Pseudo-vector Control – An Alternative Approach for Brushless DC Motor Drives Cao-Minh Ta, Senior Member, IEEE Abstract- This paper provides an alternative approach called "Pseudo-Vector Control" (PVC) to reduce torque ripple in Square-wave PM Motors (BLDC motor). Instead of conventional square-wave form, the method uses the principle of vector control to optimally design the wave-form of reference current in such a way that the torque ripple is minimal. The currents are however still regulated in phase current control manner. The proposed PVC has 2 majors advantages: a) the torque ripple is greatly reduced, and b) the flux weakening for constant-power high-speed mode can be achieved by injecting a negative d-axis current into the control system, just like for the case of PM Synchronous Motors (sine type machines). Index Terms-- Brushless DC motor, Trapezoidal type machine, PM synchronous motor, Sine type machine, Vector control, Pseudo-vector control, Torque ripple minimization. I. I NTRODUCTION A. Sine-wave PM Motor vs. Square-wave Motor: Vector Control vs. Phase Current Control With the recent development of permanent magnet (PM) materials, PM (brushless) motors have become more and more popular and find their applications in a wide range of fields: industry, office use machines, house appliances, space equipments. The use of permanent magnets has great advantage in that the created magnetic field is high-density and without loss, providing a high-efficiency operation of the whole drive. PM motors can be divided into 2 categories depending on its current wave-form [1]: • Sine-wave PM motor, which is also called PM synchronous motor (PMSM), is fed by sine-wave current supply; • Square-wave PM motor that is fed by square-wave current and it is often called Brushless DC Motor (BLDCM). The main difference between two kinds of motors from the control viewpoint resides on current control techniques and on the torque quality. The torque produced in a sine-wave PM motor is smooth as a result of the interaction between the sinusoidal stator current and the sinusoidal rotor flux. High- performance PM synchronous motor drives are regulated by the well-known vector control method, in which motor currents are controlled in synchronously rotating d-q frame (Fig. 1). Rotor position at any instant (measured by a high- resolution position sensor, or estimated by an observer) is the mandatory requirement in the high-performance vector control method. Vector control method cannot however be utilized for BLDCMs, due to the fact that terminal variables (currents, voltages) and back-EMFs wave-form are not sinusoidal. Therefore, the phase current control technique is normally employed for this kind of motor (Fig. 2). The control technique is relatively simple because the square-wave reference currents can be generated in a step manner, every 360/(m*p) electrical degrees, where m is the number of phases, p is the pole-pair number. Hence, low cost Hall-effect sensors served as position sensor are enough for the control purpose. Simplicity in control, BLDCMs suffer however from a big drawback: because of phase commutation, the torque is not as smooth as that produced by their counterpart, sine- wave PM motors. A BLDCM is, moreover, recognized as having the highest torque and power capability for a given size and weight due to its (quasi) square-wave current and trapezoidal form of back-EMF. In addition to that, a BLDCM also presents the cost advantage over the sine-wave PM motor, due to its construction and winding. Therefore, it seems to be evident that if the torque ripple in BLDCM can be overcome, this kind of motor would become a very attractive solution for many industrial and house-appliances applications. B. Torque ripple in BLDCM and its reduction techniques Torque ripple is the main concern of the BLDCM because it limits this kind of motor from many applications. There are three main sources of torque production (and therefore of the torque ripple) in BLDCMs: cogging torque, reluctance torque, and mutual torque [1]. The torque ripple can be reduced by (a) motor design; (b) control means; (c) or both of them. If in a BLDCM, either stator slots or rotor magnets are skewed by one slot pitch, the effect of the first two torque components is greatly reduced. Therefore, if the waveforms of phase back-EMFs and the phase currents are perfectly matched, torque ripple is minimized. However, perfect matching phase back-EMF and phase current is very difficult, 2011 IEEE International Electric Machines & Drives Conference (IEMDC) 978-1-4577-0061-3/11/$26.00 ©2011 IEEE 1534 considering unbalanced magnetization and/or imperfect windings [1]. Moreover, due to the finite cut-off frequency of the current control loop, the transient error of the controlled currents always occurs, especially in commutation instants, when the current profile changes drastically and also, the turn-on and turn-off characteristics of the power devices are not identical. This is one of the most critical problems in control of BLDCM drives. We can find in literature a lot of efforts to reduce the torque ripple. Ref. [2] - [6] are only some to name. Le-Huy, Perret and Feuillit [2] analyzed the torque by using Fourier series and shown that the torque ripple can be reduced by appropriately injecting selected current harmonics to eliminate the torque ripple components. Yong Liu, Z. Q. Zhu and David Howe [5] utilized DTC to reduce torque ripple in addtion to increasing torque dynamics. Haifeng Lu, Lei Zhang and Wenlong Qu [6] calculated duty cycles in the torque controller considering un-ideal back EMFs. C. Purpose of the present work The objective of the present work is to provide an alternative approach to reduce torque ripple in BLDCM drive. Instead of conventional square-wave form, the method uses the principle of vector control to optimally design the wave- form of reference current in such a way that the torque ripple is minimal. The currents are however still controlled in the phase current control manner. For this raison, we have called the proposed approach Pseudo-vector Control. II. S YSTEM C ONFIGURATION The configuration of the control system proposed in this paper is described in Fig. 3 where the part generating current references is marked by the break-line frame. For the purpose of comparison, a typical Vector Control scheme for synchronous motors (sine type machines) is depicted in Fig. 1, followed by a typical Phase Current Control scheme for brushless DC motors (trapezoidal type machines) in Fig. 2. The rotor position is normally detected by 3 Hall-effect sensors in the BLDCM drives (Fig. 2), whereas a position sensor (such as an encoder) is normally required for the case of PMSM drives (Fig. 1). It can be noted that except for the current reference generation part, the system is nearly the same as of the conventional control system for BLDC motor (Fig. 2). The motor is fed by a PWM-controlled MOSFET inverter. As we can see in Figs. 2 and 3, the currents are controlled by 3 current controllers in stationary frame. By convention, the superscript * refers to the reference variables and quantities. The reference currents generating part of the proposed Pseudo-vector Control is consisted of 3 blocks, as shown in Fig. 3, corresponding to their function that will be described latter in Sec. III. In order to perform the direct transformation and the invert one, the rotor position will be needed. The instantaneous rotor position and speed can be obtained from the Hall-sensors information using an observer. a i c i () bac iii=− + * c v e θ + − + + − − * d v * q v * a v * b v d i q i * d i * q i Fig. 1. Typical block-diagram of Vector Control for PMSM drives. 1535 * a i * b i * c i + + + − − − a i c i () bac iii=− + * T * a v * b v * c v * I 1 T k + − Fig. 2. Typical block-diagram of Phase Current Control for BLDCM drives. INVERTER BLDC motor Load PWM Current Controller Current Controller Current Controller * a i * b i * c i + + + − − − a i c i () bac iii=− + Measured currents d-q to a-b-c a-b-c to d-q a e b e c e Reference currents generation * d i * q i d e Hall-signals processing circuit Hall sensors Speed m ω Torque command * T Position e θ * a v * b v * c v Position e θ + − q e Look-up Table Fig. 3. Block-diagram of the proposed Pseudo-vector Control for BLDCM drives. III. P RINCIPLE OF P SEUDO - VECTOR C ONTROL In the conventional square-wave phase current control (Fig. 2), for a given torque command, the reference currents depend only on Hall-sensor signal showing the communication instants. The torque ripple is therefore inevitable. If we can incorporate other information of the drive into calculation of the reference currents i a * , i b * , i c * , the performance of the system should be improved. The basic idea of the proposed “Pseudo-vector control” (or PVC for short) for BLDCM is to take the back-EMFs into account in calculation of reference currents (Fig. 3). It can be derived from the power balance of the control system as follows. The electromagnetic power of motor is expressed as: ) .( ccbbaae eieieiP ++= (1) where P e is the electromagnetic power; i a , i b , i c is the motor current of the phase a, b, c, respectively; e a , e b , e c is the back-EMF of the phase a, b, c, respectively. By neglecting the motor friction loss, (1) can be rewritten: meccbbaae TeieieiP ω .) .( =++= (2) where T e is the electromagnetic torque and ω m is the motor mechanical angular speed. 1536 Therefore, the torque can be expressed as: (. . .) aa bb cc e m ie ie ie T ω ++ = (3) Equation (3) shows the basic equation of the electromechanical torque, which is a result of the interaction between the motor currents and the back-EMF produced in the motor by the permanent magnets (PM). It also explains the fact that if the currents and EMF do not match each other, torque ripple occurs. This is therefore interesting to note that: in the control configuration, if the reference currents are chosen in the manner that equation (3) is respected, there would be no torque ripple. * *** ) .( e m ccbbaa T eieiei = ++ ω (4) where the * denotes the reference and desired values. This is now to discuss how to calculate the reference currents from (4). The back-EMF can easily be estimated from the motor terminal variables, so they are considered to be known for a given speed. Let’s work in the synchronously rotating d-q frame instead of stationary a-b-c frame. The power equation (1) can be rewritten in d-q frame: 00 (. . . ) 3 .( . . . ) 2 e aabbcc dd qq P ie ie ie ie ie ie =++ =++ (5) where the indexes d , q , 0 represent the variables in d-axis, q- axis and zero-sequence, respectively. For a balance system it is desirable that the zero-sequence current i 0 is zero. Combining (4) and (5) gives for reference and desired values: ) .( 2 3 . *** qqddme eieiT += ω (6) Equation (6) means that if we can a priori select the reference current i d * , we can calculate the reference current i q * , for a given speed and desired torque T e * . q ddme q e ieT i ** * . 3 2 − = ω (7) As the desired torque contains no torque ripple, the obtained torque is theoretically free of ripple if the current controllers work properly to yield currents perfectly equal to the reference ones. The back-EMF in d-q frame e d , e q , e 0 in (5) and (6) are obtained from 3 phase EMF e a , e b , e c by Park’s transformation. As previously mentioned, the zero-sequence current i 0 is forced to be zero in our system, so only e d and e q need to be calculated using simplified Park’s transformation. ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +− +−−−− = )sin()sin()sin( )cos()cos()cos( 3 2 3 2 e 3 2 ee 3 2 e 3 2 ee ππ ππ θθθ θθθ M (8) where e θ is the rotor position: . ee t θω = , with ω e is the electrical angular speed ( ω e = ω m .p), ω m is the motor (mechanical) angular speed and p is the pole-pair number. Having calculated i d * , i q * , the 3 phase reference currents i a * , i b * , i c * are obtained by inverse Park transformation M -1 . ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ++− −−− − = − )sin()cos( )sin()cos( sincos 3 2 e 3 2 e 3 2 e 3 2 e ee 1 ππ ππ θθ θθ θθ M (9) It is worth to emphasis again here that by using the reference currents i a * , i b * , i c * to control the system, there would be theoretically no torque ripple, because the reference currents i a * , i b * , i c * was optimally designed taking back-EMFs into account. The d-q frame is utilized only for calculating these reference currents, while the phase current control principle is normally used in a-b-c frame. IV. S IMULATION R ESULTS The proposed algorithm has been simulated in Matlab/Simulink using the motor of which the parameters are shown in TABLE I. TABLE I MOTOR PARAMETER Nominal voltage 36 [V] Nominal speed 1500 [rpm] Phase resistor 0.8 [Ohm] Phase inductance 2.14 [mH] Back EMF constant 0.018 [V/rpm] Nominal current 2 [A] Nominal torque 0.344 [N.m] The working condition in simulation is as follows: the motor is starting up to 1000 rpm without load in the interval (0 - 0.1) s, then a load is applied at t = 0.15 s when the system has been already in the steady state. In order to emphasis the performance of the controlled system, the speed control loop is omitted in the Figs. 1 - 3 and the currents are controlled by hysteresis (bang-bang) techniques. A. Performance of the conventional Phase Current Control The simulation results of the Phase Current Control for 1537 BLDCM drives is reported in Fig. 4 in the order from top to bottom: (a) Motor speed, (b) Three phase currents, (c) Torque, (d) Zoomed current, (e) Zoomed torque. The torque ripple is clearly detected in the instants of commutation due to the shape slop of the currents. (a) 0 0.05 0.1 0.15 0.2 0.25 0 100 200 300 400 500 600 700 800 900 1000 Speed respond Time [s] Speed [rpm] (a) 0 0.05 0.1 0.15 0.2 0.25 0 100 200 300 400 500 600 700 800 900 1000 Speed re spond Time [s] Speed [rpm] (b) 0 0.05 0.1 0.15 0.2 0.25 -6 -4 -2 0 2 4 6 Phase curre nts Time [s] Current [A] Phase A Phase B Phase C (b) 0 0.05 0.1 0.15 0.2 0.25 -6 -4 -2 0 2 4 6 Phase currents Time [s] Current [A] Phase A Phase B Phase C (c) 0 0.05 0.1 0.15 0.2 0.25 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Torque re spond Time [s] Torque [N.m] (c) 0 0.05 0.1 0.15 0.2 0.25 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Torque re spond Time [s] Torque [N.m] (d) 0.4 0.405 0.41 0.415 0.42 0.425 -3 -2 -1 0 1 2 3 t(s) is c (A) (d) 0.4 0.405 0.41 0.415 0.42 0.425 -3 -2 -1 0 1 2 3 t(s) is c (A) (e) 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.4 0.42 0.44 0.46 0.48 0.5 0.52 Torque respond Time [s] Torque [N.m] Fig. 4. Simulation results of conventional Phase Current Control for BLDCM: (a) Speed, (b) Three phase currents, (c) Torque, (d) Zoomed phase C current, (e) Zoomed torque. (e) 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.4 0.42 0.44 0.46 0.48 0.5 0.52 Torque respond Time [s] Torque [N.m] Fig. 5. Simulation results of the proposed Pseudo-vector Control for BLDCM: (a) Speed, (b) Three phase currents, (c) Torque, (d) Zoomed phase C current, (e) Zoomed torque. 1538 B. Performance of the proposed PVC The most important signals of the proposed control system are presented in Fig. 5 in the same order as in Fig. 4. As we can see in Fig. 5 (b), the current are smooth and the torque ripple due to the 60 degrees interval commutation are eliminated, as can be seen in Fig. 5 (c). C. Performance comparison In order to compare the performance of the conventional Phase Current Control and of the proposed PVC, let’s have a closer look to the system with the zoomed current and torque curves as depicted in Figs. 4 - 5 (d), (e). The peaks at every 60 degree on the current wave form (Fig. 4 (d)) are due to phase commutation and they correspond to the torque ripple in Fig. 4 (e). On the other hand, no torque ripple is observed in the Fig. 5 (e). It is noted that high-frequency noise in the current and the torque wave forms is the natural consequence of the hysteresis band in bang-bang current control, but it does not harm the quality of produced torque. To further evaluate the performance of the system, the flux trajectory (ψ α - ψ β ) is examined. Fig. 6 (a) clearly shows the “cogs”, which have origin from the commutation at every 60 degree in case of Phase Current Control. The sharp change of current wave-form in commutation instants is reflected in the flux. Again, the torque produced in the machine can also be considered as an interaction between rotor and stator flux. The torque ripple is therefore evident in this case. In contrast, the flux trajectory of the PVC method in Fig. 6 (b) is nearly round, which can be explained that the current wave-form of PVC is smooth, as previously shown in Fig. 5 (d). This interesting feature gives another explanation on the superiority of the proposed PVC over the conventional Phase Current Control method. Note that the simulation in Figs. 4 -5 was carried out in the based-speed region with i d * = 0 in (7). The flux weakening for constant-power high-speed mode can be achieved by injecting a negative d-axis current (i d * < 0) into the control system, just like for the case of PM Synchronous Motors (sine type machines). V. C ONCLUSION The Pseudo-vector Control (PVC) has been presented in this paper in order to improve the performance of the Brushless DC Motor drives. The vector-control principle in d- q synchronously rotating frame is utilized for the generation of current references only, and the motor currents are regulated, as usual for PM trapezoidal type machines, by individual phase current control in the stationary a-b-c frame. The performance of the proposed PVC has been tested in simulation in the Matlab/Simulink environment and the results have been compared with those of the conventional phase current control method. The improvement in torque ripple reduction has been obtained. The PVC can therefore be a very promising method for applications using BLDCM. a) -0.03 -0.02 -0.01 0 0.01 0.02 0.03 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Flux trajectory Flux alpha Flux beta b) -0.03 -0.02 -0.01 0 0.01 0.02 0.03 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Flux trajectory Flux alpha Flux beta Fig. 6. The flux trajectory (ψ α - ψ β ): (a) using conventional Phase Current Control, (b) using PVC. A CKNOWLEDGEMENT A part of this work has been carried out when the author worked in NSK Co. Ltd, Japan as a Senior Engineer. The collaboration is very much appreciated. R EFERENCES [1] J. R. Hendershot Jr., T. J. E. Miller, Design of Brushless Permanent- Magnet Motors, Magna Physics Publications – Oxford Science Publications, 1994. [2] H. Le-Huy, R. Perret, and R. Feuillet, “Minimization of torque ripple in brushless DC motor drive”, IEEE Transactions on Industry Applications, vol. 22, July/Aug. 1986, pp. 748–755. [3] Joachim Holtz, “Identification and Compensation of Torque Ripple in High-Precision Permanent Magnet Motor Drives”, IEEE Transactions on Industrial Electronics, vol. 43, no.2, April 1996, pp. 309–320. [4] Thomas M. Jahns and Wen L. Soong, “Pulsating Torque Minimization Techniques for Permanent Magnet AC Motor Drives - A Review”, IEEE Tran. on Ind. Electronics, vol. 43, no. 2, April 1996, pp. 321– 330. [5] Yong Liu, Z. Q. 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