Patent Publication Number: US-7714529-B2

Title: Permanent magnet synchronous motor and controller therefor

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application is based on and claims priority under 35 U.S.C. §119 with respect to Great Britain Patent Application 0410037.6, filed on May 5, 2004, the entire content of which is incorporated herein by reference. 
   FIELD OF THE INVENTION 
   The present invention relates to a permanent magnet synchronous motor having a controller. The present invention also relates to a controller for such a motor. 
   BACKGROUND 
   The sinusoidal control of AC machines is commonly achieved by using the well-known vector control (also called field-orientated control) theory. It has been widely used for variable speed control of 3-phase asynchronous induction motors and synchronous PM AC machines. It delivers sinusoidal control of 3-phase current of AC motors with low torque ripple. Due to the involvement of extensive computation, a high-speed micro-controller, DSP or microprocessor (MCU) is usually required. Accurate rotor position information is essential for achieving high-performance vector control. However, high-resolution position sensors are usually fragile, unreliable and expensive. Therefore the development of position-sensorless sinusoidal control is driven by eliminating the cost and reliability problem caused by high-resolution position sensors. Position sensorless AC control is preferred for many industrial applications. Unfortunately accurate position determination is extremely difficult at start-up and very low speed due to the lack of position-dependent signals. Also, a high-performance MCU is required for sensorless methods due to the implementation of complex algorithms as well. 
   Low-cost switching-type Hall sensors are commonly employed to generate the commutation signals for controlling brushless DC motors in conventional six-step square-wave drives. They have also been proposed for a low-cost sine-wave drive. However, due to their rough position measurement, smooth torque control for PMSM at very low speed is not achievable. 
   Alternatively linear Hall sensors can provide more accurate position measurement continuously. Linear Hall sensors are widely utilised for measuring magnetic field strength. U.S. Pat. No. 6,522,130 proposes mounting a ring magnet on a rotor with two linear Hall sensor fixed on the stator with a phase displacement with 90 electrical degrees. The sine/cosine signal outputs from the two Hall sensors can be directly used to decode rotor position by using the well-known resolver-to-digital (RD) conversion as the process for resolver. Therefore, accurate and low-cost position determination is achievable if the magnetic flux distribution of ring magnet is sinusoidal. As a result, a sine-wave drive based on vector control theory can be implemented with a relatively low-cost solution. 
   All the sine-wave control strategies mentioned above can achieve the rapid and smooth torque control of AC machines by using a high-performance MCU. They are suitable for the high-performance servo systems. However, for some industrial applications, strictly rapid torque response is not necessary and only the torque smoothness of PMSM motor is the main concern. Furthermore the cost for high-performance MCU is intolerable in these applications. Therefore a sine-wave drive with even less cost is preferred. 
   A low-cost open-loop sine-wave solution for AC induction asynchronous motors can be implemented by well-known V/F (Voltage/Frequency) control without requiring position sensors and current sensor as well. Only low-end MCU or no MCU is required in these low-cost low-performance AC drives. The rotor magnetic field of the induction motors is induced due to the slip of rotor frequency and stator current synchronous frequency. The torque is generated by the interaction of the induced field in rotor and stator control flux. However, PMSM machines have no such feature. The rotor position of a PMSM has to be referred to for proper control of 3-phase stator currents in order to synchronise with the rotating magnetic field on the rotor of PMSM. Therefore simply applying the conventional V/F signal into the stator winding of PMSM is unable to guarantee the proper operation of a PMSM. 
   Emura T., Wang L., Yamanaka M., and Nakamura H., “A High-Precision Positioning Servo Controller Based on Phase/Frequency Detecting Technique of Two-Phase-Type PLL”,  IEEE Transactions on Industrial Applications , VOL. 47, NO. 6, pp 1298-1306, December 2000 propose the use of a Two-Phase-Type PLL in the high precision servo control of motor used in a gear grinding machine. 
   The present invention seeks to provide a controller for a PMSM which can be implemented at low cost and operate as an open loop controller (or in a closed loop control system). 
   SUMMARY OF THE INVENTION 
   According to one aspect of the present invention, there is provided a permanent magnet synchronous motor having a stator and a rotor, the stator having windings and the rotor comprising a plurality of permanent magnets and a controller, wherein the controller comprises: 
   two magnetic field sensors, each having a linear response to sensed magnetic field strength, the sensors being spaced by an angle A electrical degrees relative to the rotor, where angle A is greater than 0 degrees and less than 180 degrees or greater than 180 degrees and less then 360 degrees, for sensing the position of the rotor by sensing a magnetic field representative of the fields of the magnets of the rotor; 
   means for producing first and second signals representing normalised orthogonal components of the outputs of the sensors; and 
   energising means for producing, from the said orthogonal components, sinusoidal currents for energising the windings of the stator to drive the rotor. 
   The magnetic field sensors are preferably Hall effect sensors but other suitable sensors may be used. The following description refers to Hall sensors by way of example. 
   An embodiment of the invention relates to the sinusoidal control of a 3-phase permanent magnet synchronous AC motor (PMSM) based on the sine/cosine measurements of two linear Hall sensors, particularly regarding a low-cost open-loop PMSM sine-wave drive with low-torque ripple for PMSM motors without the phase current detection. 
   In a currently preferred embodiment the controller is a low-cost open-loop sinusoidal controller. It comprises that two linear Hall sensors fixed on the motor stator with an optimal phase displacement in 90 electric degree to detect the rotor magnetic field. An extra ring magnet with sinusoidal flux distribution and the same pole number as the controlled PMSM rotor magnet is fixed on the rotor to excite the Hall sensors. A two-phase-type phase-lock-loop (TP-PLL) is employed to normalise the Hall sensor sine/cosine signals. The normalised sine/cosine signals are subsequently transformed into 3-phase synchronous sinusoidal voltage waveforms as the inputs to pulse-width-modulator (PWM). The amplitude of the 3-phase sine-waves is normalised with a fixed voltage level. The amplitude of the PWM carrier signal is made adjustable to control the PWM average output in order to limit the phase peak current, peak output torque and peak speed. Finally 3-phase PWM output logic signals are fed into a conventional 3-phase inverter which provides power amplification. No current measurement is involved in this simple open-loop control. 
   The excitation ring magnet is mounted on the rotor of PMSM. It has exactly the same number of poles as the rotor magnet, and it has a sinusoidal-flux distribution. Its magnetic field is preferably aligned with the PMSM rotor magnetic field in order to simply the system design. Two linear Hall sensors are mounted on the stator to detect the magnetic field of the ring magnet. The Hall sensors preferably have a 90-electrical-degree displacement angle. The two Hall sensors should be mounted at a distance from the excitation ring which avoids saturation of the Hall sensor outputs. The outputs of these two Hall sensors can be defined as:
 
Hall A   =G   A ( t )sin θ Hall B   =G   B ( t )cos(θ+Δθ)  (1-a, 1-b)
 
   Ideally the output gain G A (t) and G B (t) are expected to be G A (t)=G B (t)=constant. In practice, due to the non-ideal mechanical installation, uneven magnetic field distribution, and non-uniform mounting distance between the Hall sensor and ring magnet, eventually the amplitude of sine/cosine signals from the two linear Hall sensors will deviate from their specified voltage levels. If the distorted sine/cosine signals are directly employed for open-loop control, non-uniform control and serious torque ripple will be caused. We have found that the normalisation (also referred to as unification) operation of the Hall sensor signals dramatically reduces the impact on torque ripple of deteriorated signals from the Hall sensors and results in much smoother torque control of the PMSM motor. 
   The normalisation procedure can be achieved by using: 
   
     
       
         
           
             
               
                 
                   Hall 
                   
                     A 
                     ⁢ 
                     
                       - 
                     
                     ⁢ 
                     norm 
                   
                 
                 = 
                 
                   
                     Hall 
                     A 
                   
                   
                     
                       
                         Hall 
                         A 
                         2 
                       
                       + 
                       
                         Hall 
                         B 
                         2 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   2 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
           
             
               
                 
                   Hall 
                   
                     B 
                     ⁢ 
                     
                       - 
                     
                     ⁢ 
                     norm 
                   
                 
                 = 
                 
                   
                     Hall 
                     B 
                   
                   
                     
                       
                         Hall 
                         A 
                         2 
                       
                       + 
                       
                         Hall 
                         B 
                         2 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   2 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
         
       
     
   
   It may be difficult to implement Equation 2 by either software or hardware. For the currently preferred low-cost applications, we prefer to employ a two-phase-type phase-lock-loop (TP-PLL) to fulfil this task without much computation effort by MCU software design or by low-cost hardware implementation solution. Actually the TP-PLL accomplishes the low-pass filtering and normalisation of sine/cosine input signals, which is our main concern here. In additional such a PLL provides the motor speed and position measurements continuously which may be used in a simple 2nd-order feedback loop. The stabilisation of TP-PLL can be easily achieved by adjusting the proportional-integral (PI) gains. 
   Thereafter the unified Hall sensor sine/cosine signals are transformed into 3-phase synchronous sinusoidal waveforms by the well-known Clark Transformation as follows: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         a 
                       
                     
                     
                       
                         b 
                       
                     
                     
                       
                         c 
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             
                               1 
                               2 
                             
                           
                         
                         
                           
                             
                               3 
                             
                             2 
                           
                         
                       
                       
                         
                           
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                               1 
                               2 
                             
                           
                         
                         
                           
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                                 3 
                               
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                           β 
                         
                       
                     
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                 ( 
                 3 
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   The function of the Clark Transformation in Equation (3) can be implemented by either hardware or software without difficulty. The amplitude of the 3-phase sinusoidal signals will be unity because they have been normalised. They are the inputs fed into PWM choppers. In the conventional PWM control, the amplitude of the high-frequency triangle carrier of PWM is fixed, and controlling the average output of PWM is accomplished by adjusting the amplitudes of 3-phase sinusoidal inputs simultaneously. However in an open loop low cost controller according to an embodiment of the invention, 3-phase sinusoidal input signals are maintained with a fixed unity amplitude and the required 3-phase PWM output level is achieved by adjusting the amplitude of the high-frequency carrier signal manually. With such a design, the 3-phase sinusoidal PWM control circuits are greatly simplified. 
   The 3-phase PWM logic signals are finally fed to gate drivers for controlling the switching power devices of a 3-phase inverter. The output of the inverter provides the sinusoidal control of 3-phase current of PMSM machines. There is no stringent need for detecting the stator phase to achieve this low cost open loop sine wave controller. Normally, the high performance sine wave control of a 3-phase PMSM fully relies on the accurate measurements of at least 2-phase currents and rotor position in order to control the d-q axis currents independently and fulfil the complicated field orient control algorithm of a PMSM machine. Closed loop control of 3-phase currents would be essential in such a high-end controller. The current sensor cost for a vector control based PMSM drive is usually high. However for the low cost sine wave drive proposed here of an embodiment of the invention, there is no direct closed loop feedback control of 3 phase currents to reduce system cost. Only rough measurement of the inverter DC-link current may be employed for the purpose of short circuit protection and peak current limitation to the motor and inverter. The peak phase current, peak output torque and peak speed of the PMSM machine can be controlled by adjusting the amplitude level of the PWM carrier manually. 
   Another aspect of the invention provides a controller for use with a permanent magnet synchronous motor having a stator and a rotor, the stator having windings and the rotor comprising a plurality of permanent magnets; the controller comprising; 
   inputs for connection to two magnetic field sensors, each having a linear response to sensed magnetic field strength, the sensors being spaced by an angle A electrical degrees relative to the rotor, where angle A is greater than 0 degrees and less than 180 degrees or greater than 180 degrees and less then 360 degrees, for sensing the position of the rotor by sensing a magnetic field representative of the fields of the magnets of the rotor; 
   means for producing first and second signals representing normalised orthogonal components of the outputs of the sensors; and 
   energising means for producing, from the said orthogonal components, pulse width modulated currents for application to an inverter for energising the windings of the stator to drive the rotor. 
   The invention also provides a computer program which when run on a suitable processor causes the processor to operate as the controller of said another aspect of the invention. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For a better understanding of the present invention, reference will now be made by way of example to the accompanying drawings in which: 
       FIG. 1  is a schematic diagram of an open-loop sine-wave drive, according to an embodiment of the invention, based on the output signals of two linear Hall sensors normalised by a Two Phase-Phase Locked Loop (TP-PLL); 
       FIGS. 2   a  and  b  are cross-sectional views of a 3-phase PMSM motor with 12 poles and sinusoidal flux density in air gap, and showing an example of the mounting location of linear Hall sensors; 
       FIG. 3  illustrates a Honeywell linear Hall sensor SS495A used in an embodiment of the invention; 
       FIGS. 4   a  and  b  show an example of a 12-pole permanent magnet ring used for the excitation of two linear Hall sensors and its sinusoidal flux distribution pattern; 
       FIG. 5  shows the positions of two linear Hall sensors with respect to the flux direction of the excitation ring magnet; 
       FIG. 6   a  is an example of a TP-PLL and  FIG. 6   b  is a TP-PLL algorithm; 
       FIG. 7  illustrates PWM chopping basics and PWM asymmetric/symmetric PWM comparators; 
       FIG. 8  illustrates a 3-phase inverter with the 3-phase PMSM in star connection; 
       FIGS. 9   a  and  b  show Hall sensor output signals and outputs normalised by the TP-PLL; 
       FIG. 10  is the start-up response of an open-loop sine-wave controller under no load; 
       FIG. 11  shows test results of 3-phase current and voltage waveforms under open-loop control with a load; 
       FIGS. 12   a  and  b  show a comparison of open-loop sine-wave and square-wave drive under a load; and 
       FIG. 13  is a schematic diagram of an closed-loop sine-wave drive, according to another embodiment of the invention, based on the output signals of two linear Hall sensors normalised by a Two Phase-Phase Locked Loop (TP-PLL). 
   

   DETAILED DESCRIPTION 
     FIG. 1  is a schematic diagram of an open-loop sine-wave drive according to an embodiment of the invention, comprising a PMSM  20  and an open loop controller. The controller comprises two Hall effect sensors  16  (HallA and HallB) which have a linear response to sensed magnetic field strength. In this example the sensors  16  are spaced by 90 electrical degrees but may be at any other angle except 180n where n is an integer: n may be 0, 1, 2, etc. A Two Phase Phase Locked Loop TP-PLL  22  normalises the outputs of the two Hall sensors  16  and provides normalised outputs sin θ and cos θ. A 2-to-3 Clark transformation function block  24  converts the two phases sin θ and cos θ, to three phases u a , u b  and u c  as indicated at  24 ′. A PWM chopper  28  modulates a triangular carrier produced by a carrier source  26  to output pulse width modulated phases Sa, Sb and Sc. An inverter  30  converts the PWM phases Sa, Sb and Sc to sinusoidal currents which energise the 3 phase stator of the PMSM  20 . In accordance with an embodiment of the invention, the amplitude of the carrier is adjustable by a control  260 . Adjusting the carrier amplitude allows simple adjustment of all three phases u a , u b  and u c  together avoiding the need to provide three amplitude adjusters for the respective phases. In this example the amplitude adjustment is a manual adjustment via the controller  260 . 
   The PMSM motor is illustrated in  FIG. 2 . The motor comprises a rotor with a surfaced mounted 12-pole permanent ring magnet  10 , a stator  14  of steel laminations having 18 slots for coil windings  12 , two linear Hall sensors  16  fixed on the stator with the displacement of 90 electric degrees, and an excitation ring magnet  18  fixed to the rotor. The ring has 12 poles having the same sinusoidal flux distribution as the rotor. In this example the poles of the excitation ring are aligned with those of the rotor to simplify signal processing. The Hall sensors  16  on stator, and excitation ring magnet  18  on rotor, are mounted at a distance from the stator windings in order to minimise the armature effect due to the stator current. That is they are spaced from the stator so the magnetic fields generated in the stator do not interfere with the Hall sensors and the excitation ring. 
   In an example Honeywell linear Hall sensors SS495A are used. Such a sensor is illustrated in  FIG. 3 . 
   The excitation ring magnet  18  has a 12-pole sinusoidal flux distribution. The flux pattern of the excitation ring is illustrated in  FIG. 4 . Correspondingly,  FIG. 5  shows the mounting location of two linear hall sensors with respect to the excitation magnetic flux. The two Hall sensors  16  are mounted at the same distance from the ring magnet  18 . The phase displacement angle for two linear sensors is in 90 electric-degrees. The radial direction of the ring magnet should be vertical to the flat surface of the Hall sensors in order to get the optimal detection of the magnetic field. Due to the TP-PLL-based unification process for the sine/cosine signals from the linear Hall sensors, a precise mounting distance between the Hall sensors and excitation ring magnet is not required. The only strict restriction on this mounting distance is to avoid any linear-Hall-sensor output saturation happening for a whole mechanical cycle rotation of the excitation ring magnet. 
     FIG. 6  illustrates two implementation solutions of the TP-PLL  22 . A circuit hardware solution of TP-PLL is shown in  FIG. 6(   a ). The TP-PLL  22  comprises a standard proportional-integral (PI) controller  46 , a voltage/frequency (V/F) converter  48 , an up/down counter  50 , a sine/cosine lookup table ROM  54 , and a digital-to-analogue (D/A) converter  56 . The feedback components sin θ and cos θ represents the unified results of the Hall sensor outputs Hall A  and Hall B . This feedback loop can also provide measurements θ and ω for the motor position and speed respectively if required as discussed below with reference to  FIG. 13 . The two phases sin θ and cos θ are supplied to multipliers  42  and  40  respectively. The outputs of the multipliers  42  and  409  are supplied to a differencing circuit  44  the output Δ of which is applied to the controller  46  of the TP-PLL  22 . Δ=Hall A  cos θ−Hall B  sin θ. 
   Another embodiment of the TP-PLL implements the equations of  FIG. 6(   b ). digitally with an MCU. The unified sine/cosine signals sin θ K  and cos θ K  at each sampling interval K are interpolated numerically from a sine-wave look-up table based on the calculated cyclic position from the TP-PLL algorithm. In both solutions, the phase error is detected by (Hall A   K  cos θ−Hall B   K  sin θ.) The estimated position θ should be close to the rotor position θ when the phase lock is achieved in the feedback loop. Theoretically the Laplace transformation form of TP-PLL can be approximated as 
   
     
       
         
           
             
               
                 
                   
                     θ 
                     ^ 
                   
                   ⁡ 
                   
                     ( 
                     s 
                     ) 
                   
                 
                 ≈ 
                 
                   
                     
                       
                         
                           K 
                           p 
                         
                         ⁢ 
                         s 
                       
                       + 
                       
                         K 
                         i 
                       
                     
                     
                       
                         s 
                         2 
                       
                       + 
                       
                         
                           K 
                           p 
                         
                         ⁢ 
                         s 
                       
                       + 
                       
                         K 
                         i 
                       
                     
                   
                   ⁢ 
                   
                     θ 
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   Equation 4 stands for a standard 2-order control system. The stability of the loop will be fully governed by the PI gains Kp and Ki which can be easily adjusted to meet the desired performance, such as the response time, overshoot level, and final tracking phase error. 
     FIG. 7(   a ) illustrates the relationship of the 3-phase sinusoidal waveforms  62 ,  64  and  66  (ua, ub uc) produced by the transformer  24  and the carrier signal  60  produced by the source  26 . The 3-phase sinusoidal signals of normalised fixed amplitude are fed into PWM choppers  68  with the high-frequency carrier from source  261  or  262 . The carrier amplitude is adjustable to get the desired PWM output level. Such a design results in the further simplification of the system circuit design. The carrier can be either an asymmetric ( 261 ) or symmetric ( 262 ) triangular waveform. The symmetric carrier is preferable due to its better control for electromagnetic-interference (EMI) noise. 
   The logic output signals Sa, Sb and Sc of the 3-phase PWM  28  are applied to gate drivers S 1 , S 2  and S 3  to control the upper-side power switches of the conventional 3-phase half-bridge DC-to-AC inverter, as shown in  FIG. 8 . The signals applied to the lower-side power switches are respectively complementary to those of the upper-side power switches. A certain amount of dead-time should be injected to avoid the short-circuit between the two power switches in each half-bridge of the inverter. The power switches may be IGBT or MOSFET depending on applications. The inverter output terminals are directly connected to the 3 phases of the stator of the PMSM  20 . The connection of the motor can be in either star- or delta-connection.  FIG. 8  shows a star configuration. 
     FIG. 9  shows practical test results for the unification of the two linear Hall sensor output signals by TP-PLL. As we can see from FIG.  9 - a , due to the inexact mounting of the Hall sensors, the sine/cosine signals are distorted with amplitude variation and phase displacement angle error. After the unification process by TP-PLL, the distortions are reduced. The unity amplitude for the TP-PLL sine/cosine output signals are achieved no matter how much variation the Hall-sensor sine/cosine measurements have in amplitude. By employing the 2-to-3 Clark transformations to both the Hall sensor outputs and unified sine/cosine signal of TP-PLL, as shown in  FIG. 9   b , the 3-phase sinusoidal waveforms are dramatically improved after the TP-PLL unification. The unity amplitude is obtained in all 3 phases. Apparently if the original 3-phase sinusoidal signals without unification are directly inputted to the PWM modulator, it could cause the deteriorated torque control of the PMSM motor. On the other hand the mounting distance between the Hall sensors and the ring magnet and the magnetic strength of the excitation ring magnet is difficult to maintain constant during industrial mass production. If no unification is employed, a huge mount of labour and time would have to be spent adjusting some of circuit parameters in order to achieve the uniform control performance to meet the specification for the all the whole mass produced products and this would dramatically increase the system cost. Fortunately with the help of TP-PLL, all these cost and burdens can be substantially reduced if not totally eliminated Therefore the significance of the TP-PLL-based unification to Hall sensor signals is to maintain the uniform control for mass-production PMSM motors without any effort on adjusting the controller circuits. 
     FIG. 10  shows the start-up response of the open-loop sine-wave control in a no-load condition. The amplitude of the PWM carrier is fixed with a certain value. The dq-axis current components i d  and i q  are described in the rotor synchronous rotating co-ordinates. The q-axis current i q  is proportional to the electromagnetic torque output for a surface-mounted PMSM machine in theory. The full torque can be achieved from standstill. As the back EMF gradually builds up with the increase of the motor speed, the motor current and torque output decrease steadily until a balanced peak speed is reached. The slight oscillation appears in the phase currents and output torque is dominantly caused by the inexact mounting of two Hall sensors. 
   A practical test result with load is illustrated in  FIG. 11 . The 3-phase line-to-neutral voltages are obtained through a low pass filter to eliminate the PWM carrier frequency. Basically the 3-phase voltages and currents are close to be sinusoidal. Once the load is applied, the motor automatically drops to a balanced speed from the peak speed. During the test the PMSM motor runs smoothly and silently. By adjusting the amplitude of the PWM carrier, a new speed will be reached to balance with the phase current and motor output torque. There is no closed-loop speed control offered in this simple controller. 
     FIG. 12  illustrates the control-performance comparison between open-loop sine-wave drive and square-wave drive, i.e. conventional brushless DC (BLDC) six-step control. The commutation logic for the open-loop square-wave control is determined from the measurements of the two linear Hall sensors. Due to the uniform sinusoidal flux distribution of the ring magnet, only two linear Hall sensors are enough to provide the correct synchronous commutation signals. The full starting torque at standstill can be achieved in this square-wave drive as well. Comparatively in the conventional square-wave drive at least three switching-type Hall sensors are necessary to provide the non-hesitation start-up. Obviously the sine-wave drive offers much smooth torque control than the square-wave drive for controlling a PMSM machine. Due to the large variation in the torque-related q-axis current in the square-wave drive, the resulted mechanical vibration and noise are unavoidable. By improving the mounting of the Hall sensors, even smoother torque control can be achieved in open-loop sine-wave drive. 
     FIG. 13  illustrates another embodiment of the invention, which uses closed loop control of a PMSM  20 . The elements  16 ,  20 ,  22 ,  24 ,  26 ,  28  and  30  of  FIG. 13  are identical to those of  FIG. 1  and will not be further described here. The embodiment of  FIG. 13  differs from that of  FIG. 1  in that a feedback controller  270  is provided which compares a reference value R produced by a source  260  with an actual value V to be controlled. In this example the actual value may be θ or ω for the motor position or speed as produced by the TP-PLL of  FIG. 6(   a ). 
   An embodiment of the present invention may comprise a controller C shown by the dashed box C in  FIG. 1  and which has inputs for connection to the Hall sensors  16  and to the source  260  and outputs Sa, Sb and Sc for connection to the inverter  30 . Such a controller C may be implemented by a digital processor or MCU or by a hardwired circuit. 
   The invention may also be a computer program which when run on a suitable process causes the processor to operate as the controller C of  FIG. 1 . 
   As described above the Hall sensors  16  are spaced by 90 electrical degrees. However, they may be displaced at any other angle not equal to 180n where n is an integer, n=0, 1, 2 etc. If spaced at an angle not equal to 90 degrees a circuit  222  is provided as shown in  FIG. 1  to derive from the outputs of the Hall sensors the components sin θ and cos θ. 
   As described above the Hall sensors sense the field of the excitation ring. However, the Hall sensors may sense the poles of the rotor. 
   As described above the magnetic sensors are linear Hall effect sensors. However other linear magnetic field sensors may be used.