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IEEE Transactions on Energy Conversion, Vol. 13, No. 3, September 1998 257 Commutation Torque Ripple Minimization for Permanent Magnet Synchronous Machines with Hall Effect Position Feedback Todd D. Batzel and Kwang Y. Lee, Senior Member, IEEE Department of Electrical Engineering The Pennsylvania State University University Park, PA 16802 Abstract: A permanent magnet synchronous motor (PMSM) with sinusoidal flux distribution is commonly coininutated using discrete rotor position feedback from Hall sensors. A commonly used stator current excitation strategy used in such a system is a six-step current waveform. Application of sinusoidal current waveforms is shown to produce smooth torque in the PMSM. This paper shows how a pseudo-sensorless rotor position estimator may be used with Hall sensors to provide sinusoidal current excitation in place of sixstep currents to reduce the torque ripple associated with the sixstep strategy. Performance evaluation of the rotor position estimator in a PMSM drive is provided through simulation. Keywords: torque ripple, permanent magnet syncl~ronousmotor, sensorless commutation, Hall effect sensor I. INTRODUCTION In recent years, the permanent magnet synchronous motor (PMSM) has become a popular choice for applications such as machine tool drives, computer peripherals, robotics, and electric propulsion. Much of this popularity is due to the PMSM’s reduced maintenance (no brushes), superior power density and efficiency, and low rotor inertia [I]. In a PMSM, the ability to separately control the stator current and angle allows the choice of operating re,’mimes, such as maximum torque per unit current and maximum power. In addition, separate current and angle control permits flux weakening (for operation above rated speed) and phase advancing (for reluctance torque augmentation) [2]. In order to individually control stator current and angle and thus fully exploit the characteristics of the PMSM, high resolution rotor angle information is required - usually from a device such as a resolver or encoder. These high resolution position sensors, which are usually attached to the rotor, add length to the machine, raise the system cost, increase rotor inertia, and require additional cabling. PE-959A-EC-0-05-1997 A paper recommended and approved by the IEEE Electric Machinery Committee of the IEEE Power Engineering Energy Society for publication in the IEEE Transactions on Conversion. Manuscript submitted January 2,1997; made available for printing May 23, 1997. One of the most commonly used methods for angular position sensing is the Hall effect sensor [3]. These coarse shaft position sensing devices are mounted on the magnetic axis of each stator phase winding, thus providing an aggregate resolution of 60 electrical degrees. Figs. 1 and 2 show how the discrete signals generated by Hall effect sensors may be used to decode position data as well as to generate a six-step stator excitation. Operation with six-step excitation tends to increase torque ripple [4], and due to the discrete nature of the feedback, does not easily allow some advanced PMSM operating regimes to be used [ 5 ] . Many applications may benefit from the use of sinusoidal excitation yet, due to various constraints, do not permit the use of an external position sensing device other than the built-in Hall sensors. Thus, the designer is faced with a choice of six-step stator excitation using the Hall Sensors, or using one of the many sensorless techniques that have been developed for the PMSM [6-101. The six-step approach suffers from torque ripple, and many of the sensorless techniques suffer performance limitations at low speed. In this paper, a method for obtaining high resolution shaft position information from coarse Hall sensor data, stator voltage, and stator current measurements is proposed. This method provides the reliability of the Hall sensors to commutate the machine at very low speeds, and the high resolution angle information needed for sinusoidal excitation. This pseudo-sensorless method is shown through simulation to reduce the torque ripple associated with sixstep stator currents commonly used in conjunction with Hall effect position sensors. 11. PMSM MACHINE MODEL A two pole, 3 phase salient pole machine is shown in Fig. 1. For such a machine, the flux linkage and voltage equations are given in the stationary reference frame [ 111. The flux linkages of the PMSM machine in the stationary reference frame may be represented by where v,i, 1, and R are recpective voltage. current- flux linkage. and resistance for phase a,b. or c. In addition, 0885-8969/98/$10.00 0 1997 IEEE 258 7 hm cos(8) where the flux terms are defined in (2), and di is the increment in the phase current. The torque may then be derived from the coenergy expression 2a awc where L v are the self and mutual inductances for each phase, h,is the flux due to the permanent magnet of the rotor, and 8 is the rotor angle. The self and mutual inductances for the PMSM may be written as shown in (3), where the terms LI, La,, and L, represent leakage, nominal, and position dependent inductance terms, respectively: Lll(8) =Ltl+La,-LsCOS(28) (e) = Ll+ La, - L, cos(28 + -1 - Luv 2 - (3) 2n L, cos(28 - -) 3 T = -hm[sin(8) i,+sin(8--j o 1 0 1 1 1500 to 2100 210O to 270° 0 1 0 0 270° to 330° 1 0 90 9 0 1500 +-)ic]. 3 2n 3 2n j b +sin(O+-)ic]. 3 (7) Next, we may write the generated back emf expression as [ 1 (A) expressions, the 1 Wc = -[ ~ u i+;~ 2 z i-+i ~33i3+[ Luiaib + ~ u i+ ~~ z ii iC] b~ 2 (6) 2n 2n . r - 30°to 30' 30°t0900 (5) Using (9,and assuming a non-salient rotor, the torque for the machine may now be written as 3 L12(0)=L21(0)= Using (2) and (3) to obtain the flux coenergy function can be found to be +hm[cos(8)ia+cos(8--)ib+cos(8 3 2n 3 2n ~ , , ( e )= L~+L,,- L ~ C O S --I (~B L22 T(i~,ib,i~,O)= -.a0 0 0 d axis / 1 h cos(0) hnrcos(8 --) =$xnl 3 27t cos(e + -) 3 o \ qaxis Fig. 1. PMSM analytical model. It is apparent from these equations for the PMSM machine, that the rotor angle (e> is a function of the motor voltage, current, and flux linkages, as well as the machine parameters - winding resistance, inductance, and the permanent magnet nux linkage (hI,). A. PMSM Torque Generation An expression for the torque generated by the PMSM may be obtained by using the coenergy of the electromagnetic system [8]. The coenergy is defined as 1 sin(@) 2n = -0L Slll(0 --) 3 2n sin(8 +-) 3 - (8) where e, represents the back emf of each phase and 0 1s the rotor speed in electrical radians per second. Using (8), we inay rewiite (7) in terms of the back emf as T-e,i,I a 27c r I o e&-. o P Ienia I ebib 2 O I ~ Urn I ecic ,, Om (9) where a,,, is the mechanical shaft speed and P is the number of poles for the machine. From (8) and (9), it can be seen that any mismatch between the back emf waveform and the corresponding phase currents will result in a torque ripple. Thus, for a sinusoidally wound PMSM, torque ripple may be minimized by utilizing sinusoidal phase cuirents. In the analysis of the PMSM, it is generally assumed that the stator windings may be approximated as: sinusoidally distributed windings. Since most machines are designed so that the windings produce a relatively good approximation of a sinusoidally distributed ax gap mmf , this appears to be a justified assumption. The phase currents to minimue corque ripple are 2n 2n i, = I,sin(8) ; ib = I,sin(B --) 3 ;ic = /,sin(B tT) , (10) where 1, is the amplitude of the phase currents. The resulting torque expression for the currents described by (10) is 259 amplifier. The output of the amplifier represents the voltage required to force the phase current to its reference value. - This shows that for the sinusoidal phase currents with a constant amplitude given in (lo), the resultant torque is constant and independent of shaft position. 7 ............ L.. ......................._ "0 0 4 0 2 0.6 0.8 0.6 0.8 1 time (s) 111. PMSM COMMUTATION Two techniques are commonly used for the commutation of PMSM machines - six-step and sinusoidal. A high resolution position sensor such as a resolver or encoder usually replaces Hall sensor feedback when sinusoidal commutation is used. Brushless motors exhibit back emf characteristics which may be either sinusoidal or trapezoidal, with each phase being 120" apart. .... ........... i.... 0.4 0.2 time ................. ...... (5) ....... 0 0 0.4 0.2 tin- e ( s ) Fig. 2a. Hall sensors used for six-step cumnt generation. 8' A . Six-Step Commutation 0 4 0 2 Six-step commutation is widely used for commutation of PMSM machines. For six-step commutation, Hall effect sensors are mounted as an integral part of the motor and aligned with the motor back emf at the factory. With Hall sensor feedback, an average 90" torque angle may be maintained by forcing it to remain between 60" and 120". When the torque angle has fallen to 60", the drive electronics directs the current in the stator such that the torque angle is increased to 120°, and keeps it there for the next 60" of rotation. Clearly, for six-step commutation, the forced stator currents will not match the sinusoidal back emf characteristic of the PMSM, and torque ripple' will result. The effect of this torque ripple tends to be a slight kick at the commutation points, which may be detrimental to high performance positioning and velocity regulation applications. Several simulations of the PMSM with Hall sensor feedback and six-step currents were performed, with the results shown in Figs. 2 and 3. The results shown in Fig. 2 depict an acceleration from zero speed, while Fig. 3 shows low speed operation with a high load torque. The torque ripple generated by the six-step stator currents is shown clearly in both simulations. The overall system diagram of the PMSM model, current controller, and commutation logic used for the computer simulations is shown in Fig. 4. The input to the simulation is the current command (Icmd),which the commutation logic uses with the rotor position feedback to produce the optimum phase current references. The rotor position feedback consists of simulated Hall effect signals for the six-step simulations, and high resolution angle for the sinusoidal excitation simulations. The current controller compares the actual phase current measurements to the reference currents and operates on the error to generate the input to the 0 6 0 8 time ( s ) L ........... .; ! ........................... 0.2 0 0 4 0 6 0.8 i .;.-.. Y 0 2 .. 0.2 0 4 0 6 0.8 time (s) I 2 0 \............. ....... 260 h u1 6 1 I 2 1 I 1 2, I 2 7 2 75 2 8 2 85 2 0 2 05 3 0 -1, a' . . . 06 0 8 s 1 I I -1 .......... 1 ..........i.. / i 0 4 02 , I 1 0 2 0 4 0 6 I I I , I 0 8 1 I I 2.75 2.8 2 85 time I 2.9 2.95 3 (5) Fig. 3b. Torque, velocity, and actual position for six-step current. i 1............ 3 dz I I Rotor Position (Selectable - Hall Sensor or High Resolution) Fig. 4. PMSM current controller block diagram. -0 IV. ROTOR POSITION ESTIMATOR The block diagram of a high resolution rotor position estimator developed is shown in Fig. 6. Inputs to the position estimator are the PMSM stator currents, voltages, and the Hall sensors. The output is a high resolution estimate of the rotor position. The flux linkages may be estimated by rearranging (1) and integrating: j..... 0 4 0 2 0 6 0 8 tim e (s) glo ..... - ........... ............ 1 ..I 0-, 0 0 I I 0.2 0.4 ....... I time (s) 0 6 , 0.8 Fig. 5b. Torque, m o r angle, aud shaft speed for sinusoidal current. B . Sinusoidal Commutation Sinusoidal commutation is used when very smooth torque response is desired. Typically, this is when a high degree of velocity regulation or position accuracy must be maintained while the motor is operating under heavy load torque at low speed. A much higher resolution rotor position sensor is required for sinusoidal commutation if smooth rotation at low speeds is desired. In this way, it is possible to maintain a constant torque angle very accurately, resulting in very smooth low speed rotatioil and negligible torque ripple. The results of a simulation of a sinusoidally-fed PMSM are shown in Fig. 5. The current controller shown in Fig. 4 was used for the simulations with the assumption that exact rotor position is available. As expected, there is no torque ripple. i ............ .......................... Iab<- v h,,,, Flux E m ma for z-+ Coarse ? Eqtmiator tFl" x Fig. 6 Block diagram of rotor position estimator. h,,, = j (v - iR) , (12) where v, and i are the three-phase flux linkage estimate, voltage. and current vectors defined in (1). This flux linkage estimate may then be used to generate a coarse current estimate by using (2) and the decoded Hall sensor data. The current estimate error vector, I,,, , is then calculated from the difference between measuied stator currents and the current estimate vector. A position error, Ae, which represents a measure of the error between the coarse Hall sensor position data and the actual rotor position, may then be formed. Since the flux linkage is a function of the stator currents and the rotor position, the scalar A8 may be calculated from the permanent magnet flux linkage vector, the inductance ~ 263 matrix, the current estimation error vector, as shown in (13). The flux linkage estimate is assumed to be correct when calculating A0 , A0 =[[$r[$IT[$] T [AX-LAi], rotor position estimator tracks the actual position well, the exception being at very low speeds. The tracking error at low speeds may be attributed to the lack of a developed back emf under these conditions. Recall from (12) that the back emf is integrated in the estimation of the flux vectors. 8 , I where L is the inductance matrix defined in (2) and A denotes changes in variables. A non-salient rotor is assumed in (13). The rotor position estimate is then calculated from 6 =0h,,+A8. (14) Finally, the position estimate is used to calculate the error in the estimated flux linkages. This flux correction loop is necessary to remove any accumulated errors due to the integration process, measurement errors, and uncertainty in the initial conditions of the flux linkages. The flux correction process is performed by utilizing the high resolution position estimate and the flux estimate to obtain a high resolution current estimate I* via (2). Another current estimation error I*err is then formed from the difference between the measured current and this high resolution current estimate. A measure of the flux estimation error may then be formed in a manner similar to the formation of (13). In (13), the flux estimate was assumed to be correct (AM).To calculate the flux correction, however, the position estimate is assumed to be correct (A0=0). Thus, the flux correction is calculated as ai Ai=-AI* ai The rotor position estimator shown in Fig. 6 was used in a series of system simulations. The PMSM was subjected to step changes in the current command and load torque. t i m e (s) Fig. 8 Response of rotor angle estimator to step load change Fig. 8 displays the rotor position estimation performance due to a step change in the load torque. For this test, a step load torque is applied to reverse the acceleration of the rotor at 0.5 seconds. It is seen from Fig. 8 that the step load change does not have an effect on the position estimate. V. SYSTEM SIMULATION In order to generate sinusoidal stator currents (and thus reduce torque ripple) without the use of a high resolution rotor position transducer, the rotor position estimator shown in Fig. 6 is used. A block diagram of the system used to run the simulations is shown in Fig. 9. The motor parameters used in the simulations are R = 1.91!2, L,, = 9.55 mH, L, = 0, K r =.332 * - vA , 6 poles. I 7 , Lit” c (6) Fig. 7. Actual and estimated position for step cunent reversal at 0.5 sec. Fig. 7 shows a comparison of the estimated angular position and the actual rotor position for a step change in the current command at 0.5 seconds. In this test, the rotor is initially at rest, and is accelerated through a constant torque command in the positive direction. At 0.5 seconds, the torque command is reversed. It is clear in Fig. 7 [hat the I I’ Fig. 9. Block diagram of pseudo-sensorless system. Fig. 10 shows a comparisoii of the resulting torque during an instant current reversal at 0.5 sec. for both the simulated meudo-sensorless svstem and an ideal system for which the rotor position is known exactly. The data of Fig. 10 shows 262 that the torque for the pseudo-sensorless system is very close to the ideal system, with slight deviations when the system crosses through zero-speed. The torque overshoot at 0.5 sec. in Fig. 10 is due to small overshoots in the phase currents. 0.5 0.55 0.6 0.65 0.7 0.75 0 8 0.85 0.9 lime (n) 40 I I I I I I I I I I I I , I 055 06 065 07 075 08 085 I --0 5 0.5 0.55 0.6 0.65 0.7 0 75 0.8 0 85 [2]P. Pillay and R. Krishnan, “Application characteristics of permanent magnet synchronous and brushless DC moton for servo diives,” IEEE Trans. on Ind. Appl., vo1.27,no.5, pp. 986-996, September-October 1991. [3] B. Drafts, “Selecting Hall effect devices for BLDC motor coinmutation,” Power Conversion & Intelligent Motion, vol. 22, no. 5, pp. 55-61. May 1996. [4] T.M. Jaluis, “Torque production in pennanent-magnet synchronoqs motor drives with rectangular current excitation,” IEEE Trans. on Ind. Appl., vol. 20, pp. 803-813, April 19S4. [SI S. Vlaliu,“New bmsliless AC servo drive uses isolated gate bipolar power transistors and a CPU to obtain high dynaniic performance with exceptionally high reliability ‘and efficiency,” Proc. Ind. Appl. Sociery Annual Meeting, pp. 685-690,1990. [6] T. Liu and C. Clieng, “Adaptive control for a sensorless bmshless permanent-magnet synchronous motor drive,” IEEE Trans. on Aerospace and Electronic Sys. vol. 30, no. 3, pp. 900-909, July 1994. [7] S. Matsui, “Sensorless operation of blushless DC motor drives,” IECON Proceedings vol. 2, pp. 739-744, 1993. [SI A. Consoli, “Sensorless operation of bmshless DC motor drives,” IEEE Trans. on /ndustrial Electronics vol. 41. no. 1,Pp. 91-95, Feb. 1994. [9] R. Wu, G.R. Slernon, “A pennanent magnet motor drive without a shaft sensor,”IEEE T r m . ~on . lnd. Appl. vol. 27, no. 5 , pp. 1005-1011, 1991. [10]J.S. Kin and S.K. SUI, “New aproach for high peifonnance PMSM drives without rotational position sensois,” lEEE Applied Morion Conf. And &yo, pp. 381-386. 1995. [ll]P.C. Krause and 0. Wasynzuk, Electromechanical Motion Devices, New York, McGraw-Hill. 1989. OS 09 time (sec ) Fig. lob. Velocity and angular position for pseudo-sensorless simulation. Todd D. Batzel received his B.S. degree in Electrical Engineering from the Pennsylvania State University, State College, PA, in 1984, and his M.S. degree in Electrical Engineering from the University of Pittsburgh, Pittsburgh, PA, in 1989. He is presently a Research Assistant with the Applied Research Laboratory of the Pennsylvania State University and a Ph.D. student in Electrical Engineering at the Pennsylvania State University. His research interests include machine controls; electric drives, and power electronics. VI. CONCLUSIONS In this paper, a pseudo-sensorless rotor position estimator for a PMSM has been implemented toward the goal of reducing the torque ripple for a sinusoidally wound PMSM. Simulations revealed that the pseudo-sensorless system was able to track the rotor position extremely well. This allowed the use of sinusoidal stator currents, which was found through simulation to significantly reduce the magnitude of torque ripple. The proposed system combines sensorless technology with coarse Hall sensor feedback in an innovative way to achieve torque ripple reduction. Such a system may be useful in many motion control applications - particularly where high precision is extremely important but the addition of an external position sensor is not permissible. VII. REFERENCES . “Permanent Magnet Machines” in Nasar S.A. (Ed.) Handbook of Elecrrical Machines, McGraw-Hill, New York, 1987. [l] K.J. Binns, Kwang Y. Lee received his B.S. degree in Electrical Engineering from Seoul National University, Korea, in 1964, his M.S. degree in Electrical Engineering from North Dakota State University, Fargo, ND, in 1968, and his Ph.D. degree in Systems Science from Michigan State Univei-sity, East Lansing, MI, in 1971. He has been on the faculties of Michigan State, Oregon State, University of Houston, and the Pennsylvania State University, where he is currently a Professor of Electrical Engineering. He is Director of Power Systems Control Laboratory, and a CO-Director of Intelligent Distributed Controls Research Laboratory at Penn State €€is interests include control systems, artificial intelligence, neural networks, fuzzy logic control, genetic algorithms, and their applications to power plants and power systems control, operation and planing. He is a Senior Member of IEEE, active in Power Engineering Society and Control Systems Society.