Patent Publication Number: US-9893665-B2

Title: Motor controller for position sensorless drives

Description:
CLAIM OF PRIORITY UNDER 35 U.S.C. § 120 
     This application claims the priority and benefit under 35 U.S.C. §120 of non-provisional application Ser. No. 13/761,041, entitled “PERMANENT MAGNET MOTOR WITH SINUSOIDAL BACK-EMF WAVEFORM AND RELATED MOTOR CONTROLLER FOR POSITION SENSORLESS DRIVES,” filed Feb. 6, 2013, issued as U.S. Pat. No. 9,143,066 on Sep. 22, 2015, the entirety of which is incorporated herein by reference. 
     TECHNICAL FIELD 
     This disclosure is generally directed to motors and motor controllers. More specifically, this disclosure is directed to a permanent magnet motor with a sinusoidal back-electromagnetic force (EMF) waveform and a related motor controller for position sensorless drives. 
     BACKGROUND 
     A permanent magnet motor represents a type of motor where a fixed stator causes rotation of a movable rotor. The rotor typically includes multiple magnets embedded in or connected to the rotor, and the stator typically includes multiple conductive windings. Electrical signals through the windings generate a rotating magnetic field that interacts with the magnets of the rotor, causing the rotor to rotate. 
     “Sensorless” motor control refers to an approach where one or more characteristics of a motor, such as motor speed or rotor position, are mathematically derived. Sensorless motor control typically avoids the use of separate speed and position sensors that are mechanically attached to a motor, which might otherwise detrimentally affect the performance of the motor (such as by affecting the maximum torque output per volume and drive system reliability). 
     SUMMARY 
     This disclosure provides a permanent magnet motor with a sinusoidal back-electromagnetic force (EMF) waveform and a related motor controller for position sensorless drives. 
     In a first example, a system includes a permanent magnet motor having a rotor and a stator. The rotor and the stator have a configuration that causes the motor to generate a back-electromagnetic force (EMF) waveform that is substantially sinusoidal. The system also includes a motor controller having a sliding-mode observer configured to identify the back-EMF waveform and a position observer configured to estimate at least one characteristic of the motor using the identified back-EMF waveform. 
     In a second example, an apparatus includes a permanent magnet motor having a rotor and a stator. The rotor includes multiple magnetic poles, and the stator includes multiple teeth projecting towards the rotor and multiple conductive windings. The rotor and the stator are configured so that the permanent magnet motor generates a back-electromagnetic force (EMF) waveform that is substantially sinusoidal. 
     In a third example, an apparatus includes a motor controller configured to generate signals for controlling operation of a permanent magnet motor. The motor controller includes a sliding-mode observer and a position observer. The sliding-mode observer is configured to identify a back-electromagnetic force (EMF) waveform associated with the permanent magnet motor. The position observer is configured to estimate at least one characteristic of the permanent magnet motor using the identified back-EMF waveform. 
     Other technical features may be readily apparent to one skilled in the art from the following figures, descriptions, and claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of this disclosure and its features, reference is now made to the following description, taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  illustrates an example system with a permanent magnet motor in accordance with this disclosure; 
         FIGS. 2A through 7B  illustrate example design details of a permanent magnet motor in accordance with this disclosure; 
         FIGS. 8 through 10  illustrate example design details of a motor controller for a permanent magnet motor in accordance with this disclosure; and 
         FIG. 11  illustrates an example method for sensorless control of a permanent magnet motor in accordance with this disclosure. 
     
    
    
     DETAILED DESCRIPTION 
       FIGS. 1 through 11 , discussed below, and the various examples used to describe the principles of the present disclosure in this patent document are by way of illustration only and should not be construed in any way to limit the scope of the disclosure. Those skilled in the art will understand that the principles of the present disclosure may be implemented in any suitable manner and in any type of suitably arranged device or system. 
       FIG. 1  illustrates an example system  100  with a permanent magnet motor in accordance with this disclosure. As shown in  FIG. 1 , the system  100  includes a power supply  102 , an inverter  104 , and a permanent magnet motor  106 . The power supply  102  represents a direct voltage (DC) power source that provides DC power to the inverter  104 . The power supply  102  includes any suitable structure for providing DC power, such as one or more batteries, fuel cells, solar cells, or other DC source(s). 
     The inverter  104  receives the DC power from the power source  102  and converts the DC power into an alternating voltage (AC) form. In this example, the inverter  104  represents a three-phase inverter that converts DC power into three-phase AC powers that are provided to the motor  106 . The inverter  104  includes any suitable structure for converting power from DC form to AC form. For example, the inverter  104  could include a number of transistor switches driven using pulse width modulation (PWM) signals. 
     The motor  106  represents a permanent magnet motor that operates using the voltages provided by the inverter  104 . As described in more detail below, the motor  106  includes a rotor with magnets embedded in or connected to the rotor. The motor  106  also includes a stator with multiple teeth around which conductive windings are wound. The windings are selectively energized and de-energized based on the signals from the inverter  104 , which creates a rotating magnetic field that causes the rotor to rotate. Also as described in more detail below, the motor  106  generates a back-electromagnetic force (EMF) waveform that is substantially more sinusoidal in form than conventional permanent magnet motors. 
     A motor controller  108  controls the operation of the inverter  104  to thereby control the operation of the motor  106 . For example, the motor controller  108  could generate PWM signals that drive the transistor switches in the inverter  104 . By controlling the duty cycles of the PWM signals, the motor controller  108  can control the three-phase voltages provided by the inverter  104  to the motor  106 . 
     In this example, the motor controller  108  receives as input a commanded speed signal ω e *, which identifies a desired speed of the motor  106 . The motor controller  108  may also receive as input feedback from one or more observers (such as those implemented within the motor controller  108 ), where the feedback identifies the estimated motor speed, rotor position, or other characteristic(s) of the motor  106 . The motor controller  108  uses the inputs to generate PWM signals for driving the transistor switches in the inverter  104 . 
     As described in more detail below, the motor controller  108  supports the use of sensorless field-oriented control. That is, the motor controller  108  does not receive sensor measurements from sensors mounted in or on the motor  106 . Rather, the motor controller  108  infers one or more characteristics of the motor  106 , such as motor speed or rotor position. Moreover, the motor controller  108  is designed to operate with the more sinusoidal back-EMF waveform that is generated by the motor  106 . To support these functions, the motor controller  108  uses a cascaded observer-based estimation algorithm to identify position and velocity estimates for the motor  106 , even in noisy environments. 
     In general, the performance of conventional sensorless motor drives is often limited at low speeds due to a low signal-to-noise (SNR) ratio of the back-EMF signals generated by the motors. This may not be acceptable in various applications, such as in industrial applications involving the use of electric bikes, robots, or variable-speed compressors. Designing the motor  106  to provide an improved sinusoidal back-EMF waveform and designing an appropriate control algorithm for the motor controller  108  can help to improve the low-speed operation of the motor  106  and provide better sensorless drive performance, even in very noisy environments. 
     Various benefits can be obtained using this approach, although the particular benefits depend on the specific implementation. For example, the motor controller  108  can achieve improved low-speed sensorless performance with a simpler hardware design. As described below, only two current sensors may be needed in the motor control algorithm, reducing the size and cost of using this approach. Moreover, this approach is compatible with other enhanced algorithms used in field-oriented control designs, such as initial position estimation using high-frequency signal injection (as described in U.S. Pat. No. 5,559,419). Additional details regarding the designs of the motor  106  and the motor controller  108  are provided below. 
     The components  102 - 108  here could reside within or otherwise form at least a part of any suitable larger system  110  that uses one or more motors. For example, the larger system  110  could represent a vehicle, electric scooter or bicycle, HVAC (heating, ventilation, and air conditioning) system, pump, actuator, compressor, robot, or optical disc drive of a computing device or home entertainment device. In general, any device or system that uses a motor that can operate (either temporarily or permanently) at a low speed could incorporate the designs of the motor  106  and the motor controller  108  described in this patent document. 
     Although  FIG. 1  illustrates one example of a system  100  with a permanent magnet motor, various changes may be made to  FIG. 1 . For example, various components in  FIG. 1  could be combined or further subdivided. As a particular example, one or more of the components  102 ,  104 ,  108  could be incorporated into the motor  106  itself. 
       FIGS. 2A through 7B  illustrate example design details of a permanent magnet motor in accordance with this disclosure. These design details could be incorporated into the permanent magnet motor  106  in the system  100  of  FIG. 1 . Of course, these design details could be incorporated into other permanent magnet motors operating in other systems. 
       FIGS. 2A and 2B  illustrate portions of a rotor  202  and a stator  204  in a permanent magnet motor. As shown here, the rotor  202  includes an outer ring or other area having alternating magnetic poles, such as poles  206 - 208 . For example, the poles  206  could represent magnetic “north” poles, and the poles  208  could represent magnetic “south” poles. These poles  206 - 208  can be created in the rotor  202  in any suitable manner. For instance, the poles  206 - 208  could be created using one or more magnetic structures embedded in or connected to the rotor  202 . As a particular example, the rotor  202  could include individual magnets, where each magnet forms a single one of the poles  206 - 208 . As another particular example, the rotor  202  could include a ring where individual sections are magnetized in different ways. Any other suitable structure(s) could be used to provide the alternating magnetic poles  206 - 208 . The rotor  202  could be formed from any suitable material(s) and in any suitable manner. 
     The stator  204  in this example includes multiple teeth  210  that are arranged around the rotor  202  and that project inward from an outer ring towards the rotor  202 . Although not shown here, conductive windings are placed around the teeth  210  of the stator  204 . These conductive windings are selectively energized and de-energized based on electrical signals, such as those from the inverter  104 . This creates a rotating magnetic field that interacts with the magnetic poles  206 - 208  of the rotor  202 , causing the rotor  202  to rotate. Each tooth  210  of the stator  204  can have any suitable size, shape, and dimensions. The stator  204  itself could also be formed from any suitable material(s) and in any suitable manner. 
     In a motor such as that shown in  FIGS. 2A and 2B , the geometric design of the rotor  202  and stator  204  can be used to create a back-EMF waveform that is dependent on the rotor&#39;s position and speed. This back-EMF waveform is caused by periodic changes of magnetic fluxes on the rotor  202  (where the magnetic fluxes are induced by the movement of the magnets that create the poles  206 - 208  in the rotor  202 ). As shown in  FIG. 2A , when the centerline of a stator tooth  210  is aligned with the centerline of a magnetic pole  206  or  208  of the rotor, the tooth  210  in the stator  204  can have the greatest magnetizing flux. As shown in  FIG. 2B , when the centerline of the stator tooth  210  is aligned with the inter-polar space between two magnetic poles  206 - 208  of the rotor, the tooth  210  in the stator  204  can have only a small amount of leakage magnetizing flux. Ideally, the spatial distribution of the magnetic flux is a nearly triangular waveform as shown in  FIG. 3A , which would result in a nearly trapezoidal back-EMF waveform as shown in  FIG. 3B . 
     Unfortunately, many conventional sensorless motor controllers perform position and velocity estimation calculations based on the assumption that the back-EMF waveform will be purely sinusoidal. As a result, these approaches may not be effective when used with permanent magnet motors, which typically have back-EMF waveforms that are far from sinusoidal. 
     This disclosure provides a permanent magnet motor design that utilizes the “flux leakage effect” to provide a better sinusoidal back-EMF waveform. As can be seen in  FIG. 2B , some magnetic flux enters or leaks into the stator teeth  210  when the centerlines of the stator teeth  210  are aligned with the inter-polar spaces between the rotor&#39;s magnetic poles  206 - 208 . This flux leakage is what leads to a change of magnetizing flux and the shape of the back-EMF waveform. By detecting the back-EMF waveform from current measurements, this type of motor is suitable for sensorless control. 
     A permanent magnet motor can therefore be designed as follows to provide an improved sinusoidal back-EMF waveform. First, some conventional permanent magnet motors use distributed conductive windings on the teeth  210  of a stator  204  as shown in  FIG. 4A . In this example, there are three windings  402 - 406 , and each winding  402 - 406  is wound around a pair of the stator teeth  210 . These conductive windings  402 - 406  are labeled “A,” “B,” and “C” to indicate that they are energized using different three-phase voltages from an inverter. 
     In contrast,  FIG. 4B  shows concentrated conductive windings (including windings  452 - 456 ), where each conductive winding  452 - 456  is wound around a single stator tooth  210 . Again, these conductive windings  452 - 456  are labeled “A,” “B,” and “C” to indicate that they are energized using different three-phase voltages from an inverter. The use of concentrated conductive windings (each wound around a single stator tooth) helps to increase the influence of flux leakage shown in  FIG. 2B . Thus, an improvement of the sinusoidal shape of the back-EMF waveform generated by the motor is achieved. 
     Second, some conventional permanent magnet motors have magnetic poles with large spans on their rotors. In  FIG. 4A , for example, there are only two magnetic poles on the rotor  202 , each spanning approximately 180°. The span of a magnetic pole is shown in  FIG. 5 . To help improve the sinusoidal shape of the back-EMF waveform generated by a motor, magnets  502  on a rotor (or the magnetic poles on the rotor) could span a smaller angle. For instance, in some motors, each magnet  502  (or magnetic pole) could span an angle of about 60° or less as illustrated in  FIG. 4B . As a result, a rotor could include six or more angularly-spaced magnetic poles, with each magnetic pole having an angular span of about 60° or less. 
     Third, the magnets or magnetic poles used in a rotor could be magnetized radially as shown in  FIG. 6A  or in parallel as shown in  FIG. 6B . Parallel magnetization can result in more flux leakage in a motor, which can lead to an improved sinusoidal back-EMF waveform. Thus, the use of parallel magnetization can be used to provide a more sinusoidal back-EMF waveform. 
     Using one or more of these design approaches can make the back-EMF waveform from a motor significantly more sinusoidal. An example of this is shown in  FIGS. 7A and 7B . In particular,  FIG. 7A  illustrates a back-EMF waveform  702  from a motor designed as shown in  FIG. 4A , which includes distributed windings and a 180° magnetic pole span.  FIG. 7B  illustrates a back-EMF waveform  704  from a motor designed as shown in  FIG. 4B , which includes concentrated windings and an approximately 60° magnetic pole span. As can be seen here, the back-EMF waveform  704  has a shape that is significantly more sinusoidal than the back-EMF waveform  702 . 
     Note that a perfect sinusoidal shape is not required in a back-EMF waveform. This is meant merely to show that an improved sinusoidal back-EMF waveform can be obtained using the design approaches described above. This can help lead to more accurate sensorless position and velocity estimation calculations. 
     Although  FIGS. 2A through 7B  illustrate examples of design details of a permanent magnet motor, various changes may be made to  FIGS. 2A through 7B . For example, a rotor  202  could include any number of magnetic poles  206 - 208 , and a stator  204  could include any number of teeth  210  and any number of concentrated windings  452 - 456 . Moreover, the signals shown here are for illustration only. Other motors having different design parameters may generate different back-EMF waveforms. 
       FIGS. 8 through 10  illustrate example design details of a motor controller for a permanent magnet motor in accordance with this disclosure. These design details could be incorporated into the motor controller  108  in the system  100  of  FIG. 1 . Of course, these design details could be incorporated into other motor controllers operating in other systems. 
     The motor controller here supports the use of field-oriented control, which generally includes controlling the voltages provided to a motor while representing those voltages with a vector. The motor, as a three-phase time-dependent and speed-dependent system, can be transformed via projection into a two-coordinate time-invariant synchronous system. The two coordinate axes are referred to as the d and q axes as illustrated in  FIG. 10 . The motor is controlled by generating i d   e * and i q   e * current commands for the d and q axes, respectively. The i d   e * current command is used to control the magnetizing flux of the motor, while the i q   e * current command is used to control the motor torque. These current commands are then converted to v d   e * and v q   e * voltage commands for the d and q axes, respectively. The v d   e * and v q   e * voltage commands define a voltage vector that is used to generate three-phase voltages for the motor. 
     As shown in  FIG. 8 , the motor controller includes a combiner  802 , which receives the commanded speed signal ω e * and an estimated speed signal {circumflex over (ω)} e . The estimated speed signal {circumflex over (ω)} e  represents feedback identifying an estimate of the motor&#39;s actual speed. The combiner  802  outputs a difference between these signals, which identifies the error between the commanded speed signal ω e * and the estimated speed signal {circumflex over (ω)} e . The combiner  802  includes any suitable structure for combining signals. 
     A speed controller  804  receives the output of the combiner  802 . The speed controller  804  uses the error identified by the combiner  802  to generate the current command i q   e * for the motor. The speed controller  804  includes any suitable structure for converting a speed error into a current command. 
     Another combiner  806  combines the current command i q   e * with a feedback signal i q   e , which represents a measurement of the actual current in the q axis. The combiner  806  generates an output identifying the difference or error between those signals. The combiner  806  includes any suitable structure for combining signals. 
     A current regulator  808  receives the output of the combiner  806 . The current regulator  808  uses the error identified by the combiner  806  to generate the voltage command v q   e * for the motor. The current regulator  808  includes any suitable structure for converting a current error into a voltage command. 
     A third combiner  810  combines the current command i d   e * with a feedback signal i d   e , which represents a measurement of the actual current in the d axis. The combiner  810  generates an output identifying the difference or error between those signals. The combiner  810  includes any suitable structure for combining signals. 
     A current regulator  812  receives the output of the combiner  810 . The current regulator  812  uses the error identified by the combiner  810  to generate the voltage command v d   e * for the motor. The current regulator  812  includes any suitable structure for converting a current error into a voltage command. 
     A dq/abc unit  814  receives the v d   e * and v q   e * voltage commands defining the voltage vector and converts the voltage vector into three-phase voltage signals v a , v b , and v c . These three-phase voltage signals define the voltages to be applied (to the “A,” “B,” and “C” windings of the stator) during the three phases of the motor  106 . Although not shown, the three-phase voltage signals v a , v b , and v c  can be converted into PWM signals for driving the transistor switches in the inverter  104 . The dq/abc unit  814  includes any suitable structure for converting a voltage vector into three-phase voltage signals. 
     Two phase current sensors  816   a - 816   b  capture measurements of the currents in two of the three-phase voltage signals. Among other things, the sensors  816   a - 816   b  capture information used for back-EMF estimation (which is then used to estimate the rotor position and speed). Each phase current sensor  816   a - 816   b  includes any suitable structure for measuring a current. 
     An abc/dq unit  818  receives the three-phase current measurements from the sensors  816   a - 816   b  and converts the measurements back into the d-q domain. In doing this, the abc/dq unit  818  generates the feedback signals i q   e  and i d   e , which represent the measurements of the actual currents in the q and d axes. The abc/dq unit  818  includes any suitable structure for converting current measurements associated with three-phase voltage signals into current measurements associated with d-q axes. 
     As noted above, sensorless motor control derives one or more characteristics of a motor, such as motor speed or rotor position, rather than measuring those characteristics directly. To support sensorless motor control here, the motor controller includes an EMF sliding-mode observer  820  and a position observer  822 . The sliding-mode observer  820  generally operates to estimate the back-EMF waveform of the motor, where both voltage and current information are used to obtain the back-EMF waveform. The sliding-mode observer  820  includes any suitable structure for identifying the back-EMF waveform of a motor. The position observer  822  generally operates to estimate the position of the motor&#39;s rotor, and the speed of the motor can then be estimated using the rate of change in the rotor&#39;s position. The position observer  822  includes any suitable structure for identifying the position of a rotor. 
       FIGS. 9A and 9B  illustrate example implementations of the sliding-mode observer  820  and the position observer  822 , respectively. With respect to  FIG. 9A , the use of a sliding-mode observer is beneficial since it can have better dynamic performance to reduce system noise and parameter variation, which makes it suitable for EMF estimation (particularly during low motor speeds). The sliding-mode observer  820  uses both voltage and current information to estimate the EMF waveform. 
     In some implementations, the dynamic equation of the sliding-mode observer  820  can be expressed as: 
                       d   dt     ⁢     (             i   α   s     ^                 i   β   s     ^                 e   α   s     ^                 e   β   s     ^           )       =         1       L   s     ^       ⁢     (           v   α     s   *                 v   β     s   *               0           0         )       -           R   s     ^         L   s     ^       ⁢     (             i   α   s     ^                 i   β   s     ^             0           0         )       -         K   sm         L   s     ^       ⁢     (           sign   ⁡     (     i     α_   ⁢   err       )                 sign   ⁡     (     i     β_   ⁢   err       )                   -     sign   ⁡     (       d   dt     ⁢     i     α_   ⁢   err         )         ×       L   s     ^                   -     sign   ⁡     (       d   dt     ⁢     i     β_   ⁢   err         )         ×       L   s     ^             )                 (   1   )               
Here and in  FIG. 9A , v α   s * and v β   s * represent the voltage commands, and i α   s  and i β   s  represent the measured currents. Also, i α   _   err  and i β   _   err  represent current errors calculated by two combiners  902  using the i α   s  and i β   s  measurements, and v α   _   err  and v β   _   err  represent voltage errors calculated by two sliding-mode controllers  904  using outputs of the combiners  902 . Further, î α   s , î β   s , ê α   s , and ê β   s  represent estimated currents and EMF voltages in a motor stator-referred stationary αβ frame. The î α   s  and î β   s  values can be calculated using a computation unit  906 , which can perform the calculations shown in  FIG. 9A  and provide the î α   s  and î β   s  values as feedback to the combiners  902 . In addition, the values {circumflex over (R)} s  and {circumflex over (L)} s  represent the estimated resistance and inductance of the motor, and K sm  represents the gain for the sliding-mode controllers  904 . Here, K sm  determines the estimation bandwidth of the sliding-mode observer  820  (which may be as high as possible).
 
     By manipulating the estimated currents î α   s  and î β   s  to be equal to the measured currents i α   s  and i β   s , the EMF voltage can be expressed as:
 
 ê   α   s   =K   sm sign( î   α   s )  (2)
 
 ê   β   s   =K   sm sign( î   β   s )  (3)
 
Note that ê α   s  and ê β   s  are discontinuous waveforms due to the sliding-mode control. As a result, low-pass filters (LPFs)  908  are used in the sliding-mode observer  820  to obtain continuous EMF vectors ê α   _   lpf   s  and ê β   _   lpf   s .
 
     With respect to  FIG. 9B , knowledge of the back-EMF waveform can be used to calculate the rotor position of a motor. In conventional sensorless motor control algorithms, the rotor&#39;s position is often calculated using an arc-tangent function, and the speed is estimated from the differentiation of rotor&#39;s position. This can be expressed as: 
                     θ   e     =     arc   ⁢           ⁢     tan   ⁡     (       -     e   α   s         e   β   s       )                 (   4   )                 ω   e     =       d   ⁢           ⁢     θ   e       dt             (   5   )               
The back-EMF waveform is dependent on the motor speed. However, at very low speeds, the arc-tangent calculation causes problems when the denominator in Equation (4) is near zero. This degrades conventional EMF-based sensorless drive performance because the back-EMF waveform is proportional to the motor speed.
 
     To improve the low speed performance, an observer-based estimation method can be used in place of the arc-tangent calculation in the position observer  822 . The position observer  822  in  FIG. 9B  receives |ê q   e | and ê d   e  signals, which can be defined as:
 
|{circumflex over ( e )} q   e |=√{square root over ({circumflex over ( e )} α   _   lpf   s     2     +ê   β   _   lpf   s     2   )}=ω e λ pm   (6)
 
 ê   d   e ≈ω e λ pm θ err   (7)
 
These values represent the estimated EMF voltages in the motor rotor-referred synchronous frame. The position of the motor can then be estimated based on the position error from the back-EMF voltage in the estimated dq reference frame as follows:
 
                           [             e   ^     d     r   ′                   e   ^     q     r   ′             ]     =       ⁢       [           cos   ⁢           ⁢       θ   ^     e             sin   ⁢           ⁢       θ   ^     e                   -   sin     ⁢           ⁢       θ   ^     e             cos   ⁢           ⁢       θ   ^     e             ]     ⁡     [             e   ^       α_   ⁢   1   ⁢   pf     s                 e   ^       β_   ⁢   1   ⁢   pf     s           ]                   =       ⁢       [           cos   ⁢           ⁢       θ   ^     e             sin   ⁢           ⁢       θ   ^     e                   -   sin     ⁢           ⁢       θ   ^     e             cos   ⁢           ⁢       θ   ^     e             ]     ⁡     [             -     ω   e       ⁢     λ   pm     ⁢   sin   ⁢           ⁢     θ   e                   ω   e     ⁢     λ   pm     ⁢   cos   ⁢           ⁢     θ   e             ]                   =       ⁢       [             -     ω   e       ⁢     λ   pm     ⁢   sin   ⁢           ⁢     θ   err                   ω   e     ⁢     λ   pm     ⁢   cos   ⁢           ⁢     θ   err             ]     ⁢     (       θ   err     =         θ   ^     e     ⁢     -     θ   ⁢           ⁢   e           )                     (   8   )               
It can be observed that these back-EMF waveforms contain spatial information.
 
     To implement this functionality, the position observer  822  in  FIG. 9B  includes a division unit  950 , which scales the ê d   e  signal using the |ê q   e | signal. The position observer  822  also implements a proportional-integral (PI) regulator using a proportional controller  952  and an integral controller  954 - 956 . A combiner  958  combines the outputs of the proportional controller  952  and the integral controller  954 - 956 . The output of the combiner  958  is provided to a scaling unit  960  and an integrator  962 , which generate the estimated velocity {circumflex over (ω)} m  and the estimated position {circumflex over (θ)} e  of the rotor. 
     Due to the use of a PI regulator, the position observer  822  has a low-pass filter property. As a result, high frequency noise (especially at low speeds) is reduced during the estimation. In  FIG. 9B , K p  and K i  represent the proportional and integral controller gains for the position observer  822 , and P represents the pole pairs of the motor. 
     The components shown in  FIGS. 9A and 9B  could be implemented using any suitable structure(s). For example, each of the components  902 - 908 ,  950 - 962  could be implemented using hardware modules. In other systems, various input signals could be digitized, and the components  902 - 908 ,  950 - 962  could be implemented using software or firmware instructions executed on a hardware platform. A combination of these approaches could also be used, where some of the components  902 - 908 ,  950 - 962  are implemented using only hardware and others of the components  902 - 908 ,  950 - 962  are implemented using software/firmware executed by hardware. 
     Although  FIGS. 8 through 9B  illustrate examples of design details of a motor controller for a permanent magnet motor, various changes may be made to  FIGS. 8 through 9B . For example, various components in  FIGS. 8 through 9B  could be combined or further subdivided. 
     Using the designs above for the motor  106  and the motor controller  108 , significant improvement can be obtained in the control of the permanent magnet motor  106 , particularly at low speeds. According to experimental results with particular implementations of the motor  106  and the motor controller  108 , a 2% rated speed under full-load operation can be achieved in a permanent magnet motor that provides a substantially sinusoidal back-EMF waveform. Note, however, that the design of the motor described above could be used without the design of the motor controller described above (and vice versa). 
       FIG. 11  illustrates an example method  1100  for sensorless control of a permanent magnet motor in accordance with this disclosure. As shown in  FIG. 11 , a permanent magnet motor is obtained at step  1102 . The permanent magnet motor can have the design as described above, including a stator with multiple teeth and concentrated conductive windings and a rotor with magnetic poles each spanning about 60° or less with parallel magnetization. Signals for operating the motor are generated at step  1104 . This could include, for example, the motor controller  108  generating three-phase signals for controlling the inverter  104 , which provides electrical signals to the conductive windings of the motor  106 . Because of the design of the motor  106 , the motor  106  generates a substantially sinusoidal back-EMF waveform during its operation at step  1106 . 
     During the operation of the permanent magnet motor, currents in the signals for operating the motor are measured at step  1108 . This could include, for example, using the phase current sensors  816   a - 816   b  to measure the currents in two of the three-phase signals. The back-EMF waveform of the motor is identified using a sliding-mode observer at step  1110 . This could include, for example, the sliding-mode observer  820  receiving the measured currents and voltage commands generated during operation of the motor  106 . This could also include the sliding-mode observer  820  generating continuous EMF vectors using those inputs. The position and velocity of the motor are estimated using the identified back-EMF waveform at step  1112 . This could include, for example, the position observer  822  identifying the estimated position and velocity of the rotor in the motor  106  using the continuous EMF vectors from the sliding-mode observer  820 . Feedback is generated for controlling the motor at step  1114 . This could include, for example, the motor controller  108  generating speed and current errors using the estimated velocity and the measured currents. The speed and current errors can be used to modify the signals for operating the motor. 
     Although  FIG. 11  illustrates one example of a method  1100  for sensorless control of a permanent magnet motor, various changes may be made to  FIG. 11 . For example, while shown as a series of steps, various steps in  FIG. 11  could overlap, occur in parallel, occur in a different order, or occur any number of times. As a particular example, steps for controlling the motor occur during operation of the motor and the generation of the substantially sinusoidal back-EMF waveform. 
     While this disclosure has described certain examples and generally associated methods, alterations and permutations of these examples and methods will be apparent to those skilled in the art. Accordingly, the above description of the examples does not define or constrain this disclosure. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this disclosure, as defined by the following claims.