Abstract:
The subject matter disclosed herein describes a system for controlling torque in a soft starter. In particular, torque ripple is reduced when transitioning between two different operating modes of a soft starter. A soft starter may include a first operating mode, designed for improved performance during low-speed operation of a motor, and a second operating mode, designed for improved performance during high-speed operation of the motor. However, transitioning between two different operating modes may result in significant transient currents in the motor, which, in turn, produce torque in the motor. The system described herein reduces this transient torque production in the motor.

Description:
BACKGROUND OF THE INVENTION 
     The subject matter disclosed herein relates to the control of electric motors. More specifically, the subject matter discloses an improved transition between operating modes in a soft starter. 
     Induction motors are widely used electrical machines. The most basic control of an induction motor utilizes a contactor to alternately connect and disconnect the induction motor to a fixed power supply. Contactor control provides a very inexpensive method for starting and stopping a motor. However, upon connection to the power supply, the motor will accelerate up to its rated speed in an unregulated manner, as quickly as possible, and drawing as much current as necessary from the power supply. 
     In contrast, high performance control of an induction machine is commonly achieved by using a variable frequency drive. A variable frequency drive can regulate the speed of and control the current in a motor. With the addition of a position sensing device, such as an encoder, the variable frequency drive can also regulate speed and current down to zero speed of the motor. Consequently, variable frequency drives permit controlled acceleration and deceleration of a motor, as well as operation at a wide range of operating speeds. However, variable frequency drives and encoders can add significantly to the complexity and expense of a motor control system. 
     Soft-starters provide yet another option for connecting motors to a power supply. A soft-starter provides a predefined speed profile for the motor during acceleration and deceleration, limiting the current drawn by the motor. While soft-starters typically do not provide the same level of control afforded by a variable frequency drive, they do reduce the wear on a motor and provide a simple, cost-effective means of connecting the motor to the power supply. 
     Electric soft-starters typically include a solid-state device connected in series between each phase of the power supply and the motor. The solid-state device is selectively turned on and off for a portion of the electrical cycle, controlling the voltage supplied to the motor. Many methods exist for controlling the solid-state devices in soft-starters. However, it has been found that different control methods perform better under different operating conditions. For example, control methods exist that result in improved operation at higher motor speeds while others provide better operation at lower motor speeds. 
     However, attempting to execute multiple control methods during a single operation of a motor in order to utilize the improved operation of each method at its preferred operating point is not without challenges. Switching between two control methods will result in step or very quick changes in commanded voltage, current, or speed to the motor. Typically significant levels of current and/or torque will result in the motor as the motor attempts to respond to the new control method. Consequently, it would be desirable to provide a soft-starter with an improved transition between operating modes, to facilitate operation of the soft starter in multiple operating modes. 
     BRIEF DESCRIPTION OF THE INVENTION 
     The subject matter disclosed herein describes a system for controlling torque in a soft starter. In particular, torque ripple is reduced when transitioning between two different operating modes of a soft starter. The soft starter includes a first operating mode, designed for improved performance during low-speed operation of a motor, and a second operating mode, designed for improved performance during high-speed operation of the motor. The soft starter described herein also includes a transitional operating mode to reduce the transient torque in the motor when switching between the first and second operating modes. 
     In one embodiment of the invention a method of controlling an alternating current (AC) voltage supplied to an AC motor is disclosed. The method includes determining a phase angle representing the current in the AC motor, determining a flux vector from at least one current signal and from at least one voltage signal, the flux vector having a magnitude and a phase angle representing the flux in the AC motor, and determining the polarity of the torque in the motor according to the current and flux phase angles. The conduction interval of an electronically controlled switching device used to connect the AC voltage to the AC motor is then set according to the polarity of the torque. 
     Thus it is one feature of the invention that the amount of time the motor is connected to the supply voltage during each electrical cycle may be adjusted in response to torque pulsations in the motor. 
     As one aspect of the invention, the current phase angle may be determined from at least one current signal, from at least one control signal to the switching device, or a combination thereof. Thus it is another feature of the invention that the angle of the current may be vector may be reliably determined if current signals are too small by estimating the phase angle according to firing events in the converter. 
     The method may also include the steps of determining a desired conduction interval according to either a magnitude of the current signal or a commanded speed to the motor, setting the conduction interval to a predetermined minimum value in response to a first polarity of the torque, and setting the conduction interval to the desired conduction interval in response to a second polarity of the torque. 
     In another aspect of the invention the commanded speed may be increasing. When the commanded speed is increasing, the conduction interval is set to the predetermined minimum value in response to a negative polarity of the torque, and the conduction interval is set to the desired conduction interval in response to a positive polarity of the torque. Alternately, the commanded speed may be decreasing. When the commanded speed is decreasing, the conduction interval is set to the predetermined minimum value in response to a positive polarity of the torque, and the conduction interval is set to the desired conduction interval in response to a negative polarity of the torque. 
     Thus it is another feature of the invention that the motor continues operating in the same manner it was operating during the first operating mode during the transition between operating modes. 
     The method may further include initial steps of controlling the AC motor according to a first operating mode and initiating a transition to a second operating mode. The step of initiating the transition to a second operating mode may be conditioned on reaching a predefined speed level. When the transition between operating modes is complete, the AC motor may be controlled according to a second operating mode. 
     According to another embodiment of the invention, a method of controlling an AC motor is disclosed. A program executes in a soft-starter in a first operating mode wherein an output voltage to the motor is generally proportional to a fundamental frequency of rotation of the motor. The program executes in the soft-starter in a second operating mode wherein an output voltage to the motor is generally responsive to an estimated speed of rotation of the motor. The program further executes a transitional operating mode between the first and the second operating modes wherein the output voltage to the motor is generally responsive to the polarity of the torque in the motor. 
     According to yet another embodiment of the invention, an AC power converter used to operate an AC motor includes an AC power input having at least one phase voltage, at least one electronically controlled switch selectively connecting each phase voltage to a phase of the motor according to a control signal, at least one current sensor providing a phase current signal from the motor, and a processor receiving the current signal and executing a stored program. The program may execute to estimate a flux in the motor, determine a polarity of the torque in the motor according to the current signal and the estimated flux, and set a conduction interval of the electronically controlled switching device according to the polarity of the torque. 
     These and other advantages and features of the invention will become apparent to those skilled in the art from the detailed description and the accompanying drawings. It should be understood, however, that the detailed description and accompanying drawings, while indicating preferred embodiments of the present invention, are given by way of illustration and not of limitation. Many changes and modifications may be made within the scope of the present invention without departing from the spirit thereof, and the invention includes all such modifications. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Various exemplary embodiments of the subject matter disclosed herein are illustrated in the accompanying drawings in which like reference numerals represent like parts throughout, and in which: 
         FIG. 1  is a block diagram representation of a power converter according to one embodiment of the present invention; 
         FIG. 2  is a schematic representation of a power converter according to one embodiment of the present invention; 
         FIG. 3  is a block diagram representation of the control system according to one embodiment of the present invention; 
         FIG. 4  is an example of a single cycle for one phase of input and output voltage waveforms through the power converter; 
         FIG. 5  is a flowchart illustrating a motor start-up sequence according to the control system of  FIG. 3 ; 
         FIG. 6  is a flowchart illustrating the sequence of operation during the transition mode of  FIG. 5 ; 
         FIG. 7  is a graph showing motor current and motor speed while starting the motor according to one embodiment of the present invention; and 
         FIG. 8  is an illustration of the phase relationship between a stationary, three-phase reference frame, a stationary, two-phase reference frame, and a rotating, two-phase reference frame. 
     
    
    
     In describing the various embodiments of the invention which are illustrated in the drawings, specific terminology will be resorted to for the sake of clarity. However, it is not intended that the invention be limited to the specific terms so selected and it is understood that each specific term includes all technical equivalents which operate in a similar manner to accomplish a similar purpose. For example, the word “connected,” “attached,” or terms similar thereto are often used. They are not limited to direct connection but include connection through other elements where such connection is recognized as being equivalent by those skilled in the art. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring initially to  FIG. 1 , a motor control system  10  generally includes an input power source  12 , a motor controller  16 , and a motor  20 . The input power source  12  is selected according to the type of motor  20  used and may be any suitable power source known to one skilled in the art. Preferably, the input power source  12  is a connection to a three-phase utility voltage. Input leads  14  connect the motor controller  16  to the input power source  12 . 
     The motor controller  16  typically includes a processor  28  and a memory storage device  29 . The processor  28  executes a program stored in memory  29  to provide at least one control signal  25  in response to input signals  23  and/or feedback signals  27 . An input voltage sensing section  22  may generate the input signals  23 , providing the processor  28  with magnitude and phase angle data corresponding to the input voltage. A feedback section  26  may generate feedback signals  27 , which may provide the processor  28  with current and/or voltage signals output to the motor  20 . The control signals  25  may be used to selectively energize switching devices, such as solid-state components, in the switch block  24  to regulate the current and/or voltage output to the motor  20 . Motor leads  18  connect the motor  20  to the motor controller  16 . 
     Referring next to  FIG. 2 , one embodiment of the motor control system  10  is illustrated. The input power source  12  is a three-phase, AC voltage. The power supply provides sinusoidally varying voltages, V a , V b , and V c , alternating, for example, at either 50 or 60 Hz to the motor controller  16  via input leads  14 . Optionally, other embodiments may include a power source  12  supplying voltage having a different number of phases or operating at a different frequency. 
     The input voltage sensing section  22  includes at least one voltage sensing device  30  and provides input signals  23  to the processor  28  representative of the input power source  12 . As illustrated, a separate voltage sensing device  30  may be connected to each input phase voltage, V a , V b , or V c . The voltage sensing device  30  may optionally measure either line-to-line voltage or line-to-ground voltage. 
     The feedback sensing section  26  may include a voltage feedback section  32 , a current feedback section  34 , or both as required by the control program executing on the processor  28 . The voltage feedback section  32  may include one or more voltage sensing devices  36  and provide voltage feedback signals  37  to the processor  28  representative of the voltage output to the motor  20 . As illustrated, a separate voltage sensing device  36  may be connected to each output phase voltage, V u , V v , or V w . The voltage sensing device  36  may optionally measure either line-to-line voltage or line-to-ground voltage. Similarly, a current sensing device  38  may be used to measure the output current at each motor lead  18 . The current sensing section  34  may include one or more current sensing devices  38  and provide current feedback signals  39  to the processor  28  representative of the output current drawn by the motor  20 . Optionally, one or more of the voltage feedback signals  37  and the current feedback signals  39  may be calculated based on a measured feedback signal  27 , a value calculated within the processor  28 , or a combination thereof. 
     The switch block  24  preferably includes electronically controlled switching devices to selectively connect the input leads  14  to the motor leads  18 . As illustrated, each phase includes a pair of silicon controlled rectifiers (SCRs)  40 . The first SCR  40  of the pair is connected in parallel with, and with an opposing polarity to, the other SCR in the pair. The pair of SCRs  40  is connected in series between the input leads  14  and the output leads  18 . Optionally, thyristors, power MOSFETs, IGBTs, or other solid-state switching devices, as would be known in the art, may be used. The trigger of each SCR is connected to and enabled by one of the control signals  25  generated by the processor  28 . 
     The motor  20  is connected to the motor controller  16  via the motor leads  18 . The three-phase induction motor, as shown, includes stator windings, U, V, and W. The motor controller  16  regulates at least one of the voltage or current output to the stator  19 , which, in turn, causes the rotor  21  to spin. 
     Referring then to  FIG. 3 , the processor  28  executes a control program to provide the control signals  25  to the SCR pairs  40 . The control program includes at least one, and preferably, includes three operating modes,  42 ,  44 , and  46 . A first operating mode  42  may be, for example, configured to operate at low speed and may be an open-loop algorithm, for example a volts-per-hertz like algorithm. An open-loop algorithm generates a voltage waveform to the motor according to a speed command without receiving a speed feedback signal. A volts-per-hertz algorithm varies the magnitude of the output voltage in a generally linear manner as the commanded output frequency, or speed, of the motor varies. The volts-per-hertz algorithm may vary the voltage proportionally to the output frequency or, optionally, the algorithm may vary the voltage at different rates over different ranges of output frequency. 
     A second operating mode  44 , for example, may be a closed-loop algorithm configured to operate at higher speeds. A closed-loop algorithm generates a voltage waveform to the motor according to an error signal generated by comparing a speed command to a speed feedback signal. The speed feedback signal may be provided by an encoder or other such hardware device, but preferably, the speed feedback signal is estimated within the processor  28  using voltage and current feedback signals,  37  and  39 . Optionally, the first and second operating modes,  42  and  44 , may be configured in any order and any suitable control method may be used during the first or second operating mode,  42  or  44 , as would be known in the art. 
     A transitional mode  46  is included to facilitate switching between the first and second operating modes,  42  and  44 . The transitional mode receives feedback signals  27  representing the voltage and current present at the motor. Preferably, these feedback signals  27  are initially in a three-phase reference frame, representing the voltage and current, for example, on the U, V, and W phases of the motor  20 . The three-phase voltage and current signals,  37  and  39 , are converted to two-phase signals by phase transformation blocks,  50  and  52 . The conversion may use the Park transformation, the Clarke transformation, or a derivative of either transformation, as is known in the art. The two-phase signals may be expressed as vectors having a magnitude and phase angle to represent the voltage and current present at the motor. The voltage and current vectors are then used as inputs to a flux calculator  54 . The estimated flux vector, output from the flux calculator  54 , and the current vector are used as inputs to a torque calculator  56 . The polarity of the torque, output from the torque calculator  56 , is provided as an input to the switch control function  58 . The switch control function  58  outputs the control signals  25  to the SCR pairs  40  when the transitional mode  46  is active. A mode select function  48  selectively enables one of the three operating modes,  42 ,  44 , or  46 , and passes the output of the selected mode,  42 ,  44 , or  46  as the control signals  25  to the SCRs  40 . 
     In operation, the motor controller  16  controls the motor  20  by selectively connecting the motor  20  to the input power source  12 . Referring to  FIGS. 2 and 4 , operation of a single phase of the motor controller  16  over one electrical cycle is illustrated. The SCR pairs  40  work together to enable conduction during each half cycle of the input voltage, one SCR selectively conducting during the positive half cycle  101 , and the other SCR selectively conducting during the negative half cycle  103 . Each SCR remains in a normally off state, represented by time intervals  102  and  106 , preventing conduction from the input power source  12 , represented by phase voltage V a , to the motor  20 , represented by phase voltage V u . A control signal  25  is used to turn on each SCR, for example at time  105  during the positive half cycle  101  and at time  107  during the negative half cycle  103 . Once enabled, the SCR will remain on until the voltage across the SCR reverses polarity, reverse biasing the SCR. As a result, once the SCR is turned on, it will continue conducting for the remainder of each half cycle. The shaded portions,  110  and  108 , indicate the time during which an SCR is conducting. 
     The electrical angle of the input voltage corresponding to the point at time,  105  or  107 , at which each SCR is switched on is also known as the conduction angle. The time interval,  104  or  108 , during which the SCR remains on is also known as the conduction interval. Although, the duration of the conduction intervals,  104  and  108 , for each of the positive and negative half cycles,  101  and  103 , are often the same, the duration of each conduction interval,  104  and  108 , is independently controlled and may vary from the positive to the negative half cycle,  101  or  103 . When full voltage is desired, each SCR is enabled at the start of the half cycle and conducts throughout the entire half cycle. Optionally, full conduction may be achieved by including a bypass contactor, not shown, in parallel to each SCR pair  40 , which can be energized to directly connect the input voltage, V a , to the output voltage, V u . 
     Referring next to  FIG. 5 , an exemplary start-up sequence  59 , bringing the motor from a stop up to a desired operating speed, is illustrated. The different states shown may be implemented in the mode select function  48  to selectively enable the operating modes,  42 ,  44 , or  46 , shown in  FIG. 3 . The motor controller  16  receives a start command  60  from an external source. The start command, at step  60 , may be provided by any suitable input, including but not limited to, a digital input, a serial command message, a networked command message, or by an operator interface connected to the motor controller  16 . In step  62 , the motor controller  16  begins generating control signals  25  to control the voltage supplied to the motor  20  according to a first operation mode  42 . The first operating mode  42  continues operation until a predetermined threshold, such as a time interval or commanded speed to the motor  20 , is reached, as illustrated by step  64 . Once the preset speed level is reached, the start-up sequence  59  then enters the slow speed transition mode  46 , as shown at step  66 . The slow speed transition mode  46  similarly continues operation until another predetermined threshold is reached. As shown by step  68 , the start-up sequence  59  continues operating in the transition mode  46  through two electrical cycles of the motor  20  at the preset motor speed command. The start-up sequence  59  then begins executing the normal, or second, starting mode  44  at step  70 . According to step  72 , the start-up sequence  59  continues executing according to the second operating mode  44  until the motor  20  has reached the desired operating speed. As shown in step  74 , the start-up sequence  59  is then complete. 
     As previously described, the SCR&#39;s selectively connect the input power source  12  to the motor  20  for a portion of each positive and negative half-cycle of the input voltage. Because the input power source is typically a utility grid, the input power source will operate at 50 or 60 Hz. Although operating modes may be configured to produce output voltages having fundamental output frequencies at various operating points, the voltage necessarily has an underlying 50 or 60 Hz component. The frequency components present in the output current are illustrated in  FIG. 7 . According to one embodiment of the present invention, the first operating mode  42  executes a volts-per-hertz like control algorithm between times T 0  and T 1 . The first operating mode  42  generates control signals  25  to the SCRs such that a sine-modulated current waveform results. The sine-modulated waveform has a fundamental frequency corresponding to a commanded output frequency of the motor along with a frequency component at the utility input of 50 or 60 Hz. 
     The second operating mode  44  employs a speed controlled algorithm after time T 2 . The second operating mode  44  similarly generates control signals  25  such that a sinusoidal current waveform results, having a primary frequency component at the utility input of 50 or 60 Hz. Due to the difference in control characteristics of each operating mode, a direct changeover in operating mode between the first and second operating modes,  42  and  44 , would result in a step change in commanded output voltage. A step change is an instantaneous, or very sharp, change in commanded voltage, which typically results in high currents, voltages, and/or resultant torques in the motor as the motor attempts to change operation from the first mode  42  to the second mode  44 . These high torques may produce sudden and undesirable changes in speed and deviations from the commanded speed. The transition mode  46 , operating between times T 1  and T 2 , reduces or eliminates the opportunity for high currents, voltage, and torques and produces a continued smooth speed output at the motor. 
     Referring next to  FIG. 6 , the steps of the slow speed transition mode  46  are illustrated. At step  76 , the feedback signals  27  are obtained. Voltage feedback signals  37  and current feedback signals  39  are obtained from each of the respective voltage sensors  36  and current sensors  38  present in the motor controller  16 . Alternately, voltage and current values for any phase not having a voltage or current sensor,  36  or  38 , may be calculated based on the acquired feedback signals  27  or on variables stored internally within the processor  28  and memory  29  of the controller  16 . 
     Having sampled or calculated each of the phase voltage and current signals,  37  and  39 , phase transformation functions,  50  and  52 , convert the sampled signals into a two-phase reference frame at step  78 , such as the direct and quadrature axis (d-q) reference frame. As illustrated in  FIG. 8 , the d-q reference frame may optionally be a stationary reference frame, identified by d 1 -q 1 , or a rotating reference frame, identified by d 2 -q 2 , that rotates at the electrical frequency of the motor  20 . Preferably, the phase transformation functions,  50  and  52 , convert the sampled signals into a stationary d-q reference frame. The two-phase signals may further be expressed as a vector, having a magnitude and a phase angle within the d-q reference frame. 
     At step  80 , the voltage and current values in the d-q reference frame may be used by the flux calculator  54  to determine a vector representation of the flux in the motor  20 . It is contemplated that a vector representation of the flux in either the stator  19  or the rotor  21  may be calculated. For example, the flux vector for the rotor  21  of the motor  20  is calculated according to equation 1. 
     
       
         
           
             
               
                 
                   
                     
                       K 
                       r 
                     
                     ⁢ 
                     
                       
                         
                           ψ 
                           → 
                         
                         ^ 
                       
                       r 
                     
                   
                   = 
                   
                     
                       
                         1 
                         s 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               V 
                               → 
                             
                             s 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             ⁢ 
                             
                               
                                 i 
                                 → 
                               
                               s 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       L 
                       ⁢ 
                       
                         
                           i 
                           → 
                         
                         s 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where: 
     K r =coupling factor; 
       {circumflex over (ψ)}   r =rotor flux vector; 
     {right arrow over (V)} s =stator voltage vector; 
     R s =stator resistance; 
     L=motor inductance; and 
     {right arrow over (i)} s =stator current vector. 
     Similarly, the flux vector for the stator  19  of the motor  20  is calculated according to equation 2. 
     
       
         
           
             
               
                 
                   
                     
                       ψ 
                       → 
                     
                     s 
                     ′ 
                   
                   = 
                   
                     
                       1 
                       
                         s 
                         + 
                         
                           ω 
                           c 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             V 
                             → 
                           
                           s 
                         
                         - 
                         
                           
                             R 
                             s 
                           
                           ⁢ 
                           
                             
                               i 
                               → 
                             
                             s 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where: 
     ω c =corner frequency; 
     {right arrow over (ψ)}′ s =stator flux vector; 
     As previously indicated, the flux calculator  54  obtains the voltage and current vectors used in equations 1 and 2 as input signals. The stator resistance and motor inductance are previously known or measured values stored in the memory device  29  and retrieved by the flux calculator  54  for determination of the flux vector. The flux vector output by the flux calculator  54  may be represented by a magnitude and angle. 
     After calculating the flux vector, the stator current angle is determined at step  82 . The stator current angle may be directly obtained from the current vector output by the phase transformation  52 . Optionally, the stator current angle may also be estimated according to the upcoming firing event for the SCR pairs  40 . 
     The value of the estimated stator current angle is preferably fixed between each SCR firing event. As previously discussed, the length of time during any particular half cycle that an SCR connects the input source  12  to the motor  20  varies according to the desired output voltage; however, the sequence in which the SCRs are enabled remains constant and is determined according to the phase angle and the phase sequence of the input power  14 . Preferably, the stator current angle is estimated to be one of six angles, spaced at sixty degree intervals, based on the upcoming SCR firing event, as illustrated in table 1 below. 
     
       
         
               
             
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Current Angle Estimation 
               
             
          
           
               
                   
                 Input Power 
                 Input Power 
               
               
                   
                 Positive Sequence (ABC) 
                 Negative Sequence (CBA) 
               
             
          
           
               
                 SCR firing 
                 Conducting 
                 Estimated 
                 Conducting 
                 Estimated 
               
               
                 event 
                 SCR pairs 
                 current angle 
                 SCR pairs 
                 current angle 
               
               
                   
               
             
          
           
               
                 A+ 
                 A+ B− 
                 60° 
                 A+ C− 
                 120° 
               
               
                 A− 
                 A− B− 
                 −120° 
                 A− C+ 
                 −60° 
               
               
                 B+ 
                 B+ C− 
                 180° 
                 B+ A− 
                 −120° 
               
               
                 B− 
                 B− C+ 
                 0° 
                 B− A+ 
                 60° 
               
               
                 C+ 
                 C+ A− 
                 −60° 
                 C+ B− 
                 0° 
               
               
                 C− 
                 C− A+ 
                 120° 
                 C− B+ 
                 180° 
               
               
                   
               
             
          
         
       
     
     At step  86 , the polarity of the torque is determined. The previously determined stator current angle and the flux vector, output by the flux calculator  54 , are provided as inputs to the torque calculator  56 . The torque calculator  56  may determine the value of the torque produced in the motor  20 , for example, by equation 3, given below. As may be observed in equation 3, the value of the angle, θ, determines the sign, or the polarity, of the torque in the motor  20 . 
     
       
         
           
             
               
                 
                   
                     T 
                     e 
                   
                   = 
                   
                     
                       
                         1 
                         
                           K 
                           r 
                         
                       
                       ⁢ 
                       
                         
                           
                             ψ 
                             → 
                           
                           r 
                         
                         ⊗ 
                         
                           
                             i 
                             → 
                           
                           s 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           K 
                           r 
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             ψ 
                             → 
                           
                           r 
                         
                          
                       
                       * 
                       
                          
                         
                           
                             i 
                             → 
                           
                           s 
                         
                          
                       
                       * 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where: 
     T e =electromagnetic torque produced by the motor; and 
     θ=phase difference between the phase angle of the rotor flux vector and the phase angle of the stator current vector. 
     Although equation 3 illustrates the torque being calculated using the rotor flux vector, the torque may similarly be calculated using the stator flux vector. 
     Having identified the polarity of the torque, the motor controller  16  then determines, in steps  88 - 92 , the length of the conduction interval for the next electrical cycle. First, the motor controller identifies the polarity of the torque. If the torque is positive, the conduction interval or firing angle is set to a first value. If the torque is negative, the conduction interval or firing angle is set to a second, smaller value. The first and second values may be selected according to predetermined constant values. Optionally, the first and second values may be selected according to previously calculated values, for example the speed reference or a measured current value at the end of the first operating mode  42 . The first value may be held at the previously calculated value and the second value may be a percentage of the previously calculated value. Reducing the voltage supplied to the motor  20  during time intervals where negative torque would be produced by the motor  20  allows the motor to continue increasing in speed while the transitional mode  46  is executing. The transitional mode  46  continues executing until the predefined set point is reached, as shown in step  94 . 
     The steps described above and illustrated by  FIGS. 5 and 6  for an exemplary start-up sequence  59 , may similarly be applied to a stopping sequence where the motor is slowed to a stop from a desired operating speed. During a stopping sequence, the process would transition from the second operating mode  44  to the first operating mode  42  using the transition mode  46 . Rather than limiting negative torque to the motor as down during a start-up sequence  59 , the process would limit positive torque to the motor. Just as during the start-up sequence, first and second values of the desired conduction interval or firing angle during the next electrical cycle are determined. However, in contrast to the start-up sequence, the first value, corresponding to a positive torque, is less than the second value, corresponding to a negative torque. The first and second values may be selected according to predetermined constant values. Optionally, the first and second values may be selected according to previously calculated values, for example the speed reference or a measured current value at the end of the second operating mode  44 . The second value may be held at the previously calculated value and the first value may be a percentage of the previously calculated value. Reducing the voltage supplied to the motor  20  during time intervals where positive torque would be produced by the motor  20  allows the motor to continue decreasing in speed while the transitional mode  46  is executing. 
     It should be understood that the invention is not limited in its application to the details of construction and arrangements of the components set forth herein. The invention is capable of other embodiments and of being practiced or carried out in various ways. Variations and modifications of the foregoing are within the scope of the present invention. It also being understood that the invention disclosed and defined herein extends to all alternative combinations of two or more of the individual features mentioned or evident from the text and/or drawings. All of these different combinations constitute various alternative aspects of the present invention. The embodiments described herein explain the best modes known for practicing the invention and will enable others skilled in the art to utilize the invention.