likewise, the q-axis harmonic command value Iq6n+1* is summed with the q-axis fundamental command value Iq1* in the adder 4 and output as the q-axis command value Iq*. In this way, the noise canceling current command values can be determined by simple computation. That is, each harmonic command value is easy to compute because only a single frequency component need be computed as shown in the equation (38).
The position signal processing circuit 15 computes the rotation angle θ in the stationary coordinate system based on the rotation angle signal supplied from the resolver 14, and supplies it to the delay compensating circuit 16 and the coordinate conversion circuit 17. The delay compensating circuit 16, which is a phase compensating circuit, supplies the phase-compensated rotation angle θv to the coordinate conversion circuits 24 and 25 and the coordinate conversion circuit 9 described below, to compensate for computational delays, etc. occurring in these circuits. The coordinate conversion circuit 17 performs the coordinate conversion on the U-phase current Iu and V-phase current Iv detected by the current sensors 12, and outputs a d-axis detection value Id and a q-axis detection value Iq as the current detection values defined in the rotating coordinate system.
The subtractor 5 obtains an deviation ΔId by subtracting the d-axis detection value Id from the d-axis command value Id* obtained by the above computation, while the subtractor 6 obtains a deviation ΔIq by subtracting the q-axis detection value Iq from the q-axis command value Iq*. The PI amplifier 7 applies a PI (proportional-integral) amplification operation to the deviation ΔId in order to asymptotically reduce the deviation ΔId to zero, and outputs the corresponding d-axis voltage Vd, while the PI amplifier 8 applies a PI (proportional-integral) amplification operation to the deviation ΔIq in order to asymptotically reduce the deviation ΔIq to zero, and outputs the corresponding q-axis voltage Vq.
Using the supplied phase-compensated rotation angle θv, the coordinate conversion circuit 9 converts the voltages Vd and Vq into three phase voltages Vu, Vv, and Vw defined in the rotating coordinate system; then, the PWM voltage generating circuit 10 converts the three phase voltages Vu, Vv, and Vw into PWM signal voltages Uu, Uv, and Uw, and the three-phase inverter 11 creates a three-phase AC voltage by controlling its six built-in switching devices based on the input PWM signal voltages Uu, Uv, and Uw, and applies it to each phase terminal of the three-phase synchronous generator-motor 13. The motor control apparatus described above is the same as a conventional motor control system, except for the inclusion of the harmonic command value generating circuit 2 and, as this type of PWM feedback control is well known in the art, a detailed description will not be given here.
In this way, the sixth and 12th harmonics of the magnetic noise, which are major causes of the motor noise, can be greatly reduced.
Advantages of this embodiment will be described below.
First, according to this embodiment, as the constituent elements (circuits) 5 to 11 and 17 are configured to perform feedback control on both the fundamental command value and the harmonic command value simultaneously, the circuit configuration can be simplified. As a result, the coordinate conversion circuit 17 can perform the coordinate conversion directly on the detected three-phase AC currents Iu and Iv without first separating them into the fundamental frequency component and the harmonic frequency component; this can avoid phase delays associated with the filtering for the frequency separation and can thus prevent the degradation of the noise canceling effect due to computational errors, and control delays, that would otherwise occur.
Furthermore, as the table for the amplitude and phase angle of the harmonic command value is set so as to define their correspondence to the d-axis fundamental command value Id1* and the q-axis fundamental command value Iq1* which do not vary much, the frequency of reading can be reduced.
CIRCUIT CONFIGURATION EXAMPLE 7
Another embodiment will be described below with reference to FIG. 24. In this circuit configuration example, the d-axis harmonic command value Id6k1* and its q-axis current component, i.e., the q-axis harmonic command value Iq6k1*, generated by the harmonic command value generating circuit 2 in the sixth circuit configuration example of feedback type shown in FIG. 20, are respectively input to subtractors 20 and 21 where the respective command values are subtracted from the d-axis detection value Id and the q-axis detection value Iq generated by the coordinate conversion circuit 17 as the current detection values in the rotating coordinate system, and the results Id1 and Iq1 are output as the current detection values. The values Id1 and Iq1 generated by the above means are respectively input to the subtractors 5 and 6 for subtraction, to generate ΔId and ΔIq. The process thereafter is the same as that in the sixth circuit configuration example, and will not be further described here.
CIRCUIT CONFIGURATION EXAMPLE 8
Another embodiment will be described below with reference to FIG. 25. In this circuit configuration example, the current sensors 12 and the coordinate conversion circuit 17 are omitted from the sixth circuit configuration example of feedback type shown in FIG. 20, and the subtractors 5 and 6 and the PI amplifiers 7 and 8 are replaced by a current controller 18, thus changing the circuit configuration to an open-loop control type; in this example, the current controller 18 converts the d-axis command value Id* and the q-axis command value Iq* into the d-axis voltage command value and the q-axis voltage command value, each of prescribed magnitude, and the coordinate conversion circuit 9 converts these voltage command values in the rotating coordinate system into the voltage command values Vu, Vv, and Vw in the stationary coordinate system.
CIRCUIT CONFIGURATION EXAMPLE 9
Another embodiment will be described below with reference to the flowchart shown in FIG. 26.
In this circuit configuration example, the fundamental command value generating circuit 1 and the harmonic command value generating circuit 2 shown in FIG. 21 are implemented in microcomputer software, and the harmonic command value (defined in the rotating coordinate system) is superimposed as needed on the fundamental command value (defined in the rotating coordinate system) in accordance with a prescribed condition.
First, in step S100, the fundamental command value is computed, and then, based on a prescribed decision result computed separately, a decision is made as to whether the harmonic superimposition for radial vibration reduction should be performed or not (S102). If it is decided not to perform the harmonic superimposition, then in step S103 the d-axis command value Id* is set to the d-axis command value Id1* and the q-axis command value Iq* to the q-axis command value Iq1*, and the harmonic superimposition process described hereinafter is skipped. If it is decided to perform the harmonic superimposition, the process proceeds to step S104.
Here, the decision as to whether to perform or not to perform the harmonic superimposition is made, for example, by checking whether the number of revolutions is lower than a prescribed value; if the answer is yes, the harmonic superimposition is implemented, but if no, the harmonic superimposition is skipped. By so doing, noise can be reduced by performing the harmonic superimposition in the low rpm range where radial vibration noise becomes noticeable, while in other rpm regions, reductions in power consumption, etc. can be achieved. Further, maximum motor torque can be increased by prohibiting the harmonic superimposition when a torque greater than a prescribed value is required, for example, during engine cranking.
In step S104, the obtained fundamental command value (d-axis command value Id1* and q-axis command value Iq1*) is entered into the table to obtain the amplitude and phase angle of the seventh harmonic command value and, based on the amplitude and phase angle of the seventh harmonic command value thus obtained and the compensated rotation angle θv separately input, the seventh harmonic command value in the rotating coordinate system rotating at the frequency of the fundamental command value is obtained.
In steps S108 and S110, the 13th harmonic command value is obtained in a like manner, and further, in steps S112 and S114, the (6k+1)th harmonic command value is obtained in a like manner; then, in step S116, these harmonic command values are summed together to obtain the combined d-axis harmonic command value Id6n+1* and the combined q-axis harmonic command value Iq6n+1*.
Next, in step S118, the d-axis fundamental command value Id1* is added to the combined d-axis harmonic command value Id6n+1*, and the q-axis fundamental command value Iq1* is added to the combined q-axis harmonic command value Iq6n+1*, to obtain the d-axis command value Id* and the q-axis command value Iq*. The computations in steps S116 and S118 may be performed together, or may be performed using dedicated hardware. Further, the computations in the steps S104 and S106 and the steps S108 and S110 may be performed in parallel using dedicated hardware or may be performed using analog circuits.
(Modified Mode)
In the above explanation using the mathematical equations, the sixth and 12th magnetic sound components caused by the rotor magnetomotive force Fmag containing the third, fifth, and seventh harmonics have been canceled or reduced by superimposing the seventh and 13th harmonic current components. As a result of performing calculations similar to those described above, it has been found that, by superimposing a harmonic current of a given order, at least the magnetic noise whose order is lower by one than that given order, or the magnetic noise of a different order in addition to the magnetic noise whose order is lower by one than that given order, can be canceled or reduced.
For example, by performing computations similar to those described above, the 18th magnetic sound component can be canceled or reduced by superimposing the 19th harmonic current component, and the 24th magnetic sound component can be canceled or reduced by superimposing the 25th harmonic current component. In the equations previously given, only the 6th or the 12th magnetic sound component can be selectively canceled or reduced easily by setting the corresponding terms in the equations (26), (27), and (28) to zero. As a result, the dominant magnetic noise components in the three-phase rotary electric machine, i.e., the 6th, 12th, 18th, and 24th magnetic noise components, can be effectively reduced.
Further, in the above explanation using the mathematical equations, to simplify the theoretical explanation, only the fundamental frequency component has been assumed for the stator (armature) current, and the description has been given by assuming the case where the harmonic currents for reduction of the m-th and n-th magnetic noise components are superimposed on that fundamental frequency component. However, generally (that is, when the control of the embodiment is not performed) the actual stator current contains harmonics as does the rotor magnetomotive force; therefore, if the stator current contains, for example, the j-th, k-th, and L-th harmonics, the stator currents in the equations (16), (17), and (18) can be expressed as equations (39), (40), and (41) below. When these equations are expanded by using the previously given equations, it becomes apparent that excitation forces due to the originally contained harmonic currents are generated like the excitation forces generated by the harmonic components of the rotor magnetomotive force. It is therefore apparent that the magnetic sound can be reduced or canceled by determining, with the above point in mind, the orders, amplitudes, and phases of the harmonic currents to be superimposed in the present invention. (Expansion of these equations will not be given here as one can easily understand the above by inserting the values into the above equations.)
U-Phase Current

Icmd=I1 sin(θ+s)+Ij sin j(θ+β)+Ik sin k(θ+γ)+I1 sin l(θ+δ)  (39)

V-Phase Current

Icmd=I1 sin(θ+s−120)+Ij sin j(θ+β−120)+Ik sin k(θ+γ−120)+I1 sin l(θ+δ−120)  (40)

W-Phase Current

Icmd=I1 sin(θ+s−240)+Ij sin j(θ+β−240)+Ik sin k(θ+γ−240)+I1 sin l(θ+δ−240)  (41)

When the order of a harmonic current contained in the original armature current is higher by one than that of the harmonic component of the radial vibration to be reduced, it is advantageous to subtract that harmonic current contained in the original armature current from the magnetic noise reducing harmonic current computed by the above equations and to superimpose the result on the fundamental frequency component of the original armature current.
In the case of a polyphase AC rotary electric machine different than the three-phase AC rotary electric machine, if one modifies the above equations according to the change in the number of phases, one can easily understand that the notion that when a harmonic current of a given order is superimposed, the magnetic noise (magnetic sound) whose order is lower by one than that given order can be reduced will also hold for any polyphase AC rotary electric machine.
It will also be recognized that the embodiment can also be applied to a multi-slot configuration called a distributed winding (for example, two or more slots per pole per phase).
In the above explanation, the stator current Icoil has been described as being defined in the stationary coordinate system (angle θ), but it will be appreciated that the stator current Icoil can also be described using the rotating coordinate system (d, q axes).
Furthermore, instead of computing, using the above equations, the magnetic noise reducing harmonic current to be superimposed on the armature current, it is of course possible to prestore, in a table, the variable parameters used in the above equations in association with the amplitude and phase of the magnetic noise reducing harmonic current of a prescribed order and to determine the amplitude and phase of the magnetic noise reducing harmonic current by entering the variable parameters into the table. It will also be appreciated that the above process for magnetic noise reduction can be implemented not only in hardware but also in software.
While the preferred modes of the present invention have been described in detail above, it should be understood by those skilled in the art that various modifications and changes can be made by anyone skilled in the art, and that all of such modifications and changes that come within the range of the true spirit and purpose of the present invention fall within the scope of the present invention as defined by the appended claims.

## Claims

1. A magnetic noise reduction method for an AC rotary electric machine, wherein a radial vibration reducing harmonic current, whose frequency is n times (n is a natural number) the frequency of the fundamental frequency component of a polyphase AC current fed to an armature of a polyphase AC rotary electric machine, is input into said armature by superimposing the radial reducing harmonic current on said polyphase AC current, thereby attenuating an (n−1)th harmonic component, having a frequency (n−1) times the frequency of said fundamental frequency component, that is contained in a radial vibration occurring radially about an axial center of a rotating shaft of said AC rotary electric machine due to an excitation force generated by said AC rotary electric machine or externally applied to said AC rotary electric machine.

2. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 1, wherein a radial vibration reducing harmonic current, whose frequency is 6k+1 times (k is a natural number) the frequency of the fundamental frequency component of a stator current of a three-phase AC rotary electric machine as said AC rotary electric machine, is superimposed on said stator current, thereby reducing the radial vibration of said AC rotary electric machine that has a frequency 6k times the frequency of said fundamental frequency component, as compared to the case where said superimposition is not performed.

3. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 2 wherein, by superimposing said radial vibration reducing harmonic current of order 7, a sixth harmonic component of said radial vibration is attenuated as compared to the case where said superimposition is not performed.

4. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 2 wherein, by superimposing said radial vibration reducing harmonic current of order 13, a 12th harmonic component of said radial vibration is attenuated as compared to the case where said superimposition is not performed.

5. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 2 wherein, by superimposing said radial vibration reducing harmonic currents of orders 7 and 13, sixth and 12th harmonic components of said radial vibration are simultaneously attenuated as compared to the case where said superimposition is not performed.

6. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 2, wherein the phases and amplitudes of the radial vibration reducing seventh and 13th harmonic currents are set so that the amplitude of a vector sum, taken between the sixth and 12th harmonic components of said radial vibration occurring due to the radial vibration reducing fundamental, seventh, and 13th harmonic currents and the sixth and 12th harmonic components of said radial vibration induced in said AC rotary electric machine, becomes smaller than a predetermined value.

7. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 2 wherein, by superimposing said radial vibration reducing harmonic current of order 19, an 18th harmonic component of said radial vibration is attenuated as compared to the case where said superimposition is not performed.

8. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 2 wherein, by superimposing said radial vibration reducing harmonic current of order 25, a 24th harmonic component of said radial vibration is attenuated as compared to the case where said superimposition is not performed.

9. A magnetic noise reduction method for an AC rotary electric machine as set forth in claim 1 wherein, when I1 denotes the amplitude of said fundamental frequency, In denotes the amplitude of said harmonic current, and t, x, and y respectively denote prescribed phase angles,
the fundamental frequency component of said polyphase AC current contains at least a first phase fundamental frequency component Iu1 (=I1 sin (θ)), a second phase fundamental frequency component Iv1 (=I1 sin (θ−x)), and a third phase fundamental frequency component Iw1 (=I1 sin (θ−y)), and
said radial vibration reducing harmonic current whose frequency is n times contains at least a first phase harmonic component Iun (=In sin n(θ+t)), a second phase harmonic component Ivn (=In sin {n(θ+t)−x}), and a third phase harmonic component Iwn (=In sin {n(θ+t)−y}), and wherein
said first phase harmonic component Iun is superimposed on said first phase fundamental frequency component Iu1, said second phase harmonic component Ivn is superimposed on said second phase fundamental frequency component Iv1, and said third phase harmonic component Iwn is superimposed on said third phase fundamental frequency component Iw1.

10. A motor control apparatus comprising:
a rotational position detector which detects the rotational position of an automotive M-phase synchronous AC rotary electric machine (M is a positive integer not smaller than 3); and
a motor current controller which applies, based on said detected rotational position, a prescribed phase current having a prescribed fundamental frequency and amplitude to each individual phase winding of an armature of said AC rotary electric machine, wherein
said motor current controller applies a radial vibration reducing harmonic current having a frequency n times (n is a natural number) the frequency of the fundamental frequency component of said phase current to the each individual phase winding by superimposing said radial vibration reducing harmonic current on said phase current, and thereby attenuates an (n−1)th harmonic component, having a frequency (n−1) times the frequency of said fundamental frequency component, that is contained in a radial vibration occurring radially about an axial center of a rotating shaft of said AC rotary electric machine due to an excitation force generated by said AC rotary electric machine or externally applied to said AC rotary electric machine.

11. A motor control apparatus as set forth in claim 10, wherein said motor current controller superimposes at least either one of the seventh and 13th harmonic components on said phase current, and thereby attenuates the sixth or 12th harmonic component of said radial vibration occurring in a three-phase AC rotary electric machine as said AC rotary electric machine, as compared to the case where said superimposition is not performed.

12. A motor control apparatus as set forth in claim 11, wherein by superimposing said radial vibration reducing harmonic current of order 19, said motor current controller attenuates an 18th harmonic component of said radial vibration as compared to the case where said superimposition is not performed.

13. A motor control apparatus as set forth in claim 11, wherein by superimposing said radial vibration reducing harmonic current of order 25, said motor current controller attenuates a 24th harmonic component of said radial vibration as compared to the case where said superimposition is not performed.

14. A motor control apparatus as set forth in claim 10, wherein
said AC rotary electric machine is coupled to an automotive engine so as to be decouplable therefrom, and
said motor current controller superimposes said radial vibration reducing harmonic current on said polyphase AC current when said automotive engine is in a stopped condition.

15. A motor control apparatus as set forth in claim 10, wherein
said AC rotary electric machine is coupled to an automotive engine and/or to an axle so as to be capable of regenerative braking, and
said motor current controller superimposes said radial vibration reducing harmonic current on said polyphase AC current at least either when said automotive engine is in a stopped condition or when regenerative braking is being imposed.

16. A motor control apparatus as set forth in claim 10, wherein said motor current controller
obtains said radial vibration reducing harmonic current of order n that reduces or minimizes the radial vibration having a frequency (n−1) times the frequency of said fundamental frequency component and occurring due to the application of said phase current containing at least said fundamental frequency component and a harmonic current whose frequency is n times the frequency of said fundamental frequency component, and
superimposes said radial vibration reducing harmonic current on said phase current.

17. A motor control apparatus as set forth in claim 10, wherein said motor current controller
detects as an introduced harmonic current a harmonic current contained in said phase current and having a frequency n times the frequency of said fundamental frequency component,
obtains said radial vibration reducing harmonic current of order n as a feedback control target value for reducing or minimizing the (n−1)th harmonic component of said radial vibration for the case where said phase current does not contain said introduced harmonic current but contains only said fundamental frequency component, and
performs feedback control to asymptotically reduce the difference between said introduced harmonic current and said target value to zero.

18. An AC rotary electric machine apparatus comprising:
an AC rotary electric machine having M-phase stator coils (M is a positive integer not smaller than 3);
an inverter having a transistor device for controlling on and off a stator current for each phase of said synchronous machine; and
an inverter control circuit which controls said transistor device on and off based on a detected or estimated rotation angle of said AC rotary electric machine, wherein
said inverter control circuit includes harmonic PWM signal generator for performing processing to output to said inverter a harmonic PWM signal having a frequency n times (n is an integer not smaller than 2) the frequency of the fundamental frequency component of said stator current, and thereby a harmonic current having a frequency n times greater than the frequency of said fundamental frequency component is superimposed on said stator current and input into each of said rotor coils, and thus altering the amplitude of the magnetic sound of said AC rotary electric machine that has a frequency (n−1) times the frequency of said fundamental frequency component, as compared to the case where said processing is not performed.

19. An AC rotary electric machine apparatus as set forth in claim 18, wherein
said harmonic PWM signal generator performs processing to output to said inverter a first harmonic PWM signal having a frequency n1 times the frequency of the fundamental frequency component of said stator current and a second harmonic PWM signal having a frequency n2 times the frequency of said fundamental frequency component, and thereby increases (or reduces) the magnetic sound of said AC rotary electric machine that has a frequency (n1−1) times the frequency of said fundamental frequency component, as compared to the case where said processing is not performed, while at the same time, reducing (or increasing) the magnetic sound of said AC rotary electric machine that has a frequency (n2−1) times the frequency of said fundamental frequency component, as compared to the case where said processing is not performed.

20. An AC rotary electric machine apparatus as set forth in claim 18, wherein said inverter control circuit includes fundamental PWM signal generator for outputting to said inverter a fundamental PWM signal for feeding said fundamental frequency component to each of said stator coils.

21. An AC rotary electric machine apparatus as set forth in claim 18, wherein said harmonic PWM signal generator increases the noise generated by said AC rotary electric machine when said harmonic PWM signal generator is in an operating condition, as compared to when said harmonic PWM signal generator is in an stopped condition.

22. An AC rotary electric machine apparatus as set forth in claim 18, wherein said harmonic PWM signal generator reduces the magnetic sound of said AC rotary electric machine when said harmonic PWM signal generator is in an operating condition, as compared to when said harmonic PWM signal generator is in an stopped condition.

23. An AC rotary electric machine apparatus as set forth in claim 21, wherein said harmonic PWIVI signal generator increases or reduces said magnetic sound as compared to when said harmonic PWM signal generator is in an stopped condition, by altering the phase and/or amplitude of the harmonic current to be added to said stator current in accordance with the number of revolutions.

24. An AC rotary electric machine apparatus as set forth in claim 18, wherein said harmonic PWM signal generator performs processing to output to said inverter a harmonic PWM signal having a frequency 6k+1 times (k is a natural number) the frequency of the fundamental frequency component of said stator current, and thereby alters the magnetic sound of a three-phase AC rotary electric machine as said AC rotary electric machine that has a frequency 6k times the frequency of said fundamental frequency component, as compared to the case where said processing is not performed.

25. An AC rotary electric machine apparatus as set forth in claim 24, wherein said harmonic PWM signal generator outputs a harmonic PWM signal for superimposing a seventh harmonic current on the fundamental frequency component of said stator current, and thereby alters a sixth harmonic component of said magnetic sound as compared to the case where said superimposition is not implemented.

26. An AC rotary electric machine apparatus as set forth in claim 23, wherein said harmonic PWM signal generator outputs a harmonic PWM signal for superimposing a 13th harmonic current on the fundamental frequency component of said stator current, and thereby alters a 12th harmonic component of said magnetic sound as compared to the case where said superimposition is not implemented.

27. An AC rotary electric machine apparatus as set forth in claim 23, wherein said harmonic PWM signal generator outputs a harmonic PWM signal for superimposing seventh and 13th harmonic currents on the fundamental frequency component of said stator current, and thereby alters sixth and 12th harmonic components of said magnetic sound as compared to the case where said superimposition is not implemented.

28. An AC rotary electric machine apparatus as set forth in claim 23, wherein said harmonic PWM signal generator outputs a harmonic PWM signal for superimposing 19th and/or 25th harmonic currents on the fundamental frequency component of said stator current, and thereby alters 18th and/or 24th harmonic components of said magnetic sound as compared to the case where said superimposition is not implemented.

29. An AC rotary electric machine apparatus as set forth in claim 19, wherein
said n1 is 6k1+1 (k1 is a natural number),
said n2 is 6k2+1 (k2 is a natural number), and
said harmonic PWM signal generator, as compared to the case where said processing is not performed, reduces the magnetic sound of a three-phase AC rotary electric machine as said AC rotary electric machine that has a frequency 6k1 times the frequency of said fundamental frequency component and, at the same time, increases the magnetic sound of the three-phase AC rotary electric machine as said AC rotary electric machine that has a frequency 6k2 times the frequency of said fundamental frequency component.

30. An AC rotary electric machine apparatus as set forth in claim 29, wherein k1 is 1.

31. An AC rotary electric machine apparatus as set forth in claim 29, wherein k1 is 2.

32. An AC rotary electric machine apparatus as set forth in claim 29, wherein k1 is 3.

33. An AC rotary electric machine apparatus as set forth in claim 29, wherein k1 is 4.

34. An AC rotary electric machine apparatus as set forth in claim 24, wherein said harmonic PWM signal generator performs processing to output to said inverter a harmonic PWM signal having frequencies 7 and 13 times the frequency of the fundamental frequency component of said stator current and a harmonic PWM signal having at least either one of frequencies 19 and 25 times the frequency of said fundamental frequency component, and thereby attenuates the magnetic sound of said three-phase AC rotary electric machine as said AC rotary electric machine that has frequencies 6 and 12 times the frequency of said fundamental frequency component and the magnetic sound of said three-phase AC rotary electric machine as said AC rotary electric machine that has at least either one of frequencies 18 and 24 times the frequency of said fundamental frequency component, as compared to the case where said processing is not performed.

35. An AC rotary electric machine apparatus as set forth in claim 18, wherein when I1 denotes the amplitude of the fundamental frequency of said stator current, In denotes the amplitude of the harmonic current of said stator current, and t, x, and y respectively denote prescribed phase angles,
the fundamental frequency component of said stator current contains at least a first phase fundamental frequency component Iu1 (=I1 sin (θ)), a second phase fundamental frequency component Iv1 (=I1 sin (θ−x)), and a third phase fundamental frequency component Iw1 (=I1 sin (θ−y)), and
said harmonic current whose frequency is n times contains at least a first phase harmonic component Iun (=In sin n(θ+t)), a second phase harmonic component Ivn (=In sin {n(θ+t)−x}), and a third phase harmonic component Iwn (=In sin {n(θ+t)−y}), and wherein
said first phase harmonic component Iun is superimposed on said first phase fundamental frequency component Iu1, said second phase harmonic component Ivn is superimposed on said second phase fundamental frequency component Iv1, and said third phase harmonic component Iwn is superimposed on said third phase fundamental frequency component Iw1.

36. An AC rotary electric machine apparatus as set forth in claim 18, wherein
said AC rotary electric machine is coupled to an automotive engine so as to be decouplable therefrom, and
said inverter control circuit causes said magnetic sound to vary by performing processing to output said harmonic PWM signal to said inverter when said automotive engine is in a stopped condition.

37. An AC rotary electric machine apparatus as set forth in claim 18, wherein
said AC rotary electric machine is coupled to an automotive engine and/or to an axle so as to be capable of regenerative braking, and
said inverter control circuit causes said magnetic sound to vary by performing processing to output said harmonic PWM signal to said inverter at least either when said automotive engine is in a stopped condition or when regenerative braking is being imposed.

38. A motor control apparatus comprising:
a rotation angle detecting element which detects the rotational position of an automotive m-phase synchronous AC rotary electric machine (m is a positive integer not smaller than 3); and
a motor current control element which applies, based on said detected rotational position, a phase current having a prescribed fundamental frequency and amplitude to each individual phase winding of an armature of said AC rotary electric machine, wherein
said motor control apparatus further includes a phase current detecting element which detects the phase current flowing in each phase winding, and
said motor current control element comprises:
a phase current detection value coordinate system converting element which converts the detection value (in a stationary coordinate system) of each phase current into a d-q axis coordinate system (rotating coordinate system) rotating synchronously with the fundamental frequency component of said phase current specified by an output signal of said rotation angle sensor, and outputs a d-axis detection value Id and a q-axis detection value Iq;
a fundamental command value output element which outputs a d-axis fundamental command value Id1* and a q-axis fundamental command value Iq1* as d- and q-axis components in said d-q axis coordinate system defining a fundamental command value as a target value of the fundamental frequency component of said phase current;
a harmonic command value output element which outputs a d-axis harmonic command value Id6k+1* and a q-axis harmonic command value Iq6k+1* as d- and q-axis components in said d-q axis coordinate system defining a harmonic command value as a target value of a (6k+1)th harmonic current component (k is an integer and is 0 for said fundamental frequency component);
a current deviation computing element which computes a d-axis current deviation ΔId and a q-axis current deviation ΔIq by calculating (Id1*)+(Id6k+1*)−(Id) and (Iq*)+(Iq6k+1*)−(Iq), respectively; and
a phase voltage control element which asymptotically reduces said d-axis current deviation ΔId and said q-axis current deviation ΔIq toward zero by controlling the phase current to be applied to each phase winding, based on said d-axis current deviation ΔId and said q-axis current deviation ΔIq, and wherein
said harmonic command value output element superimposes said d-axis harmonic command value Id6k+1* and said q-axis harmonic command value Iq6k+1* on said d-axis fundamental command value Id* and said q-axis fundamental command value Iq*, respectively, and thereby attenuates a 6k-th harmonic component, having a frequency 6k times the frequency of said fundamental frequency component, that is contained in a radial vibration occurring radially about an axial center of a rotating shaft of said AC rotary electric machine due to an excitation force generated by said AC rotary electric machine or externally applied to said AC rotary electric machine.

39. A motor control apparatus comprising:
a rotation angle detecting element which detects the rotational position of an automotive m-phase synchronous AC rotary electric machine (m is a positive integer not smaller than 3); and
a motor current control element which applies, based on said detected rotational position, a phase current having a prescribed fundamental frequency and amplitude to each individual phase winding of an armature of said AC rotary electric machine, wherein
said motor current control element comprises:
a fundamental command value output element which outputs a d-axis fundamental command value Id1* and a q-axis fundamental command value Iq1* as d- and q-axis components in a d-q axis coordinate system defining a fundamental command value as a target value of the fundamental frequency component of said phase current;
a harmonic command value output element which outputs a d-axis harmonic command value Id6k+1 * and a q-axis harmonic command value Iq6k+1* as d- and q-axis components in said d-q axis coordinate system defining a harmonic command value as a target value of a (6k+1)th harmonic current component (k is an integer and is 0 for said fundamental frequency component);
a current command value computing element which computes a d-axis current command value Id* and a q-axis current command value Iq* by calculating (Id1*)+(Id6k+1*) and (Iq1*)+(Iq6k+1*), respectively; and
a phase voltage control element which supplies to each phase winding the phase current corresponding to said d-axis current command value Id* and said q-axis current command value Iq* by controlling the phase current to be applied to said phase winding, based on said d-axis current command value Id* and said q-axis current command value Iq*, and wherein
said harmonic command value output element superimposes said d-axis harmonic command value Id6k+1 * and said q-axis harmonic command value Iq6k+1* on said d-axis fundamental command value Id* and said q-axis fundamental command value Iq*, respectively, and thereby attenuates a 6k-th harmonic component, having a frequency 6k times the frequency of said fundamental frequency component, that is contained in a radial vibration occurring radially about an axial center of a rotating shaft of said AC rotary electric machine due to an excitation force generated by said AC rotary electric machine or externally applied to said AC rotary electric machine.

40. A motor control apparatus as set forth in claim 38, wherein said k is 1 and/or 2.