ANALYSIS OF VOLTAGE-SOURCEINVERTER-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 H a l l sensors are widely used i n various industrial and electromechanical applications. These machines have often been considered in the literature, under one common assumption - ideal placement o f 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 H a l l sensors and propose methods to eliminate its adverse effects. The studies presented here show that misalignment o f H a l l sensors leads to unbalanced operation o f the inverter and motor phases, which in turn results i n increased low-frequency harmonics in torque ripple, possible acoustic noise, and degradation o f the overall drive performance. Thus, the first manuscript introduces the problem o f the misaligned H a 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 o f the misplaced sensors i n steady state. The second manuscript extends the discussion to the dynamic performance o f 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 o f a motor with perfectly balanced H a l l sensors. ii TABLE OF CONTENTS ii Abstract iii Table of Contents List of Tables v • List of Figures vi Acknowledgements viii Co-Authorship Statement ix 1 2 Introduction 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 Balancing Hall-Effect Signals in L o w - P r e c i s i o n B r u s h l e s s 10 Motors 3 D C 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 Improving Dynamic Performance DC Motors with Unbalanced of Low-Precision Hall S e n s o r s 3.1 Introduction 3.2 Permanent Magnet BLDC Machine Model Brushless 21 21 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 4 References D y n a m i c U n b a l a n c e d 6 of B r u s h l e s s Hall S e n s o r s D C Motors with 37 4.1 Introduction 37 4.2 Permanent-magnet B L D C M a c h i n e M o d e l 39 4.3 5 Performance 36 4.2.1 Detailed M o d e l 39 4.2.2 M o d e l Verification 41 Filtering H a l l Signals 43 4.3.1 Basic Averaging Filters 46 4.3.2 Extrapolating Filters.. 46 4.3.3 Performance of Filters 48 4.4 Reference Switching 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 Start-up Transient 52 4.5.2 Load-Step Transient 52 4.5.3 Voltage-Step Transient 54 4.5.4 Discussion 57 4.6 Conclusion 57 4.7 References r-58 Implementation 60 5.1 64 References S u m m a r y 65 6.1 Conclusion 65 6.2 Future 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 r (n) using quadratic extrapolation and subsequent averaging 31 Figure 3.14 Magnitude and phase responses of different 31 q filters 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 averagingfilters....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 f («) using quadratic extrapolation and subsequent averaging 47 Figure 4.11 Magnitude and phase responses of different 49 ; q filters Figure 4.12 Response of differentfiltersto 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 Figure 4.15 Measured phase currents without and with the proposed filtering filter. 51 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 averagingfilters....55 Figure 4.20 Measured response of phase currents to step in dc voltage Figure 5.1 Determination of the corrected time interval t corr (n) 56 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 D o m m e l and Dr. W i l l i a m 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 H a n for his help, critical remarks and valuable suggestions in our fruitful collaboration. M y profound appreciation goes to all members o f Power Systems L a b , and i n particular to: T o m DeRybel, Marcelo Tomim, Yong Zhang, and Michael Wrinch. Last but not least, I would like to thank my family, m y wife Natalia and son Fedya, as well as my parents and brothers for helping me make my dream come true. viii CO-AUTHORSHIP STATEMENT The manuscripts w h i c h constitute Chapters 2 - 4 o f this thesis were written by me i n co-authorship with Qiang H a h , and Dr. Juri Jatskevich. Although Q i a n g Han and I have common subjects o f 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 o f misaligned H a l l sensors i n the same drive. I have investigated the problem, conducted experiments and simulations, developed improvement filtering techniques, and implemented the proposed methodology i n 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 o f brushless dc motors are similar to those o f a regular brushed dc motors. Examples o f 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 i n areas where other means o f 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 o f an ac synchronous machine. U n l i k e the latter, the rotor of a B L D C motor has an assembly o f permanent magnets w h i c h 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 i n 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 i n the stator windings [5], [6], or the observer-based methods [10], [11]. These methods require significant computational resources and knowledge o f 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 l o w speeds w i t h variable and possibly unknown mechanical loads. Potentially, the high speeds may be problematic as well due to the interaction o f the back-emf-sensing circuitry with noise. The B L D C computational resources drives that use position encoders and therefore also typically require higher are more expensive, w h i c h is not justified i n many practical applications. Perhaps the most common approach to detect the rotor position i n B L D C motors is using the H a l l sensors. This approach is simple, versatile, and it uses only one H a l l sensor per phase. Several typical industrial Hall-sensor-driven B L D C motors are shown i n F i g . 1.1. In the most common three-phase configuration, the three H a l l sensors are displaced b y 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 i n F i g . 1.1 a - b) or the rotor poles, as shown i n F i g . 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 o f the motor and dividing it into six sectors. Therefore, the rotor position is known within these six sectors. T h i s 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 o f a Hall-sensor-driven B L D C motor is limited only by computational capacity o f the motor controller and the mechanical strength o f 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. W i t h most industrial ready-to-use drivers, the Hall-sensor signals are simply used as inputs. There is no need to prepare motor i n 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 i n a typical B L D C motor may be either trapezoidal or sinusoidal [14], [16], whereupon the particular type depends on the physical construction o f 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, w h i c h 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 F i g . 1.1 d)) or be placed on a circular printed circuit board ( P C B ) that is mechanically fixed to the enclosure o f the motor. T w o examples o f the P C B mounting are shown i n 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 o f 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 o f the H a l l sensors from their ideal positions may arise due to manufacturing imprecision. Figure 1.4 Hall-sensor placement in an Arrow Precision motor. Figure 1.5 Hall-sensor placement in a Maxon 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 w h i c h the H a l l sensors change their states. In this research, we have experimented with several available B L D C motors i n our lab and found that very noticeable positioning errors may exist i n many sample machines. The corresponding measured absolute errors in each phase i n mechanical degrees are summarized i n Table 1.1. TABLE 1.1 Distribution of Absolute Misplacements Among Sample Motors (deg.) ^lotor\^^hase^ A B C A r r o w Precision #1 0.8 0.0 3.6 A r r o w Precision #2 2.1 1.7 2.3 A r r o w Precision #3 3.3 4.7 1.2 A r r o w Precision #4 -0.3 0.2 -1.0 A r r o w Precision #5 -0.7 2.7 4.0 A r r o w Precision #6 4.1 -1.2 0.6 A r r o w Precision #7 2.4 -0.3 -0.3 A r r o w Precision #8 2.8 -1.9 1.2 A r r o w Precision #9 0.8 -1.8 4.0 A r r o w Precision #10 4.1 -0.6 2.3 A r r o w Precision #11 2.3 -0.6 2.6 A r r o w Precision #12 1.2 -0.7 3.4 A r r o w Precision #13 0.8 -4.0 -4.0 maxon motor 0.7 0.6 0.7 A s can be seen i n Table 1.1, in some cases, the positioning errors may exceed 4 mechanical degrees. A m o n g the motors presented in Table 1.1, the low-precision A r r o w Precision motors are 8-pole machines, i n w h i c h case the resulting error expressed i n electrical degrees is 4 times greater and may constitute a very significant portion o f the 120-degree conduction interval. The M a x o n motor is a 2-pole machine that is manufactured with much higher precision. Moreover, since this machine has only 2 magnetic poles, the errors i n 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 o f B L D C machines [14], [15], [17], [19], several assumptions are usually made. Some o f such assumptions include absence o f saturation, no cogging torque, and ideal positioning o f H a l l sensors. Although from our communications with industry and researchers in this area it appears that the problem o f 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 o f B L D C motors. Consequently, until the study described i n [19] and publications included i n this thesis, the problem o f misplaced sensors has not been addressed. In [19], the authors describe and document the Hall-sensor alignment problem that exists even i n 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 H a l l sensors is different conduction intervals among the phases, which results i n non-uniform phase currents. This, i n 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 i n 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 o f misplaced H a l l sensors not only from a technical point o f view, but also from an economical standpoint. The ideal solution must also be practical and should not require disassembly on any o f 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 H a 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 i n 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 o f 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 o f the low-level interrupt-service-routine to perform the minimal amount o f 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. 33 Conf. on Decision and Control, pp. 1844-1849, December; 1994. rd 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 BALANCING HALL-EFFECT BRUSHLESS D C M O T O R S 2.1 SIGNALS IN LOW-PRECISION 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 Hall 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 o f the first approach is a relatively simple implementation and reliable operation with variable mechanical loads even at l o w 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 H a l l sensors are placed exactly 120 electrical degrees apart on the circumference o f the rotor. However, i n low-cost motors this assumption may not hold true, and the distribution o f 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 o f H a l l sensor placement i n 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. 22 Annual IEEE Applied Power Electronics Conference (APEC 2007), Feb. 28 nd Mar. 2,2007, Anaheim CA, USA, pp 606 - 611. 10 Figure 2.2 Hall-sensor placement assembly. In this type o f B L D C machine, the sensors are mounted on a circular board outside the machine's main case, as depicted i n F i g . 2.1. To produce the rotor-position signals, the H a l l sensors react to the magnetic field o f the permanent magnet ( P M ) tablet that is mounted on the back o f the motor's shaft, as shown i n F i g . 2.2. In an ideal situation, the axes o f 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 i n positioning o f the sensors may be different for different phases. Moreover, the actual positioning errors i n 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, w h i c h is a sample industrial B L D C motor driven by a 120-degrees inverter. To analyze the effect o f H a 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 H a l l sensor positioning. 2.2 BLDC Machine Model To analyze the impact o f unbalanced H a l 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 i g . 2.3. The detailed model described herein is similar to that considered in [ l ] - [ 4 ] , except the H a 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 o f the sensors. Based on the commonly used assumptions, the stator voltage equation may be expressed as 'abcs where f abcs = \f as f bs f J cs , f s abcs dk abcs (1) dt may represent voltage, current or flux linkage vectors, and r s represents the stator resistance matrix. I f the back emf is half-wave symmetric and contains spatial harmonics, the stator-flux linkage equations may be written as [4] sin((2n-l)9 ) r abcs~ sin L i bcs s a sin where L s (2) In {2n-\U is the stator-phase self-inductance, and X' r m + is the magnitude o f the fundamental component o f the P M magnet flux linkage. A l s o , the coefficients K n the represent the magnitude o f 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 o f commercial B L D C motors and evaluated them for variation o f parameters among the samples. The parameters o f one sample machine considered i n 12 this paper are summarized i n the Appendix A . A s can be seen i n F i g . 2.2, the sensors are mounted outside the motor case and are switched by the field o f 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 K n was obtained by taking the measured back emf waveform shown in F i g . 2.4 (top) and extracting the Fourier series coefficients. The harmonic coefficients were found to be K = 0, K = 0.042 , z and K 1 5 =-0.018. If desired, more coefficients could be extracted and used i n a detailed model; however, the simulated back emf waveform depicted i n F i g . 2.4 (bottom) with just these harmonics was considered sufficient for the studies i n this paper. 30 t/ ^-^ : | Measured BEMF | 0 8 -30 0.05 30 > "a i i i 0.0525 0.055 0.0575 0.06 0.0625 0.0525 0.055 0.0575 Time, s. 0.06 0.0625 o -30 0.05 Figure 2.4 Measured and simulated phase a back emf. To demonstrate motor operation with unbalanced H a 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 V = 40V . A mechanical load o f Q.9N • m was applied, w h i c h resulted i n a speed dc of 2458 rpm. The measured phase currents were captured and are shown in F i g . 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 H a l l sensors, the relative unbalance o f the sensors results i n some phase(s) having shorter conduction intervals and other phase(s) having longer conduction intervals. 13 -15 -15 0.035 0.04 0.035 0.04 0.045 0.05 0.055 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 well as the misalignment o f the H a 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 i g . 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 i g . 2.6 shows the H a 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 o f discussion. Here <p , <p , and cp A B c denote the absolute error in displacements o f the H a 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 o f generality, cp A is assumed here to be negligible. The angle <p denotes a possible advance i n transistor firing [ 1 ] . v A waveform produced by adding all three output signals is shown on the bottom o f F i g . 2.6. I f the H a l l sensors are placed exactly 120 electrical degrees apart, this combined waveform w i l l yield a square wave with the period o f 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 i n the distortion o f 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 6 , rad. r Hall sensor outputs combined T(n-V T(n-2) r(n-l) t(n) r(n+l) J Time, s. 8(n-3) 6(n-2) e(n-\) 6 , rad. e(n) r 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 i g . 2.6, it may be observed that since the rising edge o f interval 6>(«-3) and the falling edge o f interval &(n-\) correspond to switching o f 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 H a l l sensor signal represents a 180-degree square wave, we have 0 (n) = ~ . If the combined waveform o f F i g . 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). U s i n g these time intervals, the average speed observed over the n th interval can be defined as ( n ) = ^ . Then, the average speed over three successive intervals can be expressed as ( m r T(n) r(n) (4) where the average time f ( « ) evaluated over three successive intervals is calculated as F(«) = - ( r ( « - 3 ) + r ( « - 2 ) + r ( « - l ) ) . (5) 15 Effectively, the unequal angles &(n) and intervals r(n) introduce low-frequency harmonics (sub-harmonics) i n the combined square wave (and i n 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 i g . 2.6, bottom). The averaging action o f the filter is depicted i n F i g . 2.7, where the values o f r ( « - 3 ) , r(n-2), and r(n-i) are different due to H a l l sensor misalignment. fjij r(n) ii (!) Q T(n-2) x(n) a r(n-l) 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 i n a steady state, given that each H a l 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 2Z mA - ) b n m (°) m=l where M is the order o f the filter that corresponds to the number o f previous points being averaged, and b m = \/M [9]. Although the order o f 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 H a l l sensor signals. Sixth-order is also possible but w i l l result in slower response. Since the impulse response o f the third order filter is «">-{i,i,i} co 16 it follows that -JO) ') (8) where m denotes the frequency of a discrete-time signal in radians per sample. Hence -ljm e -2ja> +e + e -jm (9) 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) = t ,if this instance corresponds to the change A 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 i n 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 H a l l signals on the torque, the calculated torque for the perfectly balanced (ideal case) and unbalanced (actual case) H a l l sensors are shown i n F i g . 2.9. These torque waveforms correspond to the same operating point at 2458 rpm and a mechanical load o f Q.9N-m. The corresponding harmonic content is shown i n F i g . 2.10 for each case, respectively. A s can be observed i n F i g . 2.10, the unbalance o f sensors results i n very significant low-frequency harmonics i n torque. These harmonics are responsible for increased vibrations, noise, and reduced motor performance. YYYYYYYYYV '0.08 1.5 0.082 I Actual case I 0.084 0.086 I , 0.084 0.086 0.088 0.09 0.088 0.09 S • 0.08 0.082 Time, s. Figure 2.9 Torque waveforms. 1 Ideal case 1 I 0.5 984 3 1968 II 2952 I 3936 11 4920 1 5904 1 1 Actual easel -a 3 •5. 4 0.5 - I I IIIIII,.III 984 1968 2952 11.1 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 o f the filter operation is shown i n Figs. 2.11 and 2.12. In particular, the considered B L D C motor operates in a steady state with V dc = 40V and mechanical load 0.9N • m . Initially, the filter is disabled, which results i n highly distorted phase current and torque. These results are similar to those shown i n Fig. 2.5 (bottom) and F i g . 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 i n F i g . 2.11 is similar to F i g . 2.9 (top). The spectrum o f torque harmonics for the case with enabled filter is depicted in Fig. 2.12, w h i c h is very close to the ideal case o f Fig. 2.10 (top). 0.13 0.13 Figure 2.11 Phase current and torque waveforms. 0.5 984 1968 i l 2952 3936 Frequency, Hz. 4920 5904 Figure 2.12 Torque harmonic content with filter. 19 2.5 Conclusion This paper presents a method o f improving characteristics o f low-precision brushless dc motors with misaligned H a l l sensors. The proposed technique is based on filtering the H a l l sensor output signals to average out the effect o f 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 o f 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 H a 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 DYNAMIC PERFORMANCE O F LOW-PRECISION BRUSHLESS SENSORS 3.1 DC MOTORS WITH UNBALANCED HALL 2 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 H a 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 H a l l sensors have been developed by many researchers under one common assumption - that the H a 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 i n the typical industrial motor considered i n this paper is illustrated i n F i g . 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 H a 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 o f the motor's shaft. The dashed axes in F i g . 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 o f sensor placement may reach several mechanical degrees. The insufficiently precise positioning o f the H a l l sensors causes unbalanced operation o f 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 i n F i g . 3.1) and a detailed model o f a B L D C drive system with unbalanced H a l l sensors. The motor is assumed to be driven using a typical 3-phase inverter as shown in F i g . 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 o f signals that is used to drive the inverter depicted i n F i g . 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 o f the B L D C machine with ideally placed H a l l sensors. The proposed method is shown to improve performance o f B L D C motor drives with inaccurately positioned H a 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 Hall sensors on B L D C motor performance, a permanent-magnet synchronous machine ( P M S M ) shown i n F i g . 3.3 is considered here. In F i g . 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 cp , <p , and cp denote the absolute errors in sensor placement. A B c H1'H1 ij-axis < 7 a x i s Figure 3.3 Permanent-magnet synchronous machine. Based on commonly used assumptions, the stator voltage equation may be expressed as follows [1H4]: dk abcs abcs where f abcs Also, r s = [f as f bs f] cs (1) dt , and f may represent the voltage, current or flux linkage vectors. represents the stator resistance matrix. In the case o f a motor with non-sinusoidal back emf, the back e m f 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)9 ) R ^•abcs ~ Ls*abcs + ^m sin (2) n=\ sin 23 cos((2n-l)c? ) r (3) COS 4 n=l COS where L s is the stator-phase self-inductance, and X m is the magnitude of the fundamental component of the P M magnet flux linkage. The coefficients K n the n th denote the normalized magnitudes of flux harmonic relative to the fundamental. The detailed model o f the system has been developed and implemented i n 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 o f industrial B L D C motors for possible variation i n the severity o f sensor unbalance parameters among the samples. The parameters o f the motor used i n the verification studies presented i n this paper are summarized i n 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 e m f waveforms are depicted i n Figs. 3.4 and 3.5 (top). 0.0525 0.055 0.0575 0.06 0.0625 Time (s) Figure 3.4 Measured line-to-line back emf at speed 2458 rpm. To improve the accuracy o f the model, the spatial harmonics according to (2) and (3) were included. The harmonic amplitudes K n were obtained by taking the measured back emf waveform shown i n F i g . 3.5 (top) and extracting the Fourier series coefficients. The most significant harmonic coefficients are summarized i n the Appendix. The measured and simulated 24 emf waveforms are compared i n F i g . 3.5, which show a very good match. I f 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 11 0.0575 ! 0.06 0.0625 0.06 0.0625 ! I Simulated BEMF 1 o a -30 0.05 0.0525 0.055 0.0575 Time (s) Figure 3.5 Measured and simulated line-to-line back emf at speed 2458 rpm. To demonstrate the effect o f unbalanced H a l 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 V =40 V , resulting i n a speed o f 2458 rpm under the given mechanical load. The measured and dc simulated phase currents for the resulting steady state operating condition are shown i n F i g . 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 o f the developed detailed model. A s can be observed in F i g . 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 i n practice, the torque waveforms were predicted using detailed simulations for the two cases: (i) ideal case - the H a 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 o f the sample motor. The predicted torque waveforms are shown in F i g . 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 i g . 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 , w h i c h 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 i n increased mechanical vibration and acoustic noise. ^ 15 I Measured currentin 1 J I CO < 1l 1 -15 0.035 15 M 1 0.04 0.045 l 0.05 1 0.055 Is mulated currcrits | Is ^1 a g i o fe s -15 0.035 0.04 0.045 0.05 0.055 Time (s) Figure 3.6 Measured and simulated phase currents. 1 Actual case 1 o w 0 0.2 1 0.202 11 0.204 0.206 0.208 0.21 Time (s) Figure 3.7 Electromagnetic torque waveforms. 26 ? 0.1 1 11 1 1 11 1 1 Ideal case I 1 •0.05 S3 0 I" 0.04 984 1 11 1968 2952 1 3936 .1 4920 5904 I Actual case I •0.02 X 0 0 984 III _1 1968 • I 2952 I I 1 • I 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 i g . 3.9. Here, the angle cp denotes a possible delay or advance i n firing [1], v and <p , cp , and cp A B c are the respective sensor-positioning errors i n each phase. W h e n the ideal motor is running, the H a l l sensors produce square wave signals displaced by exactly 120 electrical degrees relative to each other (see F i g . 3.9, dashed line). C o m b i n i n g all three ideal outputs produces a square wave (see F i g . 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 i g . 3.9, solid line), the resulting combined waveform becomes distorted, resulting i n 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 i g . 3.9, the rising edge o f interval d(n-3) 6{n-\) and the falling edge o f interval correspond to switching o f 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 9 , rad. r Hall-sensor outputs combined T(n-2) r(n) t r(n+l) r Time, s. n-3) 8(n-2) 6(n-\) e(n) 0(n+\) 9 , rad. r 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 i n r ( « ) . ... 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(»)=2V " 'M where the Fourier coefficients {c } , k = 0,1,...,N-l k frequency domain. In our case, the signal t(n) 2.7T provide the description of z(n) in the has one zero-frequency component and two AK components with frequencies of - j - and filtered out. defined as (5) ,A radians per sample. These two frequencies should be To accomplish this, a simple moving-averaging of the three previous samples may be *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 — 3 3 are filtered out, as 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 r ex (7) , (n) are then averaged to yield an analogue to f («) in (6), as follows: a i (") = J («•«_/ (")+ T f a J z {n-\)+ exJ (n - 2)) (8) The resulting equation for computing f, (n) in terms of r(«) can be written as 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 r ex ,(«) in (7) is already available at interval «-l,itcanbe used in (8). O - actual intervals • - linearly extrapolated intervals A - averaged extrapolated intervals r(n-2) J* I x(n-A) •a n-4 n-3 n-2 n-1 Sample number, n Figure 3.12 Computing r (n) using linear extrapolation and subsequent averaging. t Higher-order extrapolation is also possible. For example, the values r ex {n) q based on a three-step history and quadratic extrapolation are computed as *•«_, W = (" -1) - 3r(n - 2)+ r{n - 3) 3r Then, the three values of r ex q (10) («) are averaged as in (6), to obtain the following: ^(") = j t « _ , ( » ) + « " « _ , ( ' - ) « ^ ( ' - ) ) , 1 + r 1 2 As with linear extrapolation, r (n) can be expressed in terms of t(n) (H) as 30 (") = - (3T(« -1)+ r(« - 3 ) - 2r(« - 4)+r(n - 5)) T q A s before, since r „ ,(n) (12) is available at « - l , we can use it i n (11). The procedure o f quadratic extrapolation and subsequent averaging is depicted in F i g . 3.13. dk> </"-l) o - actual intervals • - quadratically extrapolated intervals A - averaged extrapolated intervals a o |\ r(n-5) \ (n-2) ' t •a 1 t(n)i\ • n-5 n-4 n-3 n-1 n-2 V 7 Sample number, n Figure 3.13 Computing f (n) using quadratic extrapolation and subsequent averaging. q To compare the performances o f the proposed averaging filters, their magnitude and phase responses were calculated, and are superimposed in F i g . 3.14. For completeness, the responses o f the 3-step filter as w e l 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 - 6-step averaging 5 2.0 3-step a v e r a g i n g 1 o- 1-5 - linear extrapolation • / • • • - \ • - quadratic extrapolation \ <**' I 1.0 I 0.5 00 ' ' < " VV ' i f V'.' /.'•' 1 2^3 4^/3 2K/3 4rc/3 / 2n 0 1 . 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 -10 rad/s until it reaches 320 rad/s at 3 2 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' - i d e a l Tin! -- • - 6-step a v e r a g i n g - 3-step a v e r a g i n g — — - 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) 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™='(") (") (13) +f where t(n) is the reference switching time, and f{n) may denotef (n), a f,(n) or f (n). This q 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, («)+/: where («)+/:(«) (14) is the time of the presently switching phase, and ^(«)and /"(«) are the times extrapolated from the two previous phases, as follows: tl(n)=t (n-\)+T(n) t t" (n) = t,(n- (15) 2) +2f(n) t Here, the subscript "*" may denote phase A, B, or C, respectively. 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' {n) tl{n)) + b (16) + and thus the (« +1)th switching in phase C would occur at t(n)+ f(n) instead of t (n +1). c TABLE 3.1 Switching-Event Time Determination n n-\ n-2 n-3 A tJ»-V t'Jn-2) . t\(n-\) B s ("-v t"b(n-2) x x(n-3) x(n-2) h(n-\) t'Jn-\) x(n-\) 7 T(n-V T(n-2) T(n-\) b t (n-2) C c ^next swT(n - 4) + x(n - 4) n+ 1 'a(») '> + V f (n) t"b(n + '"c(") t (" + U b V c m X(n + \) T(n) T(n + \) 7(n-y + 7(n-2) + J(n-l) + T(n) + f(n-3) x(n-2) X(n-l) m 7(n) <PA 1 % A S B t (n-\) h x(n) f (n)\ h ± t (n-2) c 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 i n 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. I j 1 260 180 100 0.18 6 0.2 0.22 0,24 0.26 0.28 If S ^2.5 It -1 0.18 0.2 0.22 0.24 Time (s) 0.26 0.28 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 i g . 3.17. A s can be observed i n F i g . 3.17, when either o f 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 i n a smaller dip i n torque and a faster recovery time than that o f the 6-step filter, due to the difference i n the memory capacities o f these two filters. The transient responses produced by the B L D C motor with extrapolating averaging filters are shown in Fig. 3.18. A s 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, w i t h 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 i n 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 o f incorrectly placed H a 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 o f a motor with ideally placed H a l 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 DYNAMIC PERFORMANCE OF BRUSHLESS WITH U N B A L A N C E D H A L L SENSORS 4.1 DC MOTORS 3 Introduction Brushless dc ( B L D C ) motors are often considered in various electromechanical applications and generally have been investigated quite well i n the literature [ l ] - [ 6 ] . The techniques used to control the inverter transistors can be placed into two major categories: those that require H a 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 o f the first approach is its relatively simple implementation and reliable operation with variable mechanical loads, even at very l o w speeds (where sensorless control may not always be effective). The theory and modeling o f B L D C motors driven by H a l l sensors have been developed by many researchers under one common assumption - that the H a 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 i n this paper is illustrated in F i g . 4.1. A s can be seen, the H a l l sensors ( H I , H 2 , and H 3 ) 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 o f the motor's shaft. In an ideal case, the axes o f the sensors should be 120 degrees apart, which i n practice is difficult to achieve with high accuracy. Moreover, the errors i n positioning o f the sensors may be different for different phases. The dashed axes in F i g . 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 o f sensor placement may reach several mechanical degrees, which translates into an even greater error in electrical degrees for machines with a high number o f magnetic poles. Although other configurations and/or mounting o f H a l l sensors are also possible i n different B L D C machines, the effect o f their misalignment leads to similar consequences. In general, insufficiently precise positioning o f the Hall sensors causes unbalanced operation o f the motor 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' 3 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 i n torque pulsation, vibrations, and acoustic noise, as w e l l as reduced overall electromechanical performance. Figure 4.1 Hall-effect sensor placement in a typical BLDC motor. Although there exists a large number o f publications on B L D C drives, only a few address the unbalanced H a l l sensors. A misalignment o f H a l l sensors was documented i n [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 o f a low-precision B L D C motor with misaligned H a l 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 i n a variety of applications, the misalignment o f H a 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 i g . 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 i g . 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 o f non-ideal placement o f 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 of a B L D C motor-drive system with significant unbalance in 38 Hall-sensor positioning. • This paper generalizes the approach o f filtering the H a 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 o f a motor with ideally placed H a 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 sensor outputs Actual Hall 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 o f unbalanced Hall sensors on B L D C motor performance, a permanent-magnet synchronous machine ( P M S M ) , shown in F i g . 4.3, is considered here. In F i g . 4.3, 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 <p , cp , and cp A B c 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 abcs v Where f aics Also, r = [/ aj f bs f J, cx s^abcs (1) r at and f may represent the voltage, current or flux linkage vectors. represents the stator resistance matrix. In the case of a motor with non-sinusoidal back s 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 L s is the stator-phase self-inductance, and A' component of the PM magnet flux linkage. The coefficients the n' h m is the magnitude of the fundamental K n denote the normalized magnitudes of flux harmonic relative to the fundamental. A detailed model of the system shown in Fig. 4.2 was developed and implemented in M A T L A B 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 i n the severity o f sensor unbalance parameters among the samples. The parameters o f the motor used i n the verification studies presented i n 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 o f 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 o f unbalanced H a l 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 V dc =40 V , resulting i n 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 Amplifier D E C 50 and A n a h e i m Automation M D C 150-050) as well 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 i g . 4.4 (middle). A s can be seen in F i g . 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 i n F i g . 4.4 (top and middle), 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. For comparison, the machine operation with ideally placed H a l l sensors was also simulated, and the resulting phase currents are plotted in F i g . 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 H a 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 i g . 4.5 and the corresponding harmonic spectrums are depicted i n F i g . 4.6, wherein a significant difference can be observed. A s can be seen i n Figs. 4.5 and 4.6 (ideal case, top), the torque waveform contains very strong harmonics at the frequency o f 984 H z , w h i c h 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 i n increased mechanical vibration and acoustic noise. The detailed analysis o f vibrations and acoustic signatures o f B L D C machines is very important [12]—[14], and i n general requires information about the machine's design and possible electromechanical resonances that is beyond the scope o f this paper. This paper focuses instead on establishing a methodology b y which the B L D C motor operation can be simply restored as close as possible to the ideal case o f balanced phase currents, depicted in F i g . 4.4 (bottom), resulting i n improved electromagnetic torque (shown i n Figs. 4.5 and 4.6 (top)). 2 15 Measured currents - Hall Sensors Unbalanced I N -15 15 0.035 0.04 0.045 0.05 0.055 Simulated currents - Hall Sensors Unbalanced I M o 3 -15 15 0.035 0.04 0.045 0.05 0.055 I .1 ! I Simulated currents - Hall Sensors Balanced I £3 c i o o 3 -15 i i I 0.035 0.04 0.045 i 0.05 ; 0.055 Time(s) Figure 4.4 Measured and simulated phase currents. 42 vvwwwvv 0.202 0.204 0.206 0.208 0.21 0.204 0.206 0.208 0.21 I Actual case I •S ¥ g O w 00.2 0.202 Time (s) Figure 4.5 Electromagnetic torque waveforms. f 0.1 1 1 1 1 1 Ideal case I 1 f 0.05 X 0 984 11 1968 1 .1 2952 3936 11 4920 5904 f 0.04 I Actual case I 1 1-0.02 X 0 0 984 A l l 1968 • • I • • 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 i g . 4.7. Here, the angle <p denotes a possible delay or advance i n firing [1], v and <p , <p , and cp A B c are the respective sensor-positioning errors i n each phase. W h e n the ideal motor is running, the H a l l sensors produce square wave signals displaced by exactly 120 electrical degrees relative to each other (see F i g . 4.7, dashed line). C o m b i n i n g all three ideal outputs produces a square wave (see F i g . 4.7 bottom, dashed line) with a period equal to one-third o f a Hall-sensor period, w h i c h is equal to 60 electrical degrees. 43 When the sensors are shifted from their ideal positions (see F i g . 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 o f intervals 9(n) are denoted here by r ( « ) . A s can be observed in F i g . 4.7, the rising edge o f interval 6{n - 3) and the falling edge o f interval 8{n-\) correspond to switching o f the same sensor (in this case the sensor o f phase A ) . Therefore, the following holds true: -{e{n-3) 9{n-2) 6{n-\)) 0{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} b y appropriately modifying (filtering) the signals from actual sensors H{l,2,3}. The proposed method works b y 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 <-H L 6 , rad. r Hall-sensor outputs combined v(n-V r(n-2) >-H H"< r(n-\) x(n) Hr* * r* J r(n+l) H J r Time, s. 6(n-l) 9(n-\) 9(n+\) -< > $ , rad. r Figure 4.7 Ideal and actual Hall-sensor output signals. For clarity, the sequence r(«) (see F i g . 4.7, bottom) is reproduced i n F i g . 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 i n phase currents and torque waveforms. The frequency content o f r(n) can be evaluated b y using the discrete-time Fourier series ( D T F S ) [15], so that the signal can be written as r[n)=2_, k k=0 c where the Fourier coefficients {c }, k (5) e A: = 0,1,...,7V-1, provide the description o f r(n) i n the frequency domain. In our case, the signal r ( « ) has one zero-frequency component and two 2TT 4TT components with frequencies o f — and — radians per sample; these two frequencies should be filtered out. a o •c 9 ... n-2 n-1 «+l n+2 ... n 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 H a l l sensor signals. Moreover, to simplify the problem o f designing the required multi-input multi-output ( M M O ) filter (see F i g . 4.2), we propose applying the filtering directly to the sequence r(n) (see F i g . 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 i n r ( « ) . A n appropriate filter may be constructed using the following general formula: M r{n)= Yb r{n-m) m (6) m = l where M is the order o f the filter corresponding to the number o f previous points taken into account, and b m are the weighting coefficients that depend on a particular filter realization and its numerical property. Without loss o f generality, i n this paper w e propose t w o families o f 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 i n (6) can be defined as b =\/M (7) . m W i t h this implementation, the 6- and 3-step filters can be represented respectively as \ ^M^-y^An-m) (8) 6 and 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, i n this case — 3 and — , should 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 o f the motor. Let us first consider a linear extrapolation approach as depicted i n F i g . 4.9. Here, each subsequent step , ( « ) is linearly extrapolated based on a two-step history, as follows: T {n) exJ = 2T{n-\)-r{n-2) (10) To ensure the cancellation o f undesirable harmonics, the values r M / ( « ) are then averaged to _ (n-2)) (11) yield an analogue to r ( « ) i n (9), as follows: fl3 T/(«) = jt-„_/(")+T„_,(»i-l)+T o t / After substituting (10) into (11), the resulting equation for computing F/(«) i n 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 i g . 4.9 shows the corresponding procedure for 46 linear extrapolation and subsequent averaging to compute (12). o - actual intervals • - linearly extrapolated intervals A - averaged extrapolated intervals T(n-2) r(n-4) / I J3 r ,(n-2) i(n) M n-4 n-3 n-2 n-1 Sample number, n 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 r ex (n) q are computed based on a three-step history and quadratic extrapolation as r„_,(«)=3r(n-l)-3r(n-2)+r(#i-3) Then, the three values of r ex (13) (n) are averaged as in (9), to obtain the following: r,(«)= j(r„_,('»)+T e t (14) _ , ( « - l ) + r„_,(/i-2)) As with linear extrapolation, F (n) can be expressed in terms of r(«) as r (n) = - (3r(n -1)+ r{n - 3)- 2r(n - A)+r{n - 5)) (15) q which also has the form of (6) and has 5-th order. o - actual intervals • - quadratically extrapolated intervals A - averaged extrapolated intervals •v r(n-S) 9- I •(n-\)'*\^ (n q J 5 n-5 n-4 n-3 n-2 n-1 * \nl V ; Sample number, n ^ /»; it' Figure 4.10 Computing F («) using quadratic extrapolation and subsequent averaging. 47 4.3.3 Performance of Filters To compare the performances o f the proposed averaging filters, their magnitude and phase responses were calculated [8], [9]. The results are superimposed i n F i g . 4.11. A s can be observed, 2.71 all o f the filters completely rejected the undesirable harmonics with frequencies o f — \K 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 o f 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 o f 255 rad/s was initially applied to all o f the filters. Then, at / = 0.02 s, the speed was linearly ramped with an acceleration o f 13 10 rad/s2 until it reached 320 rad/s at 3 t = 0.025 s, after w h i c h the speed was kept constant. The transient responses produced by the considered filters are depicted i n F i g . 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 i n F i g . 4.12, the response o f 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 o f 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 f (n), a6 f (n), a3 F ( « ) or ; r (n). For example, this reference time may be obtained by locking the switching to one o f the q phases (a phase with the smallest positioning error, i f known) [8]. Alternatively, this time may be 48 computed by averaging the switching times o f the three phases [9], as follows: '"(«)=j('.(«)+';(«)+t'M) (17) •-"»»»»•••• • linear extrapolation - quadratic extrapolation \ V.'- 1 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 Time (s) 0.03 0.032 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' (n) and ?»'(«) are the times t extrapolated from the two previous phases, as follows: t' (n)=t (n-\)+f(n) t t t:(n)=t.{n-2)+2f(n) (18) Here, the subscript " * " may denote phase A , B , or C , respectively. For the purposes o f illustration, the computation o f the switching time estimates is summarized in Table 4.1 and shown i n F i g . 4.13. Hence, i f the most recent switching occurred i n phase A , the reference time would be computed as 49 Kn) = \{t {n) + f {n) + t'M) a (19) b and thus the (« + l) th switching in phase C would occur at t(n)+r(n) T A B L E 4.1 c Switching-Event Time Determination n-3 n-2 n-l tJn-V i' ("-y fJn-X) B t' ("-V f (n-2) C t" (" - A instead o f t (n + \). a n n+ 1 t (") ? (n + I) t\(n + \) a a ttfn-V w V t (n - 2) t' (n-\) t" (n) t (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) b c 7(n - 4) + ^next_sw x(n - 4) b c c J(n-3) + f(n-3) J(n - 2) + x(n-2) c 7(n-l) + x(n-\) c T(n+\) T(n) + m T(n) 1 f1 1II =j 33 < (»-V b r(n) »ll ttfn) | • 1 1 1 c t (n c 2t(n) T i m e (s) Figure 4.13 Switching-event time relationships. 4.5 Implementation and Case Studies In order to evaluate the performance o f the proposed averaging filter, it was implemented in both the detailed model and the hardware prototype o f 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 i n (8), (9), (12) and (15), i n 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 i n a predictable amount o f time (number o f 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. A l s o , i n 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 H a l l sensors, after which the filter may be enabled again. A simplified block diagram o f the motor controller allowing automatic enabling and disabling o f the filter is shown in F i g . 4.14. Here it is assumed that one o f the proposed filters is used. To start the operation, the appropriate registers o f 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 H a l l sensor signals for the first several switching transitions. After a sufficient number o f transitions, the filter memory is ready and the "counter" variable has been incremented to pass the first IF condition. For increased safety and reliability o f 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 o f 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. 20 < -20 —£ 0.5 —• 1 — i 0.508 i—I 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-s , while the inverter was 2 supplied with 20 V D C 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 10 3 acceleration goes down below 1.0 10 3 rad/s ; by the time any filter is ready to be used, the 2 rad/s . This study demonstrates that the proposed filters do 2 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 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 ^ r 14 filter enabled a 6-step averaging filter 1 1 L «> -50 0.15 0.05 50 0.15 —r ^ filter enabled a o a | -50 linear extrapolating filter | L 0.05 0.1 0.15 50 filter enabled quadratic extrapolating 5 -50 0.1 0.05 1300 0.99-|0 rad/s 3 a & = 650 ' ^ 2 v 4.7610 rad/s 3 filter 0.15 ^ _ ^ « w f « » ! * 2 1 - 6-step averaging filter — - 3-slep averaging filter - linear extrapolating filter - quadratic extrapolating 0 0.05 0.1 filter 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 10 3 rad/s as shown in F i g . 2 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 Voltage-Step Transient To enable faster mechanical transients (similar to those considered i n F i g . 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 o f 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 o f 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 i n applied dc voltage was followed by a significant increase i n developed electromagnetic torque and subsequent rapid acceleration o f the motor. For 54 this study, the peak acceleration was found to be 13.5 • 10 3 rad/s . 2 A s was pointed out in Section III C (see F i g . 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 i g . 4.18. A s can be observed i n Fig. 4.18, when either o f the filters was used, the 2500 K - non-altered Hall signals - 3-step averaging - 6-step averaging 1000 0.14 0.14 Figure 4.18 Speed and electromagnetic torque response with 3- and 6-step averaging filters. 25001 . 1 1 , K - non-altered Hall signals • linear extrapolation - quadratic extrapolation 1000 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 i n F i g . 4.20 (first two 55 subplots). The simulated transient responses produced by the B L D C motor with extrapolating averaging filters are shown i n F i g . 4.19. A s can be seen, both extrapolating filters performed much faster than the basic moving-average filters, with almost no dip i n torque and close to ideal speed response, with the quadratic extrapolation yielding the fastest response among a l l considered cases. The corresponding measured phase currents shown i n F i g . 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 i g . 4.12. 0.12 0.14 40 1 S3 c i o t-l o § VI | -40 0.1 0.12 3-step averaging filter | 0.14 40 8 .1 a" o i 1 o -40 | linear extrapolating filter | 0.1 0.12 0.14 40 o a | quadratic extrapolating filter ^ -40 0.1 0.12 0.14 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' - 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 cp 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 cp may result in different operating modes as documented in [3]. ( ( + + v v v 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 allfiltersbecame 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 extrapolatingfiltersmay 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. Severalfiltershave 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 averagingfiltersapplied 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 MathWorks 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 To IMPLEMENTATION complement the previous Chapters, some of 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., D S P , P I C , and/or controller chip). Although the requirement o f 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 o f proposed filtering approach described in this Chapter requires very small number o f 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 H a 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]. W h e n the filter is disabled, switching o f the H a 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. W h e n the filter is enabled, the H a l l sensors triggers the input I S R 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 I S R and is stored in special temporary register. Based on this time, the so-called output I S R is scheduled to execute which switches the inverter transistors according to the new modified/filtered pattern. U n l i k e powerful computer C P U s with 64-bit floating-point units (FPUs), basic microcontrollers have low number o f bits (12 to 16), have l o w memory space and much slower clock (up to 4 0 M H z ) . 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, w h i c h 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 H a l l signals change and trigger input ISR. U s i n g this approach, the time 60 intervals between the two successive changes o f the H a 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 o f the next transistor switching can be calculated as *next_sw ='"(«) + K") = + (!) which is also the time to execute the next (upcoming) output I S R denoted here as t^ (n + \). T However, for efficient implementation, it is necessary to relate the time t®J/ (n + 1 ) to the time at T which the previous input I S R was called by the H a l l sensors. Denoting the most-recent calling o f the input I S R b y t'^{n), the next output I S R time may be expressed as tT{n^) where T c o r r (n) (2) = tg{n) r-{n) + is the appropriate correction term. Moreover, since the timer is reset to zero upon invoking o f the input ISR, the term also becomes zero. Therefore, the correction term may be calculated as t {n) = t(n)+¥{n) (3) corr where dependent on the filter realization and may denote x °"(n), r(n) r {n) c orr q c x °"(n), c r " («) c or rr , for the four considered filters, respectively. This quantity can be computed i n a straightforward manner from the available r(n) (see (8), (9), (12) and (15) i n Chapter 4.3). The calculation o f reference time t(n) needs some clarification since the timer is being reset at each time t ^ (n). ! s A c c o r d i n g to (17) (see Chapter 4.4) we have t{ ) = ±{t.{n)+t:{ ) n n + t:{n)) (4) where t' (n)=t (n-\)+T(n) t t t:(n) = t,(n-2)+2f(n) To clarify the time-interval relationships involved i n determining correction term r " (n), c the rr waveform produced b y combining the Hall-sensor outputs is shown in F i g . 5.1. This figure shows several intervals r ( « ) as w e l l as provides information on derivation o f the reference time t (n). Here, the corrected time interval T {n) corr aT { )> T r n f° { ) r rr n o r q° { )> T rr n depends on the filter and may represent r °^ {n), c r a respectively. A s can be seen i n F i g . 5.1, the timer is reset at t = t (n) and the following is valid: t 61 r,(«) = 0 t (n-\)=-T(n-\) (6) t t (n - 2) = -r(n - 2)-r(n -1) t Thereafter, it becomes possible to determine the required correction term T (n) depending on corr the value f(n), w h i c h is different for each filter. First, we derive the expression for T (n) corr assuming the 3-step averaging filter. With r ( « ) = — ^ T ( » - m) as the averaged interval duration, the terms i n (4) may be represented as o 3 *._ (») = '.(«) = 0 a 3 «3 («) = t.(n-1)+ F ' . " _ 3 l f = j (- 2r(/i -1)+ r(« - 2) + r(n - 3)) a 3 (") = t.{n- 2)+ 2 F a3 («) = I (- r(« -1)- (7) - 2)+ 2r(« - 3)) Based on results i n (7) the reference time (4) for this filter is 'o3 (") = J ('._<* (") + C„3 («) + '.*_ («)) =\{~ A " - l) + *(« - 3)) (8) fl3 Consequently, the final corrected time interval may be expressed as <T M = t (n) + r a3 fl3 («) = i (r(n - 2) + 2r(n - )) (9) 3 This final equation (9) is implemented inside the input I S R 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. input-ISR Hall-sensor outputs combined output-ISR t (. HS m t(rO_T (n) cm r(n-5) T(n-4) „ r(n-\) T(n-2) T(n-3j 2 1 — I t.(n) 7*" I r(n) j t,(n-\) f,(n) t,(n-2) 2f(n) f\(n) Time, s. 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 values for r(»). Thus, for the 6-step averaging fdter with f (n) ab 6 as the averaged = —j~\r(n-m) 6 , interval duration, the terms t,_ (n) = t (n) a6 t,(n), rj(«),and r"(») in (4) can be calculated as =0 t K («) = <.("-l) + f (n)=-(-5r(»-l)+r(«-2)+r(n-3)+r(«-4)+r(«-5)+r(»-6)) 6 fl6 (10) o6 ?;_ («) = /»(«-2)+2r («) = y(-2r(«-l)-2r(«-2)+2r(«-3)+r(«-4)+r(«-5)+r(«-6)) fl6 a6 Therefore, the corresponding reference time becomes F (n) = -(-3r(«-l)-r(«-2) + r(«-3) + r(»-4)+r(«-5)+r(«-6)) 6 The final corrected time interval may be expressed as (11) a6 <'6 («) = U (") + r 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 6 order basic averaging filter is also very simple and th 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' j{n) t = t {n-l)+T,(n) t = j(-r(«-l)+r(«-2)+r(«-3)-r(«-4)) tl_, (n) = t.(n-2)+ 2T, (n) = j (r(n - l)-r(n (13) - 2)+ 2r(n - 3)- 2r(« - 4)) Consequently F,(n) = ^ t . _ , ( « ) + ^ _ , ( n ) + /:_,(«)) = i(r(/i-3)-r(»-4)) f rr r ( ) = I, (n) + r, (n) = j (2r(n -1) + r(n - 2) + 2r(n - 3) - 2r(n - 4)) n (14) (15) This final equation (15) is implemented inside the input ISR code section o f the microcontroller. A s can be seen, the expression (15) for the 4 order linear extrapolating filter is also very simple th 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,_ (n) = t,(n) = 0 q t:_ {n) = t (n-1)+ f («) = i ( (n-3)-2r(n-4)+ q t q T r(n-5)) (16) tl_ («) = / , ( « - 2)+ 2F («) = j (3r(« -1)- 3r(« - 2)+ 2r(n - 3)- 4r(« - 4)+ 2r(« - 5)) q ? (») = j ( ' . _ » + («) + (»)) = | M « -1) - *(« - 2) + r(/i - 3) - 2r{n - 4) + r(« - 5)) (17) The corrected time interval for this filter is c T o r r q („) = 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 SUMMARY 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 i n higher quantities at manufacturing facilities all-over-the-world including A s i a . Although, in general, it is possible to improve the accuracy o f the Hall-sensor positioning through increasing the manufacturing precision at increasing costs, this thesis proposed an alternative approach o f addressing this problem at the control level, which in turn achieves performance results similar to that of high-precision-manufactured BLDC 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 i n non-equal conduction intervals among the phases, w h i c h i n turn leads to deterioration o f the motor performance. In particular, during steady state operation, a typical misplacement of the H a l l sensors from their axes produces three unequal intervals, which are repeated every half electrical revolution. This thesis proposes an innovative approach o f filtering the signals from the existing (unbalanced) Hall-sensors and producing a set o f modified (balanced) signals that are used to switch the inverter transistors. Several filters have been proposed at different stages o f this project as presented and discussed in Chapters 2 - 4. To simplify the problem o f designing the required 3-input 3-output filter, we first reduce the problem to the single-input single-output filter. T w o basic averaging filters o f 3 6 th rd and 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 o f 4 and 5 orders have been proposed as well. th th The analysis presented i n 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 well 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 proposedfilteringmethodology 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., L T D . , Model 86EMB3S98F, 36 V D C , 210 W, 2000 rpm, 8 poles, r =0.14 Q , L =0.375 mH, Z' =2\ 5 s mV-s; inertia y = 2-10" 4 m N-m-s ; 2 back emf harmonic coefficients K = 0, K = 0.042 , and K = -0.018. 3 5 7 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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Analysis of voltage-source-inverter-driven brushless dc motors with unbalanced Hall sensors Samoylenko, Nikolay 2007
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Title | Analysis of voltage-source-inverter-driven brushless dc motors with unbalanced Hall sensors |
Creator |
Samoylenko, Nikolay |
Publisher | University of British Columbia |
Date Issued | 2007 |
Description | Brushless dc motors with Hall 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 of three manuscripts which investigate the unbalance of Hall sensors and propose methods to eliminate its adverse effects. The studies presented here show that misalignment of Hall 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 Hall sensors by using a detailed model of 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 Hall sensors. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0101178 |
URI | http://hdl.handle.net/2429/32450 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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