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UBC Theses and Dissertations

Analysis of voltage-source-inverter-driven brushless dc motors with unbalanced Hall sensors Samoylenko, Nikolay 2007

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ANALYSIS OF VOLTAGE-SOURCE-INVERTER-DRIVEN BRUSHLESS DC MOTORS WITH UNBALANCED HALL SENSORS by Nikolay Samoylenko Engineer, Moscow Aviation Institute, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA August 2007 © Nikolay Samoylenko, 2007 ABSTRACT Brushless dc motors with Ha l l sensors are widely used in various industrial and electromechanical applications. These machines have often been considered in the literature, under one common assumption - ideal placement of the sensors, which is often not the case, especially for low-precision motors. This thesis is composed o f three manuscripts which investigate the unbalance o f Ha l l sensors and propose methods to eliminate its adverse effects. The studies presented here show that misalignment of H a l l sensors leads to unbalanced operation of the inverter and motor phases, which in turn results in increased low-frequency harmonics in torque ripple, possible acoustic noise, and degradation of the overall drive performance. Thus, the first manuscript introduces the problem of the misaligned Ha l l sensors by using a detailed model o f a sample brushless dc motor and proposes a simple yet practical averaging technique to significantly decrease the effects of the misplaced sensors in steady state. The second manuscript extends the discussion to the dynamic performance of a low-precision brushless dc motor by introducing extrapolating filters which specifically target the transient operation. The third manuscript generalizes the concepts presented in the first two papers by taking into account both steady-state and transient performances and provides extensive hardware studies. The presented research considers a typical industrial brushless dc motor and includes measurements, detailed models and hardware experiments to validate the analysis. The proposed averaging approach is shown to achieve performance characteristics very close to those of a motor with perfectly balanced Ha l l sensors. ii TABLE OF CONTENTS Abstract i i Table of Contents i i i List of Tables v List of Figures • v i Acknowledgements v i i i Co-Authorship Statement ix 1 I n t r o d u c t i o n 1 1.1 Brushless DC Motors 1 1.2 Problem of Misaligned Hall Sensors 4 1.3 Objectives and Contributions 6 1.4 Composition of Thesis 7 1.5 References 8 2 B a l a n c i n g H a l l - E f f e c t S i g n a l s i n L o w - P r e c i s i o n B r u s h l e s s D C M o t o r s 10 2.1 Introduction 10 2.2 BLDC Machine Model 11 2.3 Technique Description 14 2.4 Case Study 18 2.5 Conclusion 20 2.6 References 20 3 I m p r o v i n g D y n a m i c P e r f o r m a n c e o f L o w - P r e c i s i o n B r u s h l e s s D C M o t o r s w i t h U n b a l a n c e d H a l l S e n s o r s 21 3.1 Introduction 21 3.2 Permanent Magnet BLDC Machine Model 23 3.2.1 Detailed Model 23 3.2.2 Model Verification 24 3.3 Filtering Hall Signals 27 3.4 Case Study 34 3.5 Conclusion 35 iii 3.6 References 36 4 D y n a m i c P e r f o r m a n c e o f B r u s h l e s s D C M o t o r s w i t h U n b a l a n c e d H a l l S e n s o r s 37 4.1 I n t r o d u c t i o n 37 4.2 P e r m a n e n t - m a g n e t B L D C M a c h i n e M o d e l 39 4.2.1 D e t a i l e d M o d e l 39 4.2.2 M o d e l V e r i f i c a t i o n 41 4.3 F i l t e r i n g H a l l S igna l s 43 4.3.1 B a s i c A v e r a g i n g F i l t e r s 46 4.3.2 E x t r a p o l a t i n g F i l t e r s . . 46 4.3.3 P e r f o r m a n c e o f F i l t e r s 48 4.4 Reference S w i t c h i n g T i m e 48 4.5 I m p l e m e n t a t i o n a n d C a s e Studies 50 4.5.1 S ta r t -up T rans i en t 52 4.5.2 L o a d - S t e p T rans i en t 52 4.5.3 Vol tage-S tep T r a n s i e n t 54 4.5.4 D i s c u s s i o n 57 4.6 C o n c l u s i o n 57 4.7 References r-58 5 I m p l e m e n t a t i o n 60 5.1 References 64 6 S u m m a r y 65 6.1 C o n c l u s i o n 65 6.2 F u t u r e W o r k 66 Appendix 67 Vita 68 IV LIST OF TABLES TABLE 1.1 Distribution of Absolute Misplacements Among Sample Motors (deg.) 5 TABLE 3.1 Switching-Event Time Determination 33 TABLE 4.1 Switching-Event Time Determination 50 v LIST OF FIGURES Figure 1.1 BLDC motors: a) Arrow Precision motor, Model 86EMB3S98F; b) Maxon Motor, Model EC 167131; c) Maxon Motor, Model 244879; d) American Precision Industries, Model 23BLS-03S 2 Figure 1.2 Maxon Motor BLDC motor driver. 3 Figure 1.3 Anaheim Automation BLDC motor driver. 3 Figure 1.4 Hall-sensor placement in an Arrow Precision motor. 4 Figure 1.5 Hall-sensor placement in a Maxon motor. 4 Figure 2.1 Typical industrial BLDC motor with external housing of Hall sensors 10 Figure 2.2 Hall-sensor placement assembly 11 Figure 2.3 Permanent magnet synchronous machine 12 Figure 2.4 Measured and simulated phase a back emf. 13 Figure 2.5 Measured and simulated phase currents 14 Figure 2.6 Actual and ideal Hall sensor output signals 15 Figure 2.7 Filter input sequence when the machine is operated in steady state, and the produced output r{n) used for firing inverter transisitors 16 Figure 2.8 Filter magnitude and phase responses 17 Figure 2.9 Torque waveforms 18 Figure 2.10 Torque harmonic content 18 Figure 2.11 Phase current and torque waveforms 19 Figure 2.12 Torque harmonic content with filter. 19 Figure 3.1 Hall-effect sensor placement on a typical BLDC motor. 21 Figure 3.2 Brushless dc motor drive system with MIMO averaging filter. 22 Figure 3.3 Permanent-magnet synchronous machine 23 Figure 3.4 Measured line-to-line back emf at speed 2458 rpm 24 Figure 3.5 Measured and simulated line-to-line back emf at speed 2458 rpm 25 Figure 3.6 Measured and simulated phase currents 26 Figure 3.7 Electromagnetic torque waveforms 26 Figure 3.8 Electromagnetic torque harmonic content 27 Figure 3.9 Ideal and actual Hall-sensor output signals 28 Figure 3.10 Sequence of time intervals r{n) for unbalanced Hall sensors 28 Figure 3.11 Magnitude and phase response of the basic averaging filter. 29 Figure 3.12 Computing r, (n) using linear extrapolation and subsequent averaging. 30 vi Figure 3.13 Computing rq (n) using quadratic extrapolation and subsequent averaging 31 Figure 3.14 Magnitude and phase responses of different filters 31 Figure 3.15 Response of different filters to a ramp increase in speed 32 Figure 3.16 Switching-event time relationships 33 Figure 3.17 Speed and electromagnetic torque response with 3- and 6-step averaging filters 34 Figure 3.18 Speed and electromagnetic torque response with extrapolating averaging filters.... 35 Figure 4.1 Hall-effect sensor placement in a typical BLDC motor. 38 Figure 4.2Brushless dc motor drive system with filtering of Hall-sensor signals 39 Figure 4.3 Permanent-magnet synchronous machine with unbalanced Hall sensors 40 Figure 4.4 Measured and simulated phase currents 42 Figure 4.5 Electromagnetic torque waveforms 43 Figure 4.6 Electromagnetic torque harmonic content 43 Figure 4.7 Ideal and actual Hall-sensor output signals 44 Figure 4.8 Sequence of time intervals r(«) for unbalanced Hall sensors 45 Figure 4.9 Computing F ; («) using linear extrapolation and subsequent averaging 47 Figure 4.10 Computing fq («) using quadratic extrapolation and subsequent averaging 47 Figure 4.11 Magnitude and phase responses of different filters 49 Figure 4.12 Response of different filters to a ramp increase in speed 49 Figure 4.13 Switching-event time relationships 50 Figure 4.14 High-level diagram of the microcontroller including the proposed filter. 51 Figure 4.15 Measured phase currents without and with the proposed filtering 52 Figure 4.16 Measured start-up transient of BLDC motor. 53 Figure 4.17 Measured transient response due to load change 54 Figure 4.18 Speed and electromagnetic torque response with 3- and 6-step averaging filters 55 Figure 4.19 Speed and electromagnetic torque response with extrapolating averaging filters.... 55 Figure 4.20 Measured response of phase currents to step in dc voltage 56 Figure 5.1 Determination of the corrected time interval tcorr (n) 62 ACKNOWLEDGEMENTS I would like to express my deepest gratitude to Dr. Juri Jatskevich, who supervised my studies, for his invaluable guidance and support throughout my studying at U B C . I would also like to thank Dr. Hermann Dommel and Dr. Wi l l i am Dunford, who have kindly agreed to be members on my thesis committee and devoted their precious time and expertise. I would like to thank my colleague and co-author Qiang Han for his help, critical remarks and valuable suggestions in our fruitful collaboration. M y profound appreciation goes to all members of Power Systems Lab, and in particular to: Tom DeRybel, Marcelo Tomim, Yong Zhang, and Michael Wrinch. Last but not least, I would like to thank my family, my wife Natalia and son Fedya, as well as my parents and brothers for helping me make my dream come true. v i i i CO-AUTHORSHIP STATEMENT The manuscripts which constitute Chapters 2 - 4 o f this thesis were written by me in co-authorship with Qiang Hah, and Dr. Juri Jatskevich. Although Qiang Han and I have common subjects of our studies - namely the brushless dc motors, our research topics are nevertheless quite different. A s his research is concentrated primarily on developing average-value models for aforementioned type o f drives, mine is dedicated to the problem of misaligned H a l l sensors in the same drive. I have investigated the problem, conducted experiments and simulations, developed improvement filtering techniques, and implemented the proposed methodology in hardware. In addition to that, I prepared the manuscripts, which was then iteratively edited by the authors. Dr. Juri Jatskevich has supervised and directed my work. ix l INTRODUCTION 1.1 Brushless DC Motors Brushless dc ( B L D C ) motor drives are becoming widely used due to advancement in power electronics and production o f permanent magnets (PM). A s the name o f the drive implies, B L D C motors lack the brush-commutator unit which greatly contributes to their efficiency and reduces the maintenance costs. In addition, the torque-speed characteristics of brushless dc motors are similar to those of a regular brushed dc motors. Examples of using B L D C motors can be found in almost any field where other more traditional (brushed dc, synchronous, and induction) motors have been used for many years. Due to high power density, compactness, wide speed ranges, high starting torque, etc., the B L D C motor drives are becoming increasingly popular in areas where other means of providing mechanical power such as internal combustion (IC) engines and hydraulic drives dominated for a long time e.g. in propulsion in aerospace, automotive and ship industries. Other areas where B L D C motors are gaining market share include robotics, consumer appliances, power tools, and manufacturing automation. A typical B L D C motor has the structure of an ac synchronous machine. Unl ike the latter, the rotor of a B L D C motor has an assembly of permanent magnets which establish the rotor magnetic field. The stator terminals are fed from a full-bridge inverter which is supplied from the dc source. Similar to the conventional synchronous motor, the rotating magnetic field created by the stator currents interacts with the rotor field to generate the electromagnetic torque and spin the motor. To enable the motor operation from the dc source, the stator phases are commutated by the inverter transistors to produce the required torque at a given rotor position, so the inverter switching is rotor-position-dependent. The approaches to detect the rotor position in B L D C motor-drives may be roughly categorized into several groups: sensorless control [1] - [11]; position-encoder-based [12] - [13], and Hall-sensor-based [6], [14], [15]. A great variety o f sensorless control techniques have been documented in the past. The sensorless techniques are typically based on using the measurements of voltages and/or currents [2] - [4], the back emf generated in the stator windings [5], [6], or the observer-based methods [10], [11]. These methods require significant computational resources and knowledge of the motor parameters, and therefore are used only in specific industrial applications. Other challenges associated with the sensorless approaches include starting and 1 operating at low speeds with variable and possibly unknown mechanical loads. Potentially, the high speeds may be problematic as wel l due to the interaction of the back-emf-sensing circuitry with noise. The B L D C drives that use position encoders also typically require higher computational resources and therefore are more expensive, which is not justified in many practical applications. Perhaps the most common approach to detect the rotor position in B L D C motors is using the Hal l sensors. This approach is simple, versatile, and it uses only one Ha l l sensor per phase. Several typical industrial Hall-sensor-driven B L D C motors are shown in F ig . 1.1. In the most common three-phase configuration, the three Ha l l sensors are displaced by 60 or 120 degrees and react to the magnetic field produced by either a special permanent-magnet tablet assembly on the rear end of the motor's shaft as shown in F ig . 1.1 a - b) or the rotor poles, as shown in F ig . 1.1 c - d). Each sensor is producing a binary signal (0 or 1) depending whether it is under North or South magnetic pole. c) d) Figure 1.1 BLDC motors: a) Arrow Precision motor, Model 86EMB3S98F; b) Maxon Motor, Model EC 167131; c) Maxon Motor, Model 244879; d) American Precision Industries, Model 23BLS-03S 2 The three-Hall-sensor unit produces a unique sequence o f states spanning the entire electrical revolution of the motor and dividing it into six sectors. Therefore, the rotor position is known within these six sectors. This information is used to control the transistors o f the inverter. The Hall-sensor approach is inherently reliable and computationally inexpensive. It has no low-speed limitations and therefore can be used to start the motor. The high-speed performance of a Hall-sensor-driven B L D C motor is limited only by computational capacity o f the motor controller and the mechanical strength of the rotor components. In high-speed applications, the B L D C motors can easily operate above 10000 rpm. Another special point that deserves attention is the compatibility o f drivers from different manufacturers with different motors. With most industrial ready-to-use drivers, the Hall-sensor signals are simply used as inputs. There is no need to prepare motor in any way and/or input the motor parameters into the driver (which is not possible with most sensorless controls). In research presented in this thesis, two commercially drivers shown in Figs. 1.2 and 1.3 were also used alongside with a driver developed here at U B C . Figure 1.2 Maxon Motor BLDC motor driver. Figure 1.3 Anaheim Automation BLDC motor driver. The back emf in a typical B L D C motor may be either trapezoidal or sinusoidal [14], [16], whereupon the particular type depends on the physical construction of the rotor. A s the sinusoidal back emf is generally harder to achieve, the motors with trapezoidal back emf tend to be less expensive and therefore more popular. The transistors o f the inverter may be controlled using 180- or 120-degree commutation logic [14], [17], [18]. The former switching law is ideal for PWM-generating o f the sinusoidal stator currents in the motors with sinusoidal back emf. In this method, each phase is always connected 3 either to the positive or negative bus o f the inverter. The 120-degree switching is used extensively with the trapezoidal back emf machines. In this method, each stator phase is conducting for 120 electrical degrees and then left floating for 60 electrical degrees, which happens two times during one electrical revolution. The Hall-sensor-controlled trapezoidal B L D C motors are most widely used offering simplicity and robustness to many applications. 1.2 Problem of Misaligned Hall Sensors In a typical B L D C machine o f the type considered in this thesis, the H a l l sensors may either be mounted inside the motor's main case (see Fig . 1.1 d)) or be placed on a circular printed circuit board (PCB) that is mechanically fixed to the enclosure of the motor. Two examples of the P C B mounting are shown in Figs. 1.4 and 1.5. A s can be seen, the assembly look very similar. With this mounting, the H a l l sensors react to the magnetic field produced by the permanent-magnet tablet that is fixed on the rear end of the rotor shaft. Although the P C B assembly shown in Figs. 1.4 - 1.5 appear very simple and easy to manufacture, significant misalignment of the Hal l sensors from their ideal positions may arise due to manufacturing imprecision. Figure 1.4 Hall-sensor placement in an Arrow Figure 1.5 Hall-sensor placement in a Maxon motor. Precision motor. In practice, the misalignment may be easily confirmed by measuring and comparing the phases between the Hall-sensor signals and the respective line-to-line back emf waveforms. In an ideal case, with proper angle o f the sensors with respect to the stator windings, the zero-crossings of 4 the back emf waveforms should coincide with the instances at which the H a l l sensors change their states. In this research, we have experimented with several available B L D C motors in our lab and found that very noticeable positioning errors may exist in many sample machines. The corresponding measured absolute errors in each phase in mechanical degrees are summarized in Table 1.1. TABLE 1.1 Distribution of Absolute Misplacements Among Sample Motors (deg.) ^ l o t o r \ ^ ^ h a s e ^ A B C Ar row Precision #1 0.8 0.0 3.6 Ar row Precision #2 2.1 1.7 2.3 Ar row Precision #3 3.3 4.7 1.2 Ar row Precision #4 -0.3 0.2 -1.0 Ar row Precision #5 -0.7 2.7 4.0 Ar row Precision #6 4.1 -1.2 0.6 Ar row Precision #7 2.4 -0.3 -0.3 Ar row Precision #8 2.8 -1.9 1.2 Ar row Precision #9 0.8 -1.8 4.0 Ar row Precision #10 4.1 -0.6 2.3 Ar row Precision #11 2.3 -0.6 2.6 Ar row Precision #12 1.2 -0.7 3.4 Ar row Precision #13 0.8 -4.0 -4.0 maxon motor 0.7 0.6 0.7 A s can be seen in Table 1.1, in some cases, the positioning errors may exceed 4 mechanical degrees. Among the motors presented in Table 1.1, the low-precision Ar row Precision motors are 8-pole machines, in which case the resulting error expressed in electrical degrees is 4 times greater and may constitute a very significant portion of the 120-degree conduction interval. The Maxon motor is a 2-pole machine that is manufactured with much higher precision. Moreover, since this machine has only 2 magnetic poles, the errors in mechanical degrees directly translate into electrical degrees and are overall much smaller compared to the 120-degree conduction interval. Although all B L D C motors summarized in Table 1.1 have similar electromechanical 5 ratings, the higher precision motor may cost ten times more than the equivalent low-precision machine. In traditional literature on modeling and analysis of B L D C machines [14], [15], [17], [19], several assumptions are usually made. Some of such assumptions include absence o f saturation, no cogging torque, and ideal positioning o f Ha l l sensors. Although from our communications with industry and researchers in this area it appears that the problem of Hall-sensor misalignment has been noted and known for quite some time, we were unable to find any literature that would sufficiently describe the problem and/or propose solutions. Some engineers have acknowledged that in cases where high-accuracy o f the Hall-sensor positioning is required, it is often achieved manually by carefully re-adjusting the sensor assembly. However, this solution is very costly and therefore less practical for large quantities of B L D C motors. Consequently, until the study described in [19] and publications included in this thesis, the problem of misplaced sensors has not been addressed. In [19], the authors describe and document the Hall-sensor alignment problem that exists even in a medium-cost B L D C machine that is driven using a sophisticated observer-based control for mitigating the torque ripple. The immediate effect o f misplaced Hal l sensors is different conduction intervals among the phases, which results in non-uniform phase currents. This, in turn, leads to increase in torque pulsation (ripple), possible acoustic noise and vibrations, and overall deterioration o f the motor performance. Therefore, presenting effective solution to this problem may have a significant impact allowing low-cost B L D C motors to be used in much wider applications, where previously only the high-precision and costly motors were required. 1.3 Objectives and Contributions It is highly desirable to solve the problem of misplaced Ha l l sensors not only from a technical point of view, but also from an economical standpoint. The ideal solution must also be practical and should not require disassembly on any of the motor-driver components or parts. The solution algorithm should be very robust and computationally efficient for it to be implemented on readily available and commonly used motor-control chips. The work presented here focuses on low-precision and low-cost B L D C motors with non-ideal placement o f Ha l l sensors. This thesis proposes a novel approach based on filtering the existing Hall-sensor signals [20] and makes the following overall contributions: 6 • The thesis describes the phenomenon of non-ideal placement of Hall sensors based on a hardware prototype and a detailed switching model of the machine-inverter system. • It proposes a simple but very effective and practical generalized filtering technique to improve the overall performance of the BLDC motor-drive system with significant unbalance in Hall sensor positioning. • It shows that the performance of the BLDC motor with the proposed filter approaches that of a motor with ideally placed Hall sensors. • It demonstrates a possible hardware implementation and shows that the proposed methodology does not require any additional and/or special circuitry and hardware and can be implemented with a basic motor controller. 1.4 Composition of Thesis This thesis consists of three publications that present research results on low-precision BLDC motors with misplaced Hall sensors. These results were achieved during my two-year time of M.A.Sc. project at the UBC power group: Chapter 2 introduces the problem of the sensor unbalance based on a sample brushless dc motor and its detailed model. It is shown that the misalignment of the sensors leads to deterioration of the motor performance well-pronounced in the torque harmonics. Based on the theory, it is noted that in steady state there can be three different conduction intervals, which subsequently leads to development of a filtering concept based on moving-averaging over three subsequent switching intervals. The concept is then developed into a ready-to-use technique that assumes knowledge of the sensor misplacements. Simulation studies presented to verify the theoretical analysis. Chapter 3 presents a study that further investigates the misplacement of the Hall sensors and ways to mitigate this phenomenon. In particular, in addition to the simple moving-average filters of different orders, a new class of extrapolating filters is proposed with the expectation of a better performance during transients. The properties of the filters are compared in both time and frequency domains. A readily applicable methodology that does not require any previous knowledge of the Hall-sensor position errors is then proposed. Transient simulation studies verifying the extrapolating filters are presented. 7 Chapter 4 extends and generalizes the methodology introduced in Chapters 2 and 3. It is shown that any of the proposed filters can be derived from a general equation for a single-input-single-output filter with respective order and a set of weighting coefficients. This manuscript also gives details o f hardware implementation and introduces a concept of acceleration tolerance to ensure a reliable and robust operation o f the motor-drive system. Extensive simulation studies and hardware experiments are performed to validate the analysis and the proposed methodology. Chapter 5 describes the efficient implementation and gives details on how the filters are realized in the code. The proposed implementation makes use of the low-level interrupt-service-routine to perform the minimal amount of calculations and fire the inverter transistors when the filter is enabled. Chapter 6 summarizes the research and provides overall conclusions achieved by the proposed methodology. Possible extensions o f the proposed filtering technique are also presented here. 1.5 References [1] J. P. Johnson, M. Ehsani, Y. Guzelgunler, "Review of Sensorless Methods for Brushless DC, " In Proc. Industry Applications Conference, 1999. Thirty-Fourth IAS Annual Meeting. Conference Record of the 1999 IEEE, Vol. 1, pp. 143-150, Oct. 3-7, 1999. [2] T. Senjyu, K. Uezato, "Adjustable Speed Control of Brushless DC Motors without Position and Speed Sensors," In Proc. Int'l. IEEE/IAS Conf. on Industrial Automation and Control: Emerging Technologies, pp. 160-164, 1995. [3] A. Consoli, S. Musumeci, A. Raciti, A. Testa, "Sensorless Vector and Speed Control of Brushless Motor Drives," IEEE Trans, on Industrial Electronics, Vol. 41, pp. 91-96, February, 1994. [4] P. Acarnley, "Sensorless Position Detection in Permanent Magnet Drives", IEE Colloquium on Permanent Magnet Machines and Drives, pp. 1011-1014, 1993. [5] K. Iizuka, et al., "Microcomputer Control for Sensorless Brushless Motor," IEEE Transactions on Industry Applications, Vol. IA-27, pp. 595-601, May - June, 1985. [6] W. Brown, "Brushless DC Motor Control Made Easy", Microchip Technology Inc., 2002. [Online]. Available: www.microchip.com [7] K. R. Shouse, D. G. Taylor, "Sensorless Velocity Control of Permanent-Magnet Synchronous Motors", In Proc. 33rd Conf. on Decision and Control, pp. 1844-1849, December; 1994. 8 [8] N. Ertugrul, P. Acarnley, "A New Algorithm for Sensorless Operation of Permanent Magnet Motors," IEEE Transactions on Industry Applications, Vol. 30, pp. 126-133, January - February, 1994. [9] N. Matsui, "Sensorless PM Brushless DC Motor Drives," IEEE Transactions on Industrial Electronics, Vol. 43, pp. 300-308, April, 1996. [10]M Schrodl, "Sensorless Control of Permanent Magnet Synchronous Motors," Electric Machines and Power Systems, Vol. 22, pp. 173 - 185, 1994. [11]B. J. Brunsbach, G. Henneberger, T. Klepseh, "Position Controlled Permanent Magnet Excited Synchronous Motor without Mechanical Sensors," In Proc. IEE Conf on Power Electronics and Applications, Vol. 6, pp. 38-43, 1993. [12]M. Benarous, J.F. Eastham, P.C. Coles, "Sinusoidal Shaft Position Encoder," In Proc. Power Electronics, Machines and Drives (PEMD 2004), Vol. 1, Mar. 31 - Apr. 2, 2004 pp. 132-136 [13]Y. Buchnik, R. Rabinovici, "Speed and Position Estimation of Brushless DC Motor in Very Low Speeds," In Proc. Convention of Electrical and Electronics Engineers in Israel, Sept. 6 - 7, 2004 pp. 317-320 [14] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems. IEEE Press, Piscataway, NJ, 2002. [15] P. Pillay, R. Krishnan, "Modeling, simulation, and analysis of permanent-magnet motor drives. Part II. The brushless DC motor drive," IEEE Transactions on Industry Applications, Vol. 25, Iss. 2, March-April 1989, pp. 274-279. [16] P. L. Chapman, S. D. Sudhoff, C. A. Whitcomb, "Multiple Reference Frame Analysis of Non-sinusoidal Brushless DC Drives," IEEE Transactions on Energy Conversion, Vol. 14, No. 3, pp. 440^46, 1999. [17] S. D. Sudhoff, P. C. Krause, "Average-value Model of the Brushless DC 120° Inverter System," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 553-557, 1990. [18] S. D. Sudhoff, P. C. Krause, "Operation Modes of the Brushless DC Motor with a 120° Inverter," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 558-564, 1990. [19] P. B. Beccue, S. D. Pekarek, B. J. Deken, A. C. Koenig, "Compensation for Asymmetries and Misalignment in a Hall-Effect Position Observer Used in PMSM Torque-Ripple Control," IEEE Transactions on Industry Applications, Vol. 43, No. 2, pp. 560-570, 2007 [20] J. Jatskevich, N. Samoylenko, Improving Performance of Hall-Sensor-Driven Brushless DC Motors, The University of British Columbia, University Industry Liaison Office, Patent Draft 07-078, Jun. 2007. 9 2 B A L A N C I N G H A L L - E F F E C T SIGNALS IN L O W - P R E C I S I O N BRUSHLESS D C M O T O R S 1 2.1 Introduction Brushless dc ( B L D C ) motors are often considered in various electromechanical applications and in general have been investigated quite well in the literature [ l ] - [4] . The techniques used to control the inverter transistors can be placed into two major categories: those that require Hal l sensors [1], [5], [6]; and those that are based on a sensorless approach, for example, that use back emf zero-crossing [6]. The main advantage of the first approach is a relatively simple implementation and reliable operation with variable mechanical loads even at low speeds (whereas the sensorless control may not always be effective). The theory and modeling o f Hall-sensor-driven B L D C motors have been developed by many researchers under one common assumption - that the Ha l l sensors are placed exactly 120 electrical degrees apart on the circumference of the rotor. However, in low-cost motors this assumption may not hold true, and the distribution of relative displacements may be quite significant. The resulting unbalance among the phases leads to an increase in torque pulsation, vibrations, acoustic noise, and reduced overall electromechanical performance. A n example of Hal l sensor placement in a typical industrial motor is shown in Figs. 2.1 - 2.2. Figure 2.1 Typical industrial BLDC motor with external housing of Hall sensors. 1 A version of this chapter has been published. N. Samoylenko, Q. Han and J. Jatskevich, 'Balancing Hall-Effect Signals in Low-Precision Brushless DC Motors'. In proc. 22nd Annual IEEE Applied Power Electronics Conference (APEC 2007), Feb. 28 -Mar. 2,2007, Anaheim CA, USA, pp 606 - 611. 10 Figure 2.2 Hall-sensor placement assembly. In this type of B L D C machine, the sensors are mounted on a circular board outside the machine's main case, as depicted in F ig . 2.1. To produce the rotor-position signals, the Ha l l sensors react to the magnetic field o f the permanent magnet (PM) tablet that is mounted on the back o f the motor's shaft, as shown in F ig . 2.2. In an ideal situation, the axes of the sensors should be 120 degrees apart, which in practice is difficult to achieve with high accuracy. F i g . 2.2 shows the axes displacements relative to the corresponding phases. The errors in positioning o f the sensors may be different for different phases. Moreover, the actual positioning errors in mechanical degrees appear even higher in electrical degrees for B L D C machines with more poles. This paper describes this phenomenon based on a hardware prototype, which is a sample industrial B L D C motor driven by a 120-degrees inverter. To analyze the effect o f Ha l l sensor unbalance, a detailed model o f the drive system is developed and verified with the measurements. The paper also proposes a simple but effective technique to improve the overall performance o f B L D C motor drive systems, considering significant unbalance in Ha l l sensor positioning. 2.2 BLDC Machine Model To analyze the impact of unbalanced Hal l sensors on B L D C motor performance, it is instructive to consider a three-phase permanent-magnet synchronous machine ( P M S M ) , as shown in F ig . 2.3. The detailed model described herein is similar to that considered in [ l ] - [4] , except the Ha l l sensors here are allowed to have misalignment. 11 S1'S1 ay-axis Figure 2.3 Permanent magnet synchronous machine. Here, 5{l,2,3} denote the ideally placed sensors and s{l,2,3} denote the actual positions of the sensors. Based on the commonly used assumptions, the stator voltage equation may be expressed as dk 'abcs s abcs abcs dt (1) where fabcs = \fas fbs fcsJ , f may represent voltage, current or flux linkage vectors, and rs represents the stator resistance matrix. If the back emf is half-wave symmetric and contains spatial harmonics, the stator-flux linkage equations may be written as [4] abcs ~ Lsiabcs s in s in sin((2n-l)9 r) {2n-\Ur + In (2) where Ls is the stator-phase self-inductance, and X'm is the magnitude of the fundamental component o f the P M magnet flux linkage. Also , the coefficients Kn represent the magnitude of the n th spatial harmonic o f the flux linkage relative to the fundamental component. The harmonics are normalized such that A-, = 1. To verify the model, we purchased a batch of commercial B L D C motors and evaluated them for variation of parameters among the samples. The parameters of one sample machine considered in 12 this paper are summarized in the Appendix A . A s can be seen in Fig . 2.2, the sensors are mounted outside the motor case and are switched by the field of an auxiliary magnet tablet that is mounted on the back side o f the motor shaft. Overall , we found noticeable misalignment o f sensors in most of the motors. For the sample motor considered in this paper, we found the relative displacements of sensors to be + 4 . 8 ° , 0 ° , a n d -4 .8° mechanical degrees between phases A - B , B - C , and C - A , respectively. Although this mechanical unbalance may appear small, it translates into a more significant electrical angle considering the number o f poles. A set o f harmonic coefficients Kn was obtained by taking the measured back emf waveform shown in F ig . 2.4 (top) and extracting the Fourier series coefficients. The harmonic coefficients were found to be Kz = 0, K5 = 0.042 , and K1 =-0.018. If desired, more coefficients could be extracted and used in a detailed model; however, the simulated back emf waveform depicted in F ig . 2.4 (bottom) with just these harmonics was considered sufficient for the studies in this paper. 30 t / 8 0 -30 0.05 0.0525 0.055 0.0575 0.06 0.0625 30 > "a o -30 0.05 0.0525 0.055 0.0575 0.06 0.0625 Time, s. Figure 2.4 Measured and simulated phase a back emf. To demonstrate motor operation with unbalanced Ha l l sensors, we considered the sample B L D C motor with 120 degrees inverter operation. In the considered test study, the motor inverter was supplied with Vdc = 40V . A mechanical load o f Q.9N • m was applied, which resulted in a speed of 2458 rpm. The measured phase currents were captured and are shown in F ig . 2.5 (top). Analyzing the measured waveforms, the unbalance among the currents can be clearly observed. In particular, since the turn on and turn off times o f each inverter leg depends on two adjacent Ha l l sensors, the relative unbalance of the sensors results in some phase(s) having shorter conduction intervals and other phase(s) having longer conduction intervals. ^-^ : | Measured BEMF | i i i 13 -15 0.035 0.04 0.045 0.05 0.055 -15 0.035 0.04 0.045 0.05 0.055 Time, s. Figure 2.5 Measured and simulated phase currents. The detailed switch-level model o f the system has been developed and implemented in M A T L A B Simulink [7] using the toolbox [8]. Our detailed model also considered the back emf harmonics as wel l as the misalignment o f the Ha l l sensors. The 120 degrees inverter logic is implemented according to the standard table (see [2], [3]). For comparison, the simulated phase currents are plotted in F ig . 2.5 (bottom). A s can be seen, the simulated waveforms show very close agreement with the measurements, thus confirming the accuracy o f our detailed model. 2.3 Technique Description In the Hall-sensor-based B L D C motor, sensor signals determine the logic for switching the inverter transistors. F ig . 2.6 shows the Ha l l sensor output signals corresponding to the actual (solid-line) and the ideal (dashed-line) cases that are superimposed on top o f each other for the purpose of discussion. Here <pA , <pB, and cpc denote the absolute error in displacements o f the Ha l l sensors in each phase relative to the ideal case. In practice, one o f the phases w i l l have the smallest positioning error and w i l l be closer to its ideal position. Without loss of generality, cpA is assumed here to be negligible. The angle <pv denotes a possible advance in transistor firing [1 ] . A waveform produced by adding all three output signals is shown on the bottom of F ig . 2.6. If the Ha l l sensors are placed exactly 120 electrical degrees apart, this combined waveform w i l l yield a square wave with the period of pulses equal to one-third of the period o f one H a l l sensor output. In the actual case, however, the errors in sensor placement result in the distortion of this square wave wherein the interval widths between two successive switching events become unequal. Here, 9{n) denotes the actual angular distances between two successive switching events. 14 Hall sensor outputs 6r, rad. Hall sensor outputs combined J T(n-V 8(n-3) T(n-2) 6(n-2) r(n-l) e(n-\) t(n) r(n+l) e(n) Time, s. 6r, rad. Figure 2.6 Actual and ideal Hall sensor output signals. The ideal or desirable angular distance may be expressed by averaging the actual angular distances d(n) over some number o f intervals. Based on F ig . 2.6, it may be observed that since the rising edge o f interval 6>(«-3) and the falling edge of interval &(n-\) correspond to switching of the same sensor (in this case the sensor of phase A ) the following holds true: e{n) = -{e{n-i)+8{n-2)+e{n-\)) (3) Moreover, since each Ha l l sensor signal represents a 180-degree square wave, we have 0 (n) = ~ . If the combined waveform of F ig . 2.6 (bottom) is plotted against the time axis, then with each angular distance d(n) one may associate corresponding time intervals, denoted here as r(n). Using these time intervals, the average speed observed over the n th interval can be defined as m r ( n ) = ( ^ . Then, the average speed over three successive intervals can be expressed as T(n) r(n) where the average time f(«) evaluated over three successive intervals is calculated as (4) F(«) = - ( r ( « - 3 ) + r ( « - 2 ) + r ( « - l ) ) . (5) 15 Effectively, the unequal angles &(n) and intervals r(n) introduce low-frequency harmonics (sub-harmonics) in the combined square wave (and in the electromagnetic torque). The approach presented in this paper consists o f filtering out these sub-harmonics and deriving the modified signals that are balanced and can therefore be used to control the motor inverter. Moreover, it is possible to use a single-input-single-output (SISO) filter that can be applied directly to the combined waveform signal (see F ig . 2.6, bottom). The averaging action o f the filter is depicted in Fig . 2.7, where the values o f r ( « - 3 ) , r(n-2), and r(n-i) are different due to Ha l l sensor misalignment. Q T(n-2) a r(n-l) fj  r(n) i i i (!) x(n) n-3 n-2 Figure 2.7 Filter input sequence when the machine is operated in steady state, and the produced output r(p) used for firing inverter transisitors. Here, the next value w i l l be r(n), which is assumed to be the same as r (« - 3). This w i l l be the case i f the motor is in a steady state, given that each Hal l sensor signal is a 180- degree square wave. However, the average time interval w i l l be r{n), which is computed according to (5). This averaging filter may be expressed as M f(")= 2ZbmAn-m) ( ° ) m=l where M is the order o f the filter that corresponds to the number of previous points being averaged, and bm = \/M [9]. Although the order of this filter can be higher than 3, it makes sense to limit it to 3, since it gives an exact average interval #(«) = -j- for the 180-degree square wave Ha l l sensor signals. Sixth-order is also possible but w i l l result in slower response. Since the impulse response of the third order filter is « " > - { i , i , i } c o 16 it follows that -JO) ') (8) where m denotes the frequency of a discrete-time signal in radians per sample. Hence (9) e-ljm +e-2ja> + e - j m The stability of this filter is verified by the fact that the impulse response is integrable [10]. Fig. 2.8 depicts the magnitude and phase of H{CO) . It should be noted that the harmonics produced by the sensor misalignment will have a period corresponding to three intervals r(«) , and will therefore have three samples per period. As can be seen in Fig. 2.8, the proposed filter has zero magnitude at Injl, which is exactly three samples per period for the considered harmonics and what is needed to average out their effect. (O, rad/sample. Figure 2.8 Filter magnitude and phase responses. Once the average interval f(n) is available, the actual switching time used to control the inverter at the next switching state is determined as t(n + l) = / ' («)+f(«) (10) where the present switching time instance is t'{n) = tA,if this instance corresponds to the change in Hall sensor signal of phase A, and t'(n) = t(n) for any other instances. 17 2.4 Case Study Since it is difficult to measure instantaneous electromagnetic torque in practice, we calculated the predicted electromagnetic torque waveforms using the detailed simulation previously verified with the B L D C motor considered here. To show the effect o f balancing the Hal l signals on the torque, the calculated torque for the perfectly balanced (ideal case) and unbalanced (actual case) Ha l l sensors are shown in F ig . 2.9. These torque waveforms correspond to the same operating point at 2458 rpm and a mechanical load of Q.9N-m. The corresponding harmonic content is shown in F ig . 2.10 for each case, respectively. A s can be observed in F ig . 2.10, the unbalance of sensors results in very significant low-frequency harmonics in torque. These harmonics are responsible for increased vibrations, noise, and reduced motor performance. YYYYYYYYYV '0.08 0.082 0.084 0.086 0.088 0.09 1.5 S I Actual case I I , • 0.08 0.082 0.084 0.086 0.088 0.09 Time, s. Figure 2.9 Torque waveforms. I 0.5 3 1 -a 3 •5. 4 0.5 1 Ideal case 1 I I  1 1 -984 1968 2952 3936 4920 5904 - I I 1 1 Actual easel III III, .III 1 1 . 1 984 1968 2952 3936 4920 5904 Frequency, Hz. Figure 2.10 Torque harmonic content. 18 The filter was implemented with the B L D C drive system described above. The result of the filter operation is shown in Figs. 2.11 and 2.12. In particular, the considered B L D C motor operates in a steady state with Vdc = 40V and mechanical load 0.9N • m . Initially, the filter is disabled, which results in highly distorted phase current and torque. These results are similar to those shown in Fig . 2.5 (bottom) and F ig . 2.9 (bottom). However, when the filter is enabled after r = 0.12s , the switching signals become balanced and the motor performance significantly improves. The improved torque waveform in F ig . 2.11 is similar to F ig . 2.9 (top). The spectrum of torque harmonics for the case with enabled filter is depicted in Fig . 2.12, which is very close to the ideal case of F ig . 2.10 (top). 0.13 0.13 Figure 2.11 Phase current and torque waveforms. 0.5 i l 984 1968 2952 3936 4920 5904 Frequency, Hz. Figure 2.12 Torque harmonic content with filter. 19 2.5 Conclusion This paper presents a method o f improving characteristics of low-precision brushless dc motors with misaligned Ha l l sensors. The proposed technique is based on filtering the Ha l l sensor output signals to average out the effect of misaligned sensors. To verify this technique, a sample hardware B L D C motor and its detailed simulation have been considered. The presented studies show that when the averaging filter is used, the steady state performance o f the considered motor may become very close to the ideal case of perfectly positioned H a l l sensors. The proposed methodology is relatively simple and might be applicable to various B L D C motor drive systems where misalignment o f the Ha l l sensors may be anticipated. 2.6 References [1] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems. IEEE Press, Piscataway, NJ, 2002. [2] S. D. Sudhoff, P. C. Krause, "Average-value Model of the Brushless DC 120° Inverter System," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 553—557, 1990. [3] S. D. Sudhoff, P. C. Krause, "Operation Modes of the Brushless DC Motor with a 120° Inverter," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 558—564, 1990. [4] P. L. Chapman, S. D. Sudhoff, C. A. Whitcomb, "Multiple Reference Frame Analysis of Non-sinusoidal Brushless DC Drives," IEEE Transactions on Energy Conversion, Vol. 14, No. 3, pp. 440-446, 1999. [5] P. Pillay, R. Krishnan, "Modeling, simulation, and analysis of permanent-magnet motor drives. Part II. The brushless DC motor drive," IEEE Transactions on Industry Applications, Vol. 25, Iss. 2, March-April 1989, pp. 274-279. [6] W. Brown, Brushless DC Motor Control Made Easy, Microchip Technology Inc., 2002. Available: http ://www. microchip. com [7] Simulink: Dynamic System Simulation for MATLAB, Using Simulink Version 6, The MathWorks Inc., 2006. [8] Automated State Model Generator (ASMG), Reference Manual Version 2, P.C. Krause & Associates, Inc. 2003. [9] L. B. Jackson, Digital Filters and Signal Processing, Kluwer, Norwell, MA, 2002, pp. 61-62. [10] M. J. Roberts, Signals and Systems, McGraw-Hill, New York, NY, 2004, p. 164. 20 3 IMPROVING D Y N A M I C P E R F O R M A N C E O F L O W - P R E C I S I O N BRUSHLESS D C M O T O R S WITH U N B A L A N C E D H A L L SENSORS 2 3.1 Introduction Brushless dc ( B L D C ) motors are often considered in various electromechanical applications and generally have been investigated quite well in the literature [ l ] - [4] . The techniques used to control the inverter transistors can be placed into two major categories: those that require Ha l l sensors [1], [5], [6]; and those that are based on a sensorless approach, for example, that use back emf zero-crossings [6]. A n advantage of the first approach is its relatively simple implementation and reliable operation with variable mechanical loads, even at very low speeds (where sensorless control may not always be effective). The theory and modeling o f B L D C motors driven by Ha l l sensors have been developed by many researchers under one common assumption - that the Ha l l sensors are placed exactly 120 electrical degrees apart. However, in many low-cost machines this assumption may not hold true, and the distribution o f relative displacements may in fact be quite significant. A n example of Hall-sensor placement in the typical industrial motor considered in this paper is illustrated in Fig . 3 .1. Figure 3.1 Hall-effect sensor placement on a typical B L D C motor. A version of this chapter has been published. Nikolay Samoylenko, Qiang Han and Juri Jatskevich, 'Improving Dynamic Performance of Low-Precision Brushless DC Motors with Unbalanced Hall Sensors', In proc. 2007 IEEE Power Engineering Society General Meeting (PES CM'07), June 24 - 28, 2007, Tampa FL, USA. 21 A s can be seen, the Ha l l sensors ( H I , H 2 , and H3) are mounted on a P C board placed outside the motor case and react to the magnetic field produced by a permanent magnet tablet attached to the rear end of the motor's shaft. The dashed axes in F ig . 3.1 correspond to the desired positioning for the sensors and the solid lines denote their actual positions. A s can be seen, the absolute error of sensor placement may reach several mechanical degrees. The insufficiently precise positioning of the Ha l l sensors causes unbalanced operation of the motor inverter, with some phase(s) conducting for longer and other phase(s) conducting for shorter time intervals. The resulting unbalance among the phases leads to an increase in torque pulsation, vibrations, and acoustic noise, as well as reduced overall electromechanical performance. This paper describes a hardware prototype (sample motor in F ig . 3.1) and a detailed model of a B L D C drive system with unbalanced Ha l l sensors. The motor is assumed to be driven using a typical 3-phase inverter as shown in F ig . 3.2, which operates according to standard 120-degree switching logic [2], [3]. The paper presents a filtering methodology that can be applied directly to the original Hall-sensor signals to produce a modified set of signals that is used to drive the inverter depicted in F ig . 3.2. Several multi-input multi-output ( M f M O ) filters based on averaging and extrapolation are proposed that make the modified signals approach those of the B L D C machine with ideally placed H a l l sensors. The proposed method is shown to improve performance of B L D C motor drives with inaccurately positioned Ha l l sensors. Figure 3.2 Brushless dc motor drive system with MIMO averaging filter. 22 3.2 Permanent Magnet BLDC Machine Model 3.2.1 Detailed Model To analyze the impact o f unbalanced Hal l sensors on B L D C motor performance, a permanent-magnet synchronous machine ( P M S M ) shown in F ig . 3.3 is considered here. In Fig . 3.3, H{l,2,3} and H{l,2,3} denote the actual and ideal axes (positions) o f the H a l l sensors, respectively; and cpA , <pB, and cpc denote the absolute errors in sensor placement. ij-axis H1'H1 <7-axis Figure 3.3 Permanent-magnet synchronous machine. Based on commonly used assumptions, the stator voltage equation may be expressed as follows [ 1 H 4 ] : dk abcs abcs dt (1) where fabcs = [fas fbs fcs] , and f may represent the voltage, current or flux linkage vectors. Also , rs represents the stator resistance matrix. In the case of a motor with non-sinusoidal back emf, the back emf is assumed to be half-wave symmetric and contain spatial harmonics. Therefore, the stator flux linkages and electromagnetic torque may be written as [4]: ^•abcs ~ Ls*abcs + ^ m n=\ sin sin sin((2n- l)9 R ) (2) 23 4 n=l COS COS cos((2n-l)c? r) (3) where Ls is the stator-phase self-inductance, and Xm is the magnitude of the fundamental component of the P M magnet flux linkage. The coefficients Kn denote the normalized magnitudes of the nth flux harmonic relative to the fundamental. The detailed model o f the system has been developed and implemented in M A T L A B Simulink [7] using toolbox [8]. The 120-degree inverter logic was implemented according to the standard table [2], [3]. 3.2.2 Model Verification To study the phenomena o f unbalanced Hall-effect sensors, we tested a batch of industrial B L D C motors for possible variation in the severity of sensor unbalance parameters among the samples. The parameters of the motor used in the verification studies presented in this paper are summarized in the Appendix A . For the given motor, the absolute sensor positioning errors were experimentally determined to be +0 .8° , - 4 ° , and - 4 ° mechanical degrees for phases A , B , and C , respectively. Although some other motors had better or worse precision, the considered sample was assumed to be sufficiently representative. The measured back emf waveforms are depicted in Figs. 3.4 and 3.5 (top). 0.0525 0.055 0.0575 Time (s) 0.06 0.0625 Figure 3.4 Measured line-to-line back emf at speed 2458 rpm. To improve the accuracy of the model, the spatial harmonics according to (2) and (3) were included. The harmonic amplitudes Kn were obtained by taking the measured back emf waveform shown in F ig . 3.5 (top) and extracting the Fourier series coefficients. The most significant harmonic coefficients are summarized in the Appendix. The measured and simulated 24 emf waveforms are compared in F ig . 3.5, which show a very good match. If desired, additional coefficients could be considered for the detailed model; however, higher-order harmonics were found to be less significant. 30 > -30 0.05 30 : 1 Measured BEMF 1 0.0525 0.055 0.0575 0.06 0.0625 a o -30 1  ! ! I Simulated BEMF 1 0.05 0.0525 0.055 0.0575 Time (s) 0.06 0.0625 Figure 3.5 Measured and simulated line-to-line back emf at speed 2458 rpm. To demonstrate the effect o f unbalanced Ha l l sensors, an operating point determined by a mechanical load of 0.9 N - m is considered. For this study, the motor inverter was supplied with Vdc =40 V, resulting in a speed of 2458 rpm under the given mechanical load. The measured and simulated phase currents for the resulting steady state operating condition are shown in Fig . 3.6. A s can be seen, the detailed model predicts the phase currents very closely and agrees with the measured waveforms. This study confirms the accuracy of the developed detailed model. A s can be observed in F ig . 3.6, the motor phases are energized for unequal periods o f time and the currents are asymmetrically distorted. Such asymmetrical currents also distort the developed electromagnetic torque. Since it is hard to measure actual instantaneous electromagnetic torque in practice, the torque waveforms were predicted using detailed simulations for the two cases: (i) ideal case - the Ha l l sensors are precisely placed, with zero errors; and (ii) the actual case - the H a l l sensors are placed with zero errors equal to those of the sample motor. The predicted torque waveforms are shown in Fig . 3.7, wherein a significant difference can be observed. To analyze this difference, the harmonic content o f the two torque waveforms was extracted [9]. The corresponding harmonic spectrums are depicted in F ig . 3.8. A s can be seen in Figs. 3.7 and 3.8 (ideal case, top), the torque waveform contains very strong harmonics at the frequency o f 984 H z , which corresponds to the six-pulse inverter operation at the given motor speed, and is expected to dominate under normal 25 operation. However, the torque corresponding to the actual case (see Figs. 3.7 and 3.8, bottom) has a much richer spectrum, with two very strong harmonics below 984 H z . These lower harmonics are particularly undesirable as they result in increased mechanical vibration and acoustic noise. ^ 15 I CO < Is a g i o fe s -15 15 I M easured curre nti J n 1 M 1 1 l 1 l 1 0.035 0.04 0.045 0.05 0.055 -15 Is mulated currcr its | 1^ 0.035 0.04 0.045 0.05 0.055 Time (s) Figure 3.6 Measured and simulated phase currents. o w 0 1 Actual case 1 1 1 1 0.2 0.202 0.204 0.206 0.208 0.21 Time (s) Figure 3.7 Electromagnetic torque waveforms. 26 ? 0.1 •0.05 S3 0 I" 0.04 •0.02 1 1 1 1 1 1 1 1 1 1 Ideal case I 1  1 1 1 . 1 X 0 984 1968 2952 3936 4920 5904 _1 III I Actual case I • I I I 1 • I 0 984 1968 2952 3936 4920 5904 Frequency (Hz) Figure 3.8 Electromagnetic torque harmonic content. 3.3 Filtering Hall Signals To better understand how to correct the Hall-sensor signals, it is instructive to consider the diagram depicted in F ig . 3.9. Here, the angle cpv denotes a possible delay or advance in firing [1], and <pA , cpB, and cpc are the respective sensor-positioning errors in each phase. When the ideal motor is running, the Ha l l sensors produce square wave signals displaced by exactly 120 electrical degrees relative to each other (see F ig . 3.9, dashed line). Combining all three ideal outputs produces a square wave (see F ig . 3.9 bottom, dashed line) with a period equal to one-third of a Hall-sensor period, which is equal to 60 electrical degrees. When the sensors are shifted from their ideal positions (see F ig . 3.9, solid line), the resulting combined waveform becomes distorted, resulting in non-uniform angular intervals 6{n) between two successive switching events. The durations of intervals &(n) are denoted here by r(n). As can be observed in F ig . 3.9, the rising edge of interval d(n-3) and the falling edge of interval 6{n-\) correspond to switching of the same sensor (in this case the sensor o f phase A ) . Therefore, the following holds true: 0(n) = - 3) + 9(n -2)+ &{n -1)) (4) which is the average angle between two ideal successive switching events, and is equal to n/3 . 27 Hall-sensor outputs 9r, rad. Hall-sensor outputs combined T(n-2)t r(n) r(n+l) r Time, s. n-3) 8(n-2) 6(n-\) e(n) 0(n+\) 9r, rad. Figure 3.9 Ideal and actual Hall-sensor output signals. This paper presents a methodology to approximate the ideal Hall signals corresponding to H{l,2,3} by appropriately modifying (filtering) the signals from actual sensors H{l,2,3}.. The proposed method works by finding an interval duration f(«) corresponding to d(n) by means of averaging and/or extrapolating the time intervals r(n). Once f(n) is known, it is used for estimating the correct timings for commutating inverter transistors. For clarity, the sequence r{n) (see Fig. 3.9, bottom) is reproduced in Fig. 3.10 as a discrete-time signal with period N = 3 , wherein the samples are the actual values of r(n). Clearly, the non-uniform values of r(n) cause the undesirable harmonics in phase currents and torque waveforms. Therefore, it is necessary to filter out these undesirable harmonics inr(«) . ... n-2 n-l n n+1 n+2 ... Sample number, n Figure 3.10 Sequence of time intervals r(«) for unbalanced Hall sensors. 28 The frequency content of r(n) can be evaluated by using the discrete-time Fourier series (DTFS) [9], whereupon the signal can be written as r(»)=2VM",A'- (5) where the Fourier coefficients {ck} , k = 0,1,...,N-l provide the description of z(n) in the frequency domain. In our case, the signal t(n) has one zero-frequency component and two 2.7T AK components with frequencies of - j - and radians per sample. These two frequencies should be filtered out. To accomplish this, a simple moving-averaging of the three previous samples may be defined as *a («) = j H" - 0 + r{n - 2)+ r(« - 3)). (6) where the subscript "a" denotes this basic averaging procedure. The corresponding magnitude and phase responses of this basic three-step moving-average filter [10] are shown in Fig. 3.11, "2.7T ^iTC where it can be seen that the exact harmonics with frequencies — and — are filtered out, as 3 3 needed. Therefore, this filter will achieve the required balancing of the modified signals when the motor is in a steady state. Frequency a (rad/sample) Figure 3.11 Magnitude and phase response of the basic averaging filter. When the drive system undergoes a speed transient, such that r(n) may no longer be periodic, it may be advantageous to consider an extrapolation of predicted r(«) samples to better cope with the acceleration and deceleration of the motor. In this paper, we first consider a linear 29 extrapolation, in which each subsequent step r^ ,(«) is linearly extrapolated based on a two-step history, as follows: r«_/(«) = 2r(/i-l)-r(/i-2) The values rex , (n) are then averaged to yield an analogue to fa («) in (6), as follows: (7) Ti (") = J («•«_/ (")+ f a J { n - \ ) + z e x J (n - 2)) The resulting equation for computing f, (n) in terms of r(«) can be written as (8) f, (n) = j (2r(« - l)+r{n - 2)+ r{n - 3)- - 4)) (9) The corresponding procedure of linear extrapolation and subsequent averaging is depicted in Fig. 3.12. It should be noted that since rex ,(«) in (7) is already available at interval « - l , i t c a n b e used in (8). I •a O - actual intervals • - linearly extrapolated intervals A - averaged extrapolated intervals r(n-2) J* x(n-A) n-4 n-3 n-2 n-1 Sample number, n Figure 3.12 Computing rt (n) using linear extrapolation and subsequent averaging. Higher-order extrapolation is also possible. For example, the values rex q{n) based on a three-step history and quadratic extrapolation are computed as *•«_, W = 3r(" -1) - 3r(n - 2)+ r{n - 3) Then, the three values of rex q («) are averaged as in (6), to obtain the following: ^ ( " ) = j t « _ , ( » ) + « " « _ , ( ' , - 1 ) + r « ^ ( ' 1 - 2 ) ) (10) ( H ) As with linear extrapolation, r (n) can be expressed in terms of t(n) as 30 Tq (") = - (3T(« -1)+ r(« - 3 ) - 2r(« - 4)+r(n - 5)) (12) A s before, since r „ ,(n) is available at « - l , we can use it in (11). The procedure o f quadratic extrapolation and subsequent averaging is depicted in F ig . 3.13. a o •a 1 • o - actual intervals • - quadratically extrapolated intervals A - averaged extrapolated intervals dk> </"-l) \ |\ r(n-5) t(n-2) ' t(n)i\ V 7 n-5 n-4 n-3 n-2 n-1 Sample number, n Figure 3.13 Computing fq(n) using quadratic extrapolation and subsequent averaging. To compare the performances of the proposed averaging filters, their magnitude and phase responses were calculated, and are superimposed in Fig. 3.14. For completeness, the responses of the 3-step filter as wel l as the 6-step filter, which can be constructed similar to (6), are also shown. A s can be observed, all o f the filters completely reject the undesirable harmonics with frequencies 2TT 4TT of —- and — radians per sample, while perfectly retaining the dc component o f the input signal. 2.5 5 2.0 1 o- 1-5 I 1.0 I 0.5 0 0 1 . - 6 - s t e p a v e r a g i n g • / • • • - \ • - l i n e a r e x t r a p o l a t i o n 3 - s t e p a v e r a g i n g - q u a d r a t i c e x t r a p o l a t i o n \ <**' ' ' < " VV ' i f V'.'1 /.'•' / 0 2^3 4^/3 2n 2K/3 4rc/3 Frequency, <u(rad/sample) Figure 3.14 Magnitude and phase responses of different filters. 31 To compare the performances of the proposed averaging filters during speed transients, the filters were subjected to a linear acceleration assuming the same logic of the Hall sensors. In this test, the same constant speed of 255 rad/s is initially applied to all of the filters. Then, at / = 0.02 s, the speed is linearly ramped with acceleration of 13 -103 rad/s2 until it reaches 320 rad/s at t = 0.025 s, after which the speed is kept constant. The transient responses produced by the considered averaging filters are depicted in Fig. 3.15. To benchmark filters, their performance is compared to the waveform of r(«) produced by the Hall-sensor signals without any filter (ideal case, dashed line). As can be observed in Fig. 3.15, the response of various filters to the ramp test is noticeably different. The slowest response corresponds to the 6-step moving-average filter, which is attributed to its longest memory. The successive improvement is demonstrated by the 3-step filter due to its shorter memory. At the same time, the filters based on linear and quadratic extrapolation both show very close transient responses, with the quadratic extrapolation filter demonstrating a slightly faster action at the beginning and end of the speed ramp. 1.05-10' 0.9-10 - i d e a l Tin! -- • - 6 - s t e p a v e r a g i n g — — - 3-step a v e r a g i n g - l i n e a r e x t r a p o l a t i o n • - q u a d r a t i c e x t r a p o l a t i o n 0.75-10" 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 Time (s) Figure 3.15 Response of different filters to a ramp increase in speed. Once the value f(n) is established using appropriate average filtering, the actual timing for commutating the inverter transistors can be found as follows: W™='(") + f (") (13) where t(n) is the reference switching time, and f{n) may denotefa(n), f,(n) or fq(n). This reference time may be obtained by locking the switching to one of the phases (a phase with the smallest positioning error, i f known [10]), or computed by averaging the switching times of the three phases, as follows: 32 t(n)=-{t, («)+/: («)+/:(«) (14) where is the time of the presently switching phase, and ^(«)and /"(«) are the times extrapolated from the two previous phases, as follows: tl(n)=tt(n-\)+T(n) t"t(n) = t,(n- 2) +2f(n) Here, the subscript "*" may denote phase A, B, or C, respectively. (15) For the purpose of illustration, the computation of the switching time estimates is summarized in Table 3.1 and shown in Fig. 16. Hence, if the most recent switching occurred in phase A, the reference time would be computed as An) = \Un) + t'b{n) + tl{n)) and thus the (« +1)th switching in phase C would occur at t(n)+ f(n) instead of tc(n +1). (16) TABLE 3.1 Switching-Event Time Determination n - 3 n-2 n-\ n n+ 1 A tJ»-V t'Jn-2) . t\(n-\) 'a(») '> + V B sb("-v t"b(n-2) h(n-\) fb(n) t"b(n + V C tc(n-2) t'Jn-\) '"c(") tc(" + U x x(n-3) x(n-2) x(n-\) m X(n + \) 7 T(n-V T(n-2) T(n-\) T(n) T(n + \) ^next sw T(n - 4) + x(n - 4) 7(n-y + f(n-3) 7(n-2) + x(n-2) J(n-l) + X(n-l) T(n) + m % A S B 7(n) <PA 1 th(n-\) x(n) fh(n)\ ± tc(n-2) 2x(n) t''(n) Time (s) Figure 3.16 Switching-event time relationships. 33 3.4 Case Study To test the performance o f the B L D C drive system with the proposed correction technique, four different averaging filters were implemented. In the transient study considered here, the motor was assumed to operate in a steady state with a constant mechanical load o f 0.9 N - m and the motor inverter was fed with 20 V. A t t = 0.2 s, the dc voltage was stepped up to 40 V and the motor was allowed to continue to operate. The resulting transient responses produced by the B L D C motor drive with various averaging filters are shown in Figs. 3.17 and 3.18. For comparison, the transient o f the B L D C drive system controlled without the filter is also given (black solid line). A s can be seen, the increase in applied dc voltage was followed by a significant increase in developed electromagnetic torque and subsequent rapid acceleration o f the motor. 260 I j 180 1 100 0.18 0.2 0.22 0,24 0.26 0.28 6 I f S ^2.5 It -1 0.18 0.2 0.22 0.24 0.26 0.28 Time (s) Figure 3.17 Speed and electromagnetic torque response with 3- and 6-step averaging filters. The transients resulting from the 3- and 6-step averaging filter are compared in F ig . 3.17. A s can be observed in F ig . 3.17, when either of the filters was used, the developed torque had a significant dip following several switching intervals and then recovered. A s expected, the 3-step filter resulted in a smaller dip in torque and a faster recovery time than that o f the 6-step filter, due to the difference in the memory capacities of these two filters. The transient responses produced by the B L D C motor with extrapolating averaging filters are shown in Fig. 3.18. As can be seen, both extrapolating filters performed faster than the 3- and 6-step filters, with the quadratic extrapolation yielding the fastest response among all considered cases. 34 260 Time (s) Figure 3.18 Speed and electromagnetic torque response with extrapolating averaging filters. It should also be noted that all four averaging filters considered resulted in absolutely the same steady state performance, wi th complete balancing o f the phase currents and rejection o f the undesired low-frequency harmonics in torque. However, due to the averaging o f the original H a l l -sensor signals, the corrected balanced operation corresponded to the new firing advance angle, (Pv = <Pv + (<PA +(PB + (Pc)l^ • We also noticed that i f similar transient tests were performed with larger mechanical inertia o f the system (which results in a slower acceleration rate), the performance o f all filters became very similar, with even the slowest 6-step filter giving adequate transient performance. 3.5 Conclusion This paper presented a typical industrial low-precision B L D C motor and explained the phenomena o f unbalanced H a l l sensors. A detailed model o f the considered motor drive has been developed and used to determine the effect of incorrectly placed Ha l l sensors on the resulting phase currents and developed electromagnetic torque. It was shown that unbalanced sensors lead to undesirable low-frequency harmonics in developed torque. Several averaging filters have been proposed to improve steady state and dynamic performance o f such B L D C systems. A very good transient performance, approaching that of a motor with ideally placed Hal l sensors, was achieved using the extrapolating and averaging filters applied to the signals from the original misaligned sensors. 35 3.6 References [1] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems. IEEE Press, Piscataway, NJ, 2002. [2] S. D. Sudhoff, P. C. Krause, "Average-value Model of the Brushless DC 120° Inverter System," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 553-557, 1990. [3] S. D. Sudhoff, P. C. Krause, "Operation Modes of the Brushless DC Motor with a 120° Inverter," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 558-564, 1990. [4] P. L. Chapman, S. D. Sudhoff, C. A. Whitcomb, "Multiple Reference Frame Analysis of Non-sinusoidal Brushless DC Drives," IEEE Transactions on Energy Conversion, Vol. 14, No. 3, pp. 440-446, 1999. [5] P. Pillay, R. Krishnan, "Modeling, simulation, and analysis of permanent-magnet motor drives. Part II. The brushless DC motor drive," IEEE Transactions on Industry Applications, Vol. 25, Iss. 2, March-April 1989, pp. 274-279. [6] W. Brown, "Brushless DC Motor Control Made Easy", Microchip Technology Inc., 2002. Available: http://www.microchip.com [7] Simulink: Dynamic System Simulation for MATLAB, Using Simulink Version 6, The MathWorks Inc., 2006. [8] Automated State Model Generator (ASMG), Reference Manual Version 2, P.C. Krause & Associates, Inc. 2003. [9] J. G. Proakis, D. G. Manolakis, Digital Signal Processing. Prentice Hall, Upper Saddle River, NJ, 1996, p. 235,p.248. [10]N. Samoylenko, Q. Han and J. Jatskevich, 'Balancing Hall-Effect Signals in Low-Precision Brushless DC Motors', In Proc. IEEE Applied Power Electronics Conference (APEC 2007), Feb. 28-Mar. 2, 2007, Anaheim CA, USA, pp 606-611. 36 4 D Y N A M I C P E R F O R M A N C E O F BRUSHLESS D C M O T O R S WITH U N B A L A N C E D H A L L SENSORS 3 4.1 Introduction Brushless dc ( B L D C ) motors are often considered in various electromechanical applications and generally have been investigated quite well in the literature [ l ] - [6] . The techniques used to control the inverter transistors can be placed into two major categories: those that require Hal l sensors [1], [5], [6]; and those that are based on a sensorless approach, for example, that use back emf zero-crossings [6]. A n advantage of the first approach is its relatively simple implementation and reliable operation with variable mechanical loads, even at very low speeds (where sensorless control may not always be effective). The theory and modeling of B L D C motors driven by H a l l sensors have been developed by many researchers under one common assumption - that the Ha l l sensors are placed exactly 120 electrical degrees apart. However, in many low-cost machines, this assumption may not hold true, and the distribution o f relative displacements may in fact be quite significant. A n example o f Hall-sensor placement in the typical industrial motor considered in this paper is illustrated in F ig . 4.1. A s can be seen, the H a l l sensors ( H I , H2 , and H3) are mounted on a P C board placed outside the motor case and react to the magnetic field produced by a permanent magnet tablet attached to the rear end of the motor's shaft. In an ideal case, the axes of the sensors should be 120 degrees apart, which in practice is difficult to achieve with high accuracy. Moreover, the errors in positioning of the sensors may be different for different phases. The dashed axes in F ig . 4.1 correspond to the desired positioning o f the sensors, and the solid lines denote their actual positions. A s can be seen, the absolute error of sensor placement may reach several mechanical degrees, which translates into an even greater error in electrical degrees for machines with a high number of magnetic poles. Although other configurations and/or mounting o f H a l l sensors are also possible in different B L D C machines, the effect of their misalignment leads to similar consequences. In general, insufficiently precise positioning of the Hal l sensors causes unbalanced operation of the motor 3 A version of this chapter has been submitted for publication in IEEE Transactions on Energy Conversion, Manuscript No. TEC-00306-2007. Nikolay Samoylenko, Qiang Han and Juri Jatskevich, 'Dynamic Performance of Brushless DC Motors with Unbalanced Hall Sensors' 37 inverter, with some phase(s) conducting for longer and other phase(s) conducting for shorter time intervals. The resulting unbalance among the phases leads to a number o f adverse phenomena, such as an increase in torque pulsation, vibrations, and acoustic noise, as wel l as reduced overall electromechanical performance. Figure 4.1 Hall-effect sensor placement in a typical BLDC motor. Although there exists a large number of publications on B L D C drives, only a few address the unbalanced Hal l sensors. A misalignment of Ha l l sensors was documented in [7], where the authors investigated a relatively sophisticated (expensive) B L D C motor drive with an advanced observer-based torque ripple mitigation control. The operation of a low-precision B L D C motor with misaligned Hal l sensors was described in [8], where a simple averaging o f the time intervals was proposed to improve the steady state operation. This work was further developed to consider the motor operation during the transients in [9]. A s low-cost/low-precision B L D C motors are now becoming widely available and used in a variety of applications, the misalignment of Ha l l sensors requires more detailed attention. This paper focuses on a typical 3-phase B L D C motor-inverter system, as shown in F ig . 4.2. We present a filtering methodology that can be applied directly to the original Hall-sensor signals to produce a modified set of signals that is used to drive the inverter depicted in F ig . 4.2. The present manuscript extends the work reported by the authors in [9] and makes the following overall contributions: • The paper describes the phenomenon of non-ideal placement of H a l l sensors based on a hardware prototype and a detailed switching model. • We propose a simple but very effective and practical filtering technique to improve the overall performance o f a B L D C motor-drive system with significant unbalance in 38 Hall-sensor positioning. • This paper generalizes the approach of filtering the Ha l l sensor signals presented in [9] and provides the experimental results. We show that the performance o f the B L D C motor with the proposed filters approaches that of a motor with ideally placed Ha l l sensors. • The proposed methodology does not require any additional and/or special circuitry or hardware. Our solution can be implemented (programmed) with a basic motor controller. Modified Hall Actual Hall sensor outputs sensor outputs Figure 4.2Brushless dc motor drive system with filtering of Hall-sensor signals. 4.2 Permanent-magnet BLDC Machine Model 4.2.1 Detailed Model To analyze the impact of unbalanced Hal l sensors on B L D C motor performance, a permanent-magnet synchronous machine ( P M S M ) , shown in F ig . 4.3, is considered here. In F ig . 4.3, H{l,2,3} and H{l,2,3} denote the actual and ideal axes (positions) of the Ha l l sensors, respectively; and <pA, cpB, and cpc denote the absolute errors in sensor placement in electrical degrees. Based on commonly used assumptions, the stator voltage equation may be expressed as follows [ l ]-[4]: 39 vabcs rs^abcs (1) at Where faics = [/aj fbs fcxJ, and f may represent the voltage, current or flux linkage vectors. Also, rs represents the stator resistance matrix. In the case of a motor with non-sinusoidal back emf, the back emf is assumed to be half-wave symmetric and contain spatial harmonics. Therefore, the stator flux linkages and electromagnetic torque may be written as [4]: sin((2n-l)f?r) cs-axis Figure 4.3 Permanent-magnet synchronous machine with unbalanced Hall sensors, where Ls is the stator-phase self-inductance, and A'm is the magnitude of the fundamental component of the PM magnet flux linkage. The coefficients Kn denote the normalized magnitudes of the n'h flux harmonic relative to the fundamental. A detailed model of the system shown in Fig. 4.2 was developed and implemented in MATLAB Simulink [10] using the toolbox [11]. The 120-degree inverter logic was implemented according to the 40 standard table [3], [5], [6]. 4.2.2 Model Verification To study the phenomena o f unbalanced Hall-effect sensors, we tested a batch o f industrial B L D C motors for possible variation in the severity of sensor unbalance parameters among the samples. The parameters o f the motor used in the verification studies presented in this paper are summarized in the Appendix. For the given motor, the absolute sensor positioning errors were determined experimentally to be +0 .8° , - 4 ° , and - 4 ° mechanical degrees for phases A , B , and C, respectively. Although some other motors had better or worse precision, the considered sample was assumed to be sufficiently representative. The measured back emf waveforms for this motor have been included in [9] and are not repeated here due to space limitations. To improve the accuracy of the model, the spatial harmonics according to (2) and (3) were included. The most significant harmonic coefficients are summarized in the Appendix. If desired, additional coefficients could be considered for the detailed model; however, higher-order harmonics were found to be less significant. To demonstrate the effect of unbalanced Hal l sensors, an operating point determined by a mechanical load o f 0.9 N - m is considered. For this study, the motor inverter was supplied with Vdc =40 V, resulting in a speed of 2458 rpm under the given mechanical load.-The measured phase currents were captured and are shown in Fig . 4.4 (top). The experiments were carried out using several commercially available B L D C Hall-sensor-based drivers (Maxon E C Amplif ier D E C 50 and Anaheim Automation M D C 150-050) as wel l as our own prototype driver (see Section V ) , all producing the same results. The simulated phase currents for the same steady state operating condition are shown in F ig . 4.4 (middle). A s can be seen in F ig . 4.4, the detailed model predicts the phase currents very closely and agrees with the measured waveforms. This study confirms the accuracy o f the developed detailed model. A s can be observed in F ig . 4.4 (top and middle), the motor phases are energized for unequal periods of time, and the currents are asymmetrically distorted. Such asymmetrical currents also distort the developed electromagnetic torque. For comparison, the machine operation with ideally placed Ha l l sensors was also simulated, and the resulting phase currents are plotted in F ig . 4.4 (bottom). A s can be seen from the figure, the conduction intervals and current waveforms should be balanced among the phases. Since it is hard to measure actual instantaneous electromagnetic torque in practice, the torque waveforms were predicted using detailed simulations for the two cases: (i) ideal case - the Ha l l 41 sensors are precisely placed, with zero errors; and (ii) the actual case - the H a l l sensors are placed with errors equal to those of the sample motor. The predicted torque waveforms are shown in F ig . 4.5 and the corresponding harmonic spectrums are depicted in F ig . 4.6, wherein a significant difference can be observed. A s can be seen in Figs. 4.5 and 4.6 (ideal case, top), the torque waveform contains very strong harmonics at the frequency o f 984 H z , which corresponds to the six-pulse inverter operation at the given motor speed, and is expected to dominate under normal operation. However, the torque corresponding to the actual case (see Figs. 4.5 and 4.6, bottom) has a much richer spectrum, with two very strong harmonics below 984 H z . These lower harmonics are particularly undesirable as they result in increased mechanical vibration and acoustic noise. The detailed analysis of vibrations and acoustic signatures o f B L D C machines is very important [12]—[14], and in general requires information about the machine's design and possible electromechanical resonances that is beyond the scope of this paper. This paper focuses instead on establishing a methodology by which the B L D C motor operation can be simply restored as close as possible to the ideal case of balanced phase currents, depicted in F ig . 4.4 (bottom), resulting in improved electromagnetic torque (shown in Figs. 4.5 and 4.6 (top)). 2 1 5 o 3 £3 c i o o 3 -15 15 Measured currents - Hall Sensors Unbalanced I 0.035 0.04 0.045 0.05 -15 15 Simulated currents - Hall Sensors Unbalanced I M -15 N 0.055 0.035 0.04 0.045 0.05 0.055 I .1 ! I Simulated currents - Hall Sensors Balanced I i i I i ; 0.035 0.04 0.045 0.05 0.055 Time(s) Figure 4.4 Measured and simulated phase currents. 42 v v w w w v v 0.202 0.204 0.206 0.208 0.21 •S ¥ g O w 0 I Actual case I 0.2 0.202 0.204 0.206 0.208 0.21 Time (s) Figure 4.5 Electromagnetic torque waveforms. f 0.1 1 f 0.05 X 0 f 0.04 1 1-0.02 1 1 1 1 1 Ideal case I 1  1 . 1 1 1 984 1968 2952 3936 4920 5904 X 0 A l l • • I • • I Actual case I 0 984 1968 2952 3936 4920 5904 Frequency (Hz) Figure 4.6 Electromagnetic torque harmonic content. 4.3 Filtering Hall Signals To better understand how to correct the Hall-sensor signals, it is instructive to consider the diagram depicted in F ig . 4.7. Here, the angle <pv denotes a possible delay or advance in firing [1], and <pA, <pB, and cpc are the respective sensor-positioning errors in each phase. When the ideal motor is running, the Ha l l sensors produce square wave signals displaced by exactly 120 electrical degrees relative to each other (see F ig . 4.7, dashed line). Combining all three ideal outputs produces a square wave (see F ig . 4.7 bottom, dashed line) with a period equal to one-third of a Hall-sensor period, which is equal to 60 electrical degrees. 43 When the sensors are shifted from their ideal positions (see F ig . 4.7, solid line), the resulting combined waveform becomes distorted, resulting in non-uniform angular intervals d(n) between two successive switching events. The durations of intervals 9(n) are denoted here by r (« ) . A s can be observed in F ig . 4.7, the rising edge of interval 6{n - 3) and the falling edge of interval 8{n-\) correspond to switching of the same sensor (in this case the sensor o f phase A ) . Therefore, the following holds true: 0{n) = -{e{n-3)+9{n-2)+6{n-\)) (4) which is the average angle between two ideal successive switching events, and is equal to n\3 . This paper presents a methodology to approximate the ideal H a l l signals corresponding to H{l,2,3} by appropriately modifying (filtering) the signals from actual sensors H{l,2,3}. The proposed method works by finding an interval duration f(n) corresponding to d(n) by means of averaging and/or extrapolating the time intervals r(n). Once f(n) is known, it is used for estimating the correct timings for commutating inverter transistors. Hall-sensor outputs L<-H 6r, rad. Hall-sensor outputs combined J v(n-V r(n-2) r(n-\) x(n) r(n+l) >-H H"< Hr* *Jr* H r Time, s. 6(n-l) 9(n-\) 9(n+\) -< > $r, rad. Figure 4.7 Ideal and actual Hall-sensor output signals. For clarity, the sequence r(«) (see F ig . 4.7, bottom) is reproduced in F ig . 4.8 as a discrete-time signal with period N = 3, wherein the samples are the actual values o f r(«). Clearly, the 44 non-uniform values o f r(«) cause undesirable harmonics in phase currents and torque waveforms. The frequency content o f r(n) can be evaluated by using the discrete-time Fourier series (DTFS) [15], so that the signal can be written as r[n)=2_,cke (5) k=0 where the Fourier coefficients {ck}, A: = 0,1,...,7V-1, provide the description o f r(n) in the frequency domain. In our case, the signal r(«) has one zero-frequency component and two 2TT 4TT components with frequencies of — and — radians per sample; these two frequencies should be filtered out. a o •c 9 ... n-2 n-1 n «+l n+2 ... Sample number, n Figure 4.8 Sequence of time intervals r{n) for unbalanced Hall sensors. In this paper, we present a methodology for removing the undesirable harmonics based on filtering the original Ha l l sensor signals. Moreover, to simplify the problem o f designing the required multi-input multi-output ( M M O ) filter (see F ig . 4.2), we propose applying the filtering directly to the sequence r(n) (see Fig . 4.8), which internally reduces the problem to the single-input single-output (SISO) filter. Therefore, it is necessary to filter out the undesirable harmonics in r ( « ) . A n appropriate filter may be constructed using the following general formula: M r{n)= Ybmr{n-m) (6) m = l where M is the order of the filter corresponding to the number of previous points taken into account, and bm are the weighting coefficients that depend on a particular filter realization and its numerical property. Without loss o f generality, in this paper we propose two families of suitable filters: (i) basic averaging filters, and (ii) extrapolating filters, whereas other filters may also be derived based on (6). 45 4.3.1 Basic Averaging Filters In this approach, the coefficients in (6) can be defined as bm=\/M . (7) With this implementation, the 6- and 3-step filters can be represented respectively as \ 6 and ^M^-y^An-m) (8) Tai{n)=^yr(n-m) (9) Here, the subscript " a " denotes this basic averaging procedure. The order o f the filter should be selected with care considering that the undesirable harmonics, in this case — and — , should 3 3 be suppressed. 4.3.2 Extrapolating Filters When the drive system experiences a speed transient, such that r(n) may no longer be periodic, it may be advantageous to consider an extrapolation (prediction) o f samples r(n) to better cope with the acceleration and deceleration of the motor. Let us first consider a linear extrapolation approach as depicted in F ig . 4.9. Here, each subsequent step ,(«) is linearly extrapolated based on a two-step history, as follows: TexJ{n) = 2T{n-\)-r{n-2) (10) To ensure the cancellation of undesirable harmonics, the values r M /(«) are then averaged to yield an analogue to r f l 3 (« ) in (9), as follows: T / ( « ) = j t - „ _ / ( " ) + T „ _ , ( » i - l ) + T o t _ / ( n - 2 ) ) (11) After substituting (10) into (11), the resulting equation for computing F/(«) in terms o f r(«) can be written as F, («) = j (2r(w -1)+ r(« - 2) + r(« - 3 ) - r(n - 4)) (12) which has the form o f (6) and has 4-th order. F ig . 4.9 shows the corresponding procedure for 46 linear extrapolation and subsequent averaging to compute (12). J3 o - actual intervals • - linearly extrapolated intervals A - averaged extrapolated intervals T(n-2) r(n-4) / I rM ,(n-2) i(n) n-4 n-3 n-2 n-1 n Sample number, n Figure 4.9 Computing f) («) using linear extrapolation and subsequent averaging. Higher-order extrapolation is also possible. For example, the procedure of quadratic extrapolation and subsequent averaging is depicted in Fig. 4.10. Using this approach, the values rex q(n) are computed based on a three-step history and quadratic extrapolation as r „ _ , ( « ) = 3 r ( n - l ) - 3 r ( n - 2 ) + r ( # i - 3 ) (13) Then, the three values of rex (n) are averaged as in (9), to obtain the following: r , ( « ) = j ( r „ _ , ( ' » ) + T e t _ , ( « - l ) + r „ _ , ( / i - 2 ) ) As with linear extrapolation, F (n) can be expressed in terms of r(«) as (14) rq (n) = - (3r(n -1)+ r{n - 3)- 2r(n - A)+r{n - 5)) which also has the form of (6) and has 5-th order. 5 o - actual intervals • - quadratically extrapolated intervals A - averaged extrapolated intervals •v r(n-S) 9- I •(n-\)'*\^q(nJ n-5 n-4 n-3 n-2 n-1 * \nl V ; ^ /»; it' (15) Sample number, n Figure 4.10 Computing F («) using quadratic extrapolation and subsequent averaging. 47 4.3.3 Performance of Filters To compare the performances of the proposed averaging filters, their magnitude and phase responses were calculated [8], [9]. The results are superimposed in F ig . 4.11. A s can be observed, 2.71 \K all of the filters completely rejected the undesirable harmonics with frequencies of — and — radians per sample, while perfectly retaining the dc component o f the input signal. Therefore, all o f these filters w i l l achieve the required balancing of the modified Hall-sensor signals when the motor is in a steady state. To compare the performances o f the proposed averaging filters during speed transients, the filters were subjected to a linear acceleration assuming the same logic o f the H a l l sensors. In this test, a constant speed of 255 rad/s was initially applied to all of the filters. Then, at / = 0.02 s, the speed was linearly ramped with an acceleration of 13 10 3 rad/s2 until it reached 320 rad/s at t = 0.025 s, after which the speed was kept constant. The transient responses produced by the considered filters are depicted in F ig . 4.12. To benchmark the filters, their performance was compared to the waveform o f r{n) produced by the Hall-sensor signals without any filter (ideal case, dashed line). A s can be observed in F ig . 4.12, the response of various filters to the ramp test is noticeably different. The slowest response corresponds to the 6-step moving-average filter (8), which is attributed to its, longest memory. The successive improvement is demonstrated by the 3-step filter (9) due to its shorter memory. A t the same time, the filters based on linear and quadratic extrapolation [(12) and (15), respectively] both show very close transient responses, with the quadratic extrapolation filter demonstrating a slightly faster action at the beginning and end of the speed ramp. 4.4 Reference Switching Time Once the value f(n) is established using the appropriate filter, the actual timing for commutating the inverter transistors can be found as follows: t*ext_sw=Kn)+f{") 06) where t(n) is the reference switching time, and f(«) may denote fa6(n), fa3(n), F ; («) or rq (n). For example, this reference time may be obtained by locking the switching to one of the phases (a phase with the smallest positioning error, i f known) [8]. Alternatively, this time may be 48 computed by averaging the switching times of the three phases [9], as follows: ' " ( « ) = j ( ' . ( « ) + ' ; ( « ) + t ' M ) •-"»»»»•••• • linear extrapolation - quadratic extrapolation 1 \ V.'-V \ N \ 2^ /3 4n/3 2fC Frequency, <D(rad/sample) Figure 4.11 Magnitude and phase responses of different filters. 1.05-10' - ideal t(a) - 6-step averaging - 3-step averaging - linear extrapolation - quadratic extrapolation 0.9 10 0.75-10" 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 Time (s) (17) Figure 4.12 Response of different filters to a ramp increase in speed . where t*(n) is the time of the presently switching phase, and t't(n) and ?»'(«) are the times extrapolated from the two previous phases, as follows: (18) t't(n)=tt(n-\)+f(n) t:(n)=t.{n-2)+2f(n) Here, the subscript " * " may denote phase A , B , or C , respectively. For the purposes of illustration, the computation of the switching time estimates is summarized in Table 4.1 and shown in F ig . 4.13. Hence, i f the most recent switching occurred in phase A , the reference time would be computed as 4 9 Kn) = \{ta{n) + fb{n) + t'M) (19) and thus the (« + l) th switching in phase C would occur at t(n)+r(n) instead of tc(n + \). T A B L E 4.1 Switching-Event Time Determination n-3 n-2 n-l n n+ 1 A tJn-V i'a("-y fJn-X) ta(") ?a(n + I) B t'b("-V fb(n-2) ttfn-V w t\(n + \) C t"c(" - V tc(n - 2) t'c(n-\) t"c(n) tc(n + \) x x(n - 3) x(n - 2) f(n - 1) x(n) X(n + \) T T(n-3) T(n-2) T(n-\) T(n) T(n+\) ^next_sw 7(n - 4) + x(n - 4) J(n-3) + f(n-3) J(n - 2) + x(n-2) 7(n-l) + x(n-\) T(n) + m T(n) 1 f 1 1 II =j <b(»-V »ll r(n) ttfn) | • 33 1 c 1 1 tc(n 2t(n) Time (s) Figure 4.13 Switching-event time relationships. 4.5 Implementation and Case Studies In order to evaluate the performance of the proposed averaging filter, it was implemented in both the detailed model and the hardware prototype of the B L D C motor-inverter system. A programmable integrated circuit microcontroller PIC18F2331 [16] was used to allow flexibility in the filter implementation. A popular choice for motor drive applications, this microcontroller is often used for Hall-sensor-driven brushless dc motors [17]. The filters proposed in (8), (9), (12) and (15), in conjunction with (16), were coded inside the section o f the program that is triggered by a hardware interrupt coming from the Hall-sensor readings. This way it is possible to perform all of the necessary filter calculations in a predictable amount o f time (number of instructions) as well as determine the timings o f firing the inverter transistors and schedule the corresponding interrupts. 50 Since all presented filters have memory, their usage imposes conditions on when the filters may be activated. For example, it is not possible to start a motor with the filter enabled, since at the beginning there is no previous history. Also , in the case o f a very fast acceleration/deceleration transient, there potentially may be a need to deactivate the filter for some brief time, thereby defaulting to the existing Ha l l sensors, after which the filter may be enabled again. A simplified block diagram of the motor controller allowing automatic enabling and disabling o f the filter is shown in F ig . 4.14. Here it is assumed that one o f the proposed filters is used. To start the operation, the appropriate registers of the microcontroller have to be initialized. The variable "counter" counts the number o f Hall-sensor transitions, whereas the "threshold" is set to the filter order plus one. After initialization, the controller checks the first IF condition. The purpose of this condition is to ensure that the filter is not used before its memory has stored sufficient data, and the motor starts using the original Ha l l sensor signals for the first several switching transitions. After a sufficient number of transitions, the filter memory is ready and the "counter" variable has been incremented to pass the first IF condition. For increased safety and reliability of the drive, the second IF condition checks to see i f the motor is in any adverse transient by comparing the estimated acceleration/deceleration with some specified acceleration tolerance. If both conditions are satisfied, the control of inverter transistors is performed using the modified (filtered) signals. YES using original Hall sensor signals Switch transistors using modified Hall sensor signals I Figure 4.14 High4evel diagram of the microcontroller including the proposed filter. In the test implementation, the filter could also be enabled or disabled manually. To demonstrate 51 the operation of the proposed filters in steady state, Fig. 4.15 shows a fragment of the measured stator currents corresponding to the dynamometer torque of about 1.4 N m . Here, in the first part of the plot, the filter is disabled and the waveforms are clearly unbalanced - similar to those depicted in Fig. 4.4 (top). The filter is then enabled in the middle of Fig. 4.15, thereafter making the conduction intervals equal and the waveforms balanced among the phases - very similar to Fig. 4.4 (bottom). A similar improvement of the phase currents was observed for every filter considered here at different steady state operating conditions. - 2 0 1 — £ — • — i i—I 0.5 0.508 0.516 Figure 4.15 Measured phase currents without and with the proposed filtering. 4.5.1 Start-up Transient To illustrate the concept of automatic enabling of the filter, we ran experimental start-up studies. To illustrate the performance of the motor in typical working conditions, the motor was mechanically coupled to a dynamometer with a combined inertia of 12 • 10 - 4 N-m-s2, while the inverter was supplied with 20 V DC to avoid over-current operation. For better comparisons among the filters, the initial position of the rotor was approximately aligned to the same reference position. The recorded transients are shown in Fig. 4.16. As can be seen, initially the motor operates with disabled filters producing very similar unbalanced currents. The filters are enabled at different times depending on the filter order, after which balanced operation among the motor phases is maintained. In each case, the motor accelerates following very similar speed trajectories, shown in Fig. 4.16 (bottom), wherein the initial acceleration is around 4.78 103 rad/s2; by the time any filter is ready to be used, the acceleration goes down below 1.0 103 rad/s2. This study demonstrates that the proposed filters do not compromise the startup performance of the drive. 4.5.2 Load-Step Transient To investigate the dynamic performance of the BLDC motor with the proposed filters, we consider a transient cased by changing the dynamometer load. Since the dynamometer is a DC machine, the load change was implemented by changing the load resistor connected to its armature terminals. In this study, the inverter was supplied with 40 V dc, while the effective load was increased from about 20 < 52 0.2 N-m to 1.4 N-m. The transient responses recorded without and with the proposed filtering are shown in Fig. 4.17. As Fig. 4.17 shows, when the filter is disabled (top subplot) the phase currents are unbalanced and spiky, similar to those depicted in Fig. 4.4 (top). 50 a «> -50 50 a o a -50 50 5 -50 1300 a & = 650 ^ filter e n a b l e d r 14 6 - s t e p a v e r a g i n g filter 1 L 1 0.15 0.05 0.15 — r ^ filter e n a b l e d L | l i n e a r e x t r a p o l a t i n g filter | 0.05 0.1 0.15 filter e n a b l e d q u a d r a t i c e x t r a p o l a t i n g filter 0.05 0.1 0.15 4.76103rad/s2 v 0.99-|03rad/s2 ' ^ ^ _ ^ « w f « » ! * 1 - 6 - s t e p a v e r a g i n g filter — - 3 - s l e p a v e r a g i n g filter - l i n e a r e x t r a p o l a t i n g f i l t e r - q u a d r a t i c e x t r a p o l a t i n g filter 0 0.05 0.1 0.15 Time (s) Figure 4.16 Measured start-up transient of BLDC motor. In this study, all the previously described filters resulted in the same transient performance, achieving the desired balancing of the phase currents as shown in Fig. 4.17 (middle). For the 53 given total inertia o f the system and the peak deceleration o f 0.827 103 rad/s 2 as shown in F ig . 4.17 (bottom), even the slowest 6-step filter performed adequately. 0.18 0.24 Time (s) Figure 4.17 Measured transient response due to load change. 4.5.3 V o l t a g e - S t e p T r a n s i e n t To enable faster mechanical transients (similar to those considered in F ig . 4.12) and emulate the motor operation with small inertia, in the following studies the dynamometer was disconnected, leaving the B L D C motor with a bare coupling. Initially, the machine was assumed to run in a steady state fed from 20 V dc with a total mechanical loss torque of about 0.1 N-m. A t / = 0.1 s, the dc voltage was stepped up to 35 V dc, and the motor accelerated and continued to operate. Since in this test direct measurement of speed and/or torque was not possible, both the detailed simulations and the hardware measurements were carried out. The corresponding simulated speed and torque responses are shown in Figs. 4.18 and 4.19. For comparison, the transient o f the B L D C drive system controlled without the filter is also given (black solid line). A s can be seen, the increase in applied dc voltage was followed by a significant increase in developed electromagnetic torque and subsequent rapid acceleration of the motor. For 54 this study, the peak acceleration was found to be 13.5 • 103 rad/s 2. A s was pointed out in Section III C (see Fig . 4.12), the proposed filters w i l l perform differently at very rapid changes o f speed. The transients resulting from the 3- and 6-step averaging filters are compared in F ig . 4.18. A s can be observed in Fig . 4.18, when either o f the filters was used, the 2500 K 1000 - non-altered Hall signals - 3-step averaging - 6-step averaging 0.14 0.14 Figure 4.18 Speed and electromagnetic torque response with 3- and 6-step averaging filters. 25001 . 1 1 , K 1000 - non-altered Hall signals • linear extrapolation - quadratic extrapolation 0.14 non-altered Hall signals linear extrapolation quadratic extrapolation 0.11 0.12 0.13 0.14 Time (s) Figure 4.19 Speed and electromagnetic torque response with extrapolating averaging filters. developed torque had a noticeable dip following several switching intervals, and then recovered. A s expected, the 3-step filter resulted in a smaller dip in torque and a faster recovery time than did the 6-step filter, due to the difference in the memory capacities o f these two filters. The corresponding delays are also noticed in the measured phase currents shown in Fig . 4.20 (first two 55 subplots). The simulated transient responses produced by the B L D C motor with extrapolating averaging filters are shown in F ig . 4.19. A s can be seen, both extrapolating filters performed much faster than the basic moving-average filters, with almost no dip in torque and close to ideal speed response, with the quadratic extrapolation yielding the fastest response among all considered cases. The corresponding measured phase currents shown in F ig . 4.20 (third and fourth subplots) completely agree with this observation, with the quadratic extrapolating filter yielding the best performance. This is an expected result, as the extrapolating filters where shown to cope very well with similar acceleration, as depicted in F ig . 4.12. 40 S3 c i o t-l o § VI 8 .1 a" o i o 1 -40 0.1 40 -40 0.1 40 o a -40 0.1 0.12 0.14 1 | 3-step averaging filter | 0.12 0.14 | linear extrapolating filter | 0.12 0.14 | quadratic extrapolating filter ^  0.14 0.12 Time (s) Figure 4.20 Measured response of phase currents to step in dc voltage. 56 4.5.4 Discussion It should be noted that all four filters described here resulted in absolutely the same steady state performance, with complete balancing of the phase currents and rejection of the undesired low-frequency harmonics in torque, and therefore performance approaching that of the ideally placed Hall sensors. However, due to the averaging of the original Hall-sensor signals, the corrected balanced operation corresponded to the new firing advance angle, (P'v-(Pv + (<PA +<PB + VcV^ • This is a good result since the average of the absolute errors should be smaller that each individual error. In general, changes in firing advance angle cpv affect the static torque-speed characteristic [1, Chap. 6], but small deviations should have minimal effect and the overall result should still be better than using the original unbalanced Hall sensors directly. Large deviations in cpv may result in different operating modes as documented in [3]. As has been observed in the studies of Figs. 4.16 and 4.17, with larger mechanical inertia of the system (which results in a slower acceleration rate), the performance of all filters became very similar, with even the slowest 6-step filter giving adequate transient performance. This approach therefore can be used in a large number of practical electromechanical and servo applications. For the systems with small inertia and/or very fast acceleration/deceleration requirements, the proposed extrapolating filters may offer a good solution. 4.6 Conclusion This paper presented a typical industrial low-precision BLDC motor and explained the phenomena of unbalanced Hall sensors. A detailed model of the considered motor drive has been developed and used to determine the effect of inaccurately placed Hall sensors on the resulting phase currents and developed electromagnetic torque. It was shown that unbalanced sensors lead to undesirable low-frequency harmonics in developed torque. Several filters have been proposed to improve steady-state and dynamic performance of such BLDC machine systems. Detailed simulations and hardware measurements were conducted to support the analysis. A very good transient performance, approaching that of a motor with ideally placed Hall sensors, was achieved using the extrapolating and averaging filters applied to the signals from the original misaligned sensors. 57 4.7 References [I] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems. IEEE Press, Piscataway, NJ, 2002. [2] S. D. Sudhoff, P. C. Krause, "Average-value Model of the Brushless DC 120° Inverter System," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 553-557, 1990. [3] S. D. Sudhoff, P. C. Krause, "Operation Modes of the Brushless DC Motor with a 120° Inverter," IEEE Transactions on Energy Conversion, Vol. 5, No. 3, pp. 558-564, 1990. [4] P. L. Chapman, S. D. Sudhoff, C. A. Whitcomb, "Multiple Reference Frame Analysis of Non-sinusoidal Brushless DC Drives," IEEE Transactions on Energy Conversion, Vol. 14, No. 3, pp. 440^146, 1999. [5] P. Pillay, R. Krishnan, "Modeling, Simulation, and Analysis of Permanent-Magnet Motor Drives. Part II. The brushless DC motor drive," IEEE Transactions on Industry Applications, Vol. 25, Iss. 2, March-April 1989, pp. 274-279. [6] W. Brown, Brushless DC Motor Control Made Easy, Microchip Technology Inc., 2002. Available: http://www.microchip.com [7] P. B. Beccue, S. D. Pekarek, B. J. Deken, A. C. Koenig, "Compensation for Asymmetries and Misalignment in a Hall-Effect Position Observer Used in PMSM Torque-Ripple Control," IEEE Transactions on Industry Applications, Vol. 43, No. 2, pp. 560-570, 2007 [8] N. Samoylenko, Q. Han, J. Jatskevich, "Balancing Hall-Effect Signals in Low-Precision Brushless DC Motors," In Proc. IEEE Applied Power Electronics Conference (APEC 2007), Feb. 28 - Mar. 2, 2007, Anaheim CA, USA, pp. 606-611. [9] N. Samoylenko, Q. Han, J. Jatskevich, "Improving Dynamic Performance of Low-Precision Brushless DC Motors with Unbalanced Hall Sensors," In Proc. IEEE Power Engineering Society General Meeting (PES GM'07), June 24-28, 2007, Tampa FL, USA [10] .Simulink: Dynamic System Simulation for MATLAB, Using Simulink Version 6, The Math Works Inc., 2006. [ I I ] Automated State Model Generator (ASMG), Reference Manual Version 2, PC Krause & Associates, Inc. 2003. (available: www.pcka.com) [12] M. Brackley, C. Pollock, "Analysis and reduction of acoustic noise from a brushless DC drive," IEEE Transactions on Industry Applications, Vol. 36, No. 3, pp. 772—777, 2000. [13] A. Hartman, W. Lorimer, "Undriven Vibrations in Brushless DC Motors," IEEE Transactions on Magnetics, Vol. 37, No. 2, pp. 789-792, 2001. [14] T. Yoon, "Magnetically induced vibration in a permanent-magnet brushless DC motor with symmetric pole-slot configuration," IEEE Transactions on Magnetics, Vol. 41, No. 6, pp. 2173-2179, 2005. 58 [15]J. G. Proakis, D. G. Manolakis, Digital Signal Processing, Prentice Hall, Upper Saddle River, NJ, 1996, p. 248. [16]PIC18F2331/2431/4331/4431 Data Sheet, 28/40/44-Pin Enhanced Flash Microcontrollers with nanoWatt Technology, High Performance PWM and A/D. Microchip Technology Inc., 2003. Available: http://www.microchip.com [17]Padmaraja Yedamale "Brushless DC Motor Control Using PIC18FXX31 MCUs, AN899," Microchip Technology Inc. Available: http://wwl.microchip.com/downloads/en/AppNotes/ 00899a.pdf 59 5 I M P L E M E N T A T I O N To complement the previous Chapters, some o f the important details regarding the implementation o f the proposed methodology are described here. These details have not been covered in any o f the manuscripts primarily due to space limitations imposed on the I E E E papers. In general, more sophisticated control algorithms tent to require significant computational resources and appropriately powerful C P U (e.g., DSP, PIC, and/or controller chip). Although the requirement of high power C P U may be easily accommodated in research and/or industry labs on an experimental basis, it often prevents many potentially very beneficial control solutions from wider use. In this regard, the efficient implementation of proposed filtering approach described in this Chapter requires very small number of instructions. Simplicity o f the algorithm combined with efficient numerical realization allows the proposed approach to be readily implemented even on a very basic microcontroller, which in turn, as we hope, w i l l lead to its wider adoption and use with practical drive systems. Since the computational complexity o f the proposed filtering algorithm is very low, in this thesis we used a programmable integrated circuit microcontroller PIC18F2331. To read the Ha l l sensor signals and to switch the inverter transistors according to the modified/filtered pattern, the algorithm was implemented using an interrupt-based approach [1]. When the filter is disabled, switching of the Ha l l sensors triggers the hardware (input) interrupt service routines (ISR) to perform necessary calculations. In this mode, the transistors are switched according to the logic table and the inverter topology corresponding to the given rotor position. When the filter is enabled, the Ha l l sensors triggers the input ISR as before, but switching o f the inverter transistors is performed at a possibly different time. This transistor switching time is previously calculated during the preceding input ISR and is stored in special temporary register. Based on this time, the so-called output ISR is scheduled to execute which switches the inverter transistors according to the new modified/filtered pattern. Unl ike powerful computer C P U s with 64-bit floating-point units (FPUs), basic microcontrollers have low number of bits (12 to 16), have low memory space and much slower clock (up to 40MHz) . Therefore, to make possible indefinite continuous operation o f the motor drive, it is necessary to eliminate any possibility o f overflow during the ran-time, which is achieved by performing periodic reset of all counters and timers. In our implementation, the timer is reset back to zero at each time the Ha l l signals change and trigger input ISR. Using this approach, the time 60 intervals between the two successive changes of the Ha l l signals denoted by r(n) are readily available as the timer counts between the Hall-sensor transitions. A s shown in Chapters 3, 4 the modified time of the next transistor switching can be calculated as *next_sw ='"(«) + K") = + (!) which is also the time to execute the next (upcoming) output ISR denoted here as t^T(n + \). However, for efficient implementation, it is necessary to relate the time t®J/T(n +1) to the time at which the previous input ISR was called by the Hal l sensors. Denoting the most-recent calling o f the input ISR by t'^{n), the next output ISR time may be expressed as tT{n^) = tg{n)+r-{n) (2) where T c o r r ( n ) is the appropriate correction term. Moreover, since the timer is reset to zero upon invoking of the input ISR, the term also becomes zero. Therefore, the correction term may be calculated as tcorr{n) = t(n)+¥{n) (3) where r(n) dependent on the filter realization and may denote xc°"(n), xc°"(n), rc"rr(«) or rcqorr{n) , for the four considered filters, respectively. This quantity can be computed in a straightforward manner from the available r(n) (see (8), (9), (12) and (15) in Chapter 4.3). The calculation of reference time t(n) needs some clarification since the timer is being reset at each time t!^s(n). According to (17) (see Chapter 4.4) we have t{n) = ±{t.{n)+t:{n) + t:{n)) (4) where t't(n)=tt(n-\)+T(n) t:(n) = t,(n-2)+2f(n) To clarify the time-interval relationships involved in determining correction term rc"rr(n), the waveform produced by combining the Hall-sensor outputs is shown in F i g . 5.1. This figure shows several intervals r(«) as wel l as provides information on derivation of the reference time t (n). Here, the corrected time interval Tcorr{n) depends on the filter and may represent rca°^r{n), TaTr{n)> rf°rr{n) o r Tq°rr{n)> respectively. A s can be seen in F i g . 5.1, the timer is reset at t = tt (n) and the following is valid: 61 r , («) = 0 tt(n-\)=-T(n-\) (6) tt (n - 2) = -r(n - 2)-r(n -1) Thereafter, it becomes possible to determine the required correction term Tcorr(n) depending on the value f(n), which is different for each filter. First, we derive the expression for Tcorr(n) assuming the 3-step averaging filter. With r o 3 ( « ) = — ^ T ( » - m) as the averaged interval duration, the terms in (4) may be represented as * . _ a 3 ( » ) = ' . ( « ) = 0 «3 («) = t.(n-1)+ F a 3 = j (- 2r(/i -1)+ r(« - 2) + r(n - 3)) '."_fl3 (") = t.{n- 2)+ 2 F a 3 («) = I (- r(« -1)- - 2)+ 2r(« - 3)) Based on results in (7) the reference time (4) for this filter is (7) 'o3 (") = J ('._<* (") + C„3 («) + '.*_fl3 («)) =\{~ A " - l) + *(« - 3)) Consequently, the final corrected time interval may be expressed as (8) <T M = ta3 (n) + r f l 3 («) = i (r(n - 2) + 2r(n - 3)) (9) This final equation (9) is implemented inside the input ISR code section o f the microcontroller. A s can be seen, the expression (9) to obtain the corrected time interval for the 3 r d order basic averaging filter is very simple and computationally efficient. 2 1 —I r(n-5) T(n-4) Hall-sensor outputs combined „ T(n-3j input-ISR output-ISR tm(. HS T(n-2) r(n-\) t(rO_ Tcm(n) 7 * " t.(n) I r(n) j t,(n-\) f,(n) t,(n-2) Time, s. 2f(n) f\(n) Figure 5.1 Determination of the corrected time interval r c o r r (n). 62 Similar calculations can be performed for the remaining filters taking into account their respective 1 6 values for r(»). Thus, for the 6-step averaging fdter with fab(n) = —j~\r(n-m) as the averaged 6 , interval duration, the terms t,(n), r j (« ) , and r"(») in (4) can be calculated as t,_a6(n) = tt(n) = 0 K fl6(«) = <.("-l) + fo6(n)=-(-5r(»-l)+r(«-2)+r(n-3)+r(«-4)+r(«-5)+r(»-6)) (10) 6 ?;_fl6(«) = /»(«-2)+2ra6(«) = y(-2r(«-l)-2r(«-2)+2r(«-3)+r(«-4)+r(«-5)+r(«-6)) Therefore, the corresponding reference time becomes F a 6(n) = -(-3r(«-l)-r(«-2) + r(«-3) + r(»-4)+r(«-5)+r(«-6)) (11) 6 The final corrected time interval may be expressed as <'6r («) = U (") + f « 6 M = j (- r(n -1) + x(n - 3) + r (n - 4) + r(n - 5) + r(« - 6)) (12) This final equation (12) is implemented inside the input ISR code section o f the microcontroller. A s can be seen, the expression (12) for the 6th order basic averaging filter is also very simple and computationally efficient. The linear extrapolating filter is considered next. For this filter we have Ti(n) = ^ -(2r(«-l)+r(«-2)+r(«-3)-r(«-4)). The extrapolated switching times become / . _ / ( n ) = / . ( » ) = 0 t'tj{n) = tt{n-l)+T,(n) = j(-r(«-l)+r(«-2)+r(«-3)-r(«-4)) (13) tl_, (n) = t.(n-2)+ 2T, (n) = j (r(n - l)-r(n - 2)+ 2r(n - 3)- 2r(« - 4)) Consequently F,(n) = ^ t . _ , (« )+^_ , (n ) + /:_,(«)) = i(r(/i-3)-r(»-4)) (14) rfrr (n) = I, (n) + r, (n) = j (2r(n -1) + r(n - 2) + 2r(n - 3) - 2r(n - 4)) (15) This final equation (15) is implemented inside the input ISR code section of the microcontroller. As can be seen, the expression (15) for the 4th order linear extrapolating filter is also very simple and computationally efficient. 63 Finally, the quadratic extrapolating filter has r ? ( « ) = i ( 3 r ( « - l ) + r ( n - 3 ) - 2 r ( « - 4 ) + r ( « - 5 ) ) . The corresponding extrapolated times and the reference time are calculated as t,_q(n) = t,(n) = 0 t:_q{n) = tt(n-1)+ fq («) = i (T(n-3)-2r(n-4)+ r(n-5)) (16) tl_q («) = / , ( « - 2)+ 2F? («) = j (3r(« -1)- 3r(« - 2)+ 2r(n - 3)- 4r(« - 4)+ 2r(« - 5)) (») = j ( ' . _ » + («) + (»)) = | M « -1) - *(« - 2) + r(/i - 3) - 2r{n - 4) + r(« - 5)) (17) The corrected time interval for this filter is Tcqo r r („) = F, (w) + F, («) = - j (4r(/i -1) - r (n - 2) + 2r (« - 3) - 4r(» - 4) + 2T(/I - 5)) (18) This final equation (18) is implemented inside the input ISR code section of the microcontroller. As can be seen, the expression (18) for the 5 th order quadratic extrapolating filter is also very efficient. 5.1 References [1] PIC18F2331/2431/4331/4431 Data Sheet, 28/40/44-Pin Enhanced Flash Microcontrollers with nanoWatt Technology, High Performance PWM and A/D. Microchip Technology Inc., 2003. Available: http://www.microchip.com 64 6 S U M M A R Y Brushless dc motors are relatively new compared with traditional induction and synchronous motors. Initially, B L D C motors were introduced in high-end military and special-purpose commercial applications, where the precision and accuracy o f the Hall-sensor positioning was addressed simply through the high-tech manufacturing at appropriate costs. Today, these motors are finding wider application and are produced in higher quantities at manufacturing facilities all-over-the-world including As ia . Although, in general, it is possible to improve the accuracy of the Hall-sensor positioning through increasing the manufacturing precision at increasing costs, this thesis proposed an alternative approach of addressing this problem at the control level, which in turn achieves performance results similar to that of high-precision-manufactured B L D C motors. 6.1 Conclusion In a typical low-precision Hall-sensor-controlled B L D C motor, the H a l l sensors may be significantly misplaced from their ideal positions. This results in non-equal conduction intervals among the phases, which in turn leads to deterioration of the motor performance. In particular, during steady state operation, a typical misplacement of the Ha l l sensors from their axes produces three unequal intervals, which are repeated every half electrical revolution. This thesis proposes an innovative approach of filtering the signals from the existing (unbalanced) Hall-sensors and producing a set of modified (balanced) signals that are used to switch the inverter transistors. Several filters have been proposed at different stages of this project as presented and discussed in Chapters 2 - 4. To simplify the problem of designing the required 3-input 3-output filter, we first reduce the problem to the single-input single-output filter. Two basic averaging filters of 3 r d and 6 t h orders have been designed to completely cancel the undesirable harmonics due to the considered misalignment o f the Hall-sensors. Moreover, to make the proposed approach effective even during rapid electromechanical transients (accelerations/decelerations), the linear and quadratic extrapolating filters of 4 t h and 5 t h orders have been proposed as wel l . The analysis presented in this thesis contains the detailed modeling and simulation as well as the experimental (hardware) verification of the proposed methodology on a typical industrial B L D C motor. The proposed methodology is simple to implement and is shown to be very effective in both steady state operation as wel l as during transients. For very fast accelerations/decelerations, 65 the extrapolating filters are shown to be more effective than the basic averaging filters proposed earlier in the project. Overall, the performance of BLDC motor with significantly misaligned Hall-sensors and the proposed filtering is demonstrated to approach that of the motor with ideally placed sensors. In this regard, I feel that the original objectives set forth in the beginning of this project have been met. 6.2 Future Work The research on filtering the Hall-sensor signals presented in this thesis may be extended in several directions. One important extension can be made based on calculation of the reference time as described in Chapter 4. In particular, it is possible to calculate the set of modified and balanced signals even when one or two Hall sensors have failed. This additional feature of increased fault-tolerance may be particularly useful in applications where the BLDC motors are subjected to harsh conditions but the reliability of the drive is critically important. Another important direction may be to extent the proposed filtering methodology to the rotary position encoders, which are also known to introduce errors in speed measurement due to their low precision. 66 APPENDIX B L D C Parameters: Arrow Precision Motor Co., LTD. , Model 86EMB3S98F, 36 V D C , 210 W, 2000 rpm, 8 poles, r5=0.14 Q, Ls =0.375 mH, Z'm=2\ mV-s; inertia y = 2-10"4 N-m-s2; back emf harmonic coefficients K3 = 0, K5 = 0.042 , and K7 = -0.018. 67 VITA Nikolay Samoylenko (IEEE S'06) received the degree of Engineer in 2002 from the Moscow Aviation Institute, Russia. Currently he is pursuing the M.A.Sc. in Electrical Engineering at the University of British Columbia, Vancouver, BC, Canada. His research interests include electric machines and power electronic systems. 68 

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