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Dynamic average-value modeling of the 120° VSI-commutated brushless dc motors with non-sinusoidal back… Tabarraee, Kamran 2011

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Dynamic Average-Value Modeling of the 120° VSI-Commutated Brushless DC Motors with Non-Sinusoidal Back EMF by Kamran Tabarraee B.Sc., The Amirkabir University of Technology, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (ELECTRICAL & COMPUTER ENGINEERING) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  June 2011 © Kamran Tabarraee, 2011  Abstract For large and small signal analysis of electromechanical systems with power electronic devices such as Brushless DC (BLDC) motor-inverter drives, average-value models (AVMs) are indisputable. Average-value models are typically orders of magnitude faster than the corresponding detailed models. This advantage makes AVMs ideal for representing motordrive components in system level studies. Derivation of accurate dynamic average-value model of BLDC motor-drive system is generally challenging and requires careful averaging of the stator phase voltages and currents over a prototypical switching interval (SI) to find the corresponding average-value relationships for the state variables and the resulting electromagnetic torque. The so-called 120° voltage source inverter (VSI) driven brushless dc (BLDC) motors are very common in many commercial and industrial applications. This thesis extends the previous work and presents a new and improved dynamic average-value model (AVM) for such BLDC motor-drive systems. The new model is explicit and uses a proper qd model of the permanent magnet synchronous machine with non-sinusoidal rotor flux. The model utilizes multiple reference frame theory to properly include the back EMF harmonics as well as commutation and conduction intervals into the averaged voltage and torque relationships. The commutation angle is readily obtained from the detailed simulation. The proposed model is then demonstrated on two typical industrial BLDC motors with differently-shaped back EMF waveforms (i.e. trapezoidal and close to sinusoidal). The results of studies are compared with experimental measurements as well as previously established state-of-the-art models, whereas the new model is shown to provide appreciable improvement especially for machines with pronounced trapezoidal back EMF.  ii  Preface A version of Chapter 2 has been published in the following manuscript: Kamran Tabarraee, Jaishankar Iyer, Sina Chiniforoosh, and Juri Jatskevich, “Comparison of Brushless DC Motors with Trapezoidal and Sinusoidal Back EMF,” In proc. IEEE Canadian Conference on Electrical and Computer Engineering, May 2011, Niagara Falls, Canada. I developed the models, performed the tests and wrote most of the manuscript, while the conducted research was supervised by Dr. Juri Jatskevich, and revised and assisted by my supervisor and Jaishankar Iyer, and Sina Chiniforoosh. A version of Chapter 3 has also been published in the following manuscript: K. Tabarraee, J. Iyer, and J. Jatskevich, “Average-Value Modeling of Brushless DC Motors with Trapezoidal Back EMF,” In proc. IEEE International Symposium on Industrial Electronics, June 2011, Gdansk, Poland. I developed the models, performed the tests and wrote most of the manuscript, while the conducted research was supervised by Dr. Juri Jatskevich, and revised and assisted by my supervisor and Jaishankar Iyer. A version of Chapter 4 has been submitted for publication: K. Tabarraee, J. Iyer, H. Atighechi and J. Jatskevich, “Dynamic Average-Value Modeling of 120° VSI-Commutated Brushless DC Motors with Trapezoidal Back EMF”. I developed and implemented the model, performed and designed the tests and wrote most of the manuscript. The research has been supervised by Dr. Juri Jatskevich, and assisted by Jaishankar Iyer and Hamid Atighechi.  iii  Table of Contents  Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iii Table of Contents ................................................................................................................... iv List of Figures ......................................................................................................................... vi Acknowledgements .............................................................................................................. viii Dedication ............................................................................................................................... ix 1  2  Introduction ..................................................................................................................... 1 1.1  Why Average-Value Modeling? ........................................................................................... 1  1.2  Literature Review.................................................................................................................. 3  1.3  Brushless DC Motor-Inverter System ................................................................................... 4  1.4  Contributions......................................................................................................................... 5  1.5  Thesis Composition............................................................................................................... 7  Detailed Modeling of Brushless DC Motors with Non-Sinusoidal Back EMF .......... 8 2.1  Model Description................................................................................................................. 8  2.2  Effect of Back EMF Harmonics in Detailed Model ............................................................ 13  2.2.1  Model Verification in Steady-State ................................................................................ 13  2.2.2  Model Verification in Transient ..................................................................................... 17  2.3  3  4  Case studies ......................................................................................................................... 18  2.3.1  ASMG vs. SIMPOWER Model ...................................................................................... 18  2.3.2  Torque-Speed Characteristic .......................................................................................... 19  Average-Value Modeling of Brushless DC Motors with Trapezoidal Back EMF .. 21 3.1  Model Description............................................................................................................... 21  3.2  Case Studies ........................................................................................................................ 29  3.2.1  Start-Up Transient .......................................................................................................... 30  3.2.2  Steady-State .................................................................................................................... 31  Dynamic Average-Value Modeling of 120° VSI-Commutated Brushless DC Motors  with Trapezoidal Back EMF ................................................................................................ 35 4.1  Model Description............................................................................................................... 35 iv  5  4.2  Model Implementation ........................................................................................................ 40  4.3  Case Studies ........................................................................................................................ 43  4.3.1  Steady-State .................................................................................................................... 44  4.3.2  Transient Response to Mechanical Load Change ........................................................... 47  4.3.3  Start-Up Transient .......................................................................................................... 48  4.3.4  Transient Response to Input Voltage Change................................................................. 50  Conclusion ..................................................................................................................... 52 5.1  Summary ............................................................................................................................. 52  5.2  Future Research Topics....................................................................................................... 53  References .............................................................................................................................. 54 Appendices ............................................................................................................................. 58 Appendix A : Prototype Parameters ................................................................................................. 58 A.1  Motor-A Parameters ....................................................................................................... 58  A.2  Motor-B Parameters........................................................................................................ 58  Appendix B : BLDC Motor Controller ............................................................................................ 59  v  List of Figures Figure 1.1  A typical power-electronic-based electro-mechanical system ............................. 1  Figure 1.2  Prototype Motor A Set-up Including Drive Circuit and Mechanical Load. ......... 6  Figure 1.3  Prototype Motor B Set-up Including Drive Circuit and Mechanical Load. ......... 6  Figure 2.1  Schematic diagram of a typical VSI-driven BLDC motor-drive system. ............ 9  Figure 2.2  Switching sequence of the inverter according to the 120° switching logic. ....... 11  Figure 2.3  Steady state waveforms of phase back EMF, phase current and electromagnetic  torque predicted by various models for Motor A.................................................................... 15 Figure 2.4  Steady state waveforms of phase back EMF, phase current and electromagnetic  torque predicted by various models for Motor B. ................................................................... 15 Figure 2.5  Measured and simulated waveforms of phase back EMF, phase current, and  phase voltage for Motor A. ..................................................................................................... 16 Figure 2.6  Measured and simulated waveforms of phase back EMF, phase current, and  phase voltage for Motor B. ..................................................................................................... 16 Figure 2.7  Measured and simulated source current and stator phase current waveforms for  Motor A................................................................................................................................... 17 Figure 2.8  Measured and simulated source current and stator phase current waveforms for  Motor B. .................................................................................................................................. 18 Figure 2.9  Steady-state waveforms of phase current, back-EMF, and electromagnetic  torque as predicted by models with trapezoidal back-EMF implemented in different simulation packages. ............................................................................................................... 19 Figure 2.10  Steady-state torque-speed characteristic predicted by various models and  simulation packages. ............................................................................................................... 20 Figure 3.1  Start-up transient response as predicted by various models for the Motor A. ... 30  Figure 3.2  Start-up transient response as predicted by various models for the Motor B. .... 31  Figure 3.3  Steady-state torque as predicted by various models for the Motor A. ............... 32  Figure 3.4  Steady-state torque as predicted by various models for the Motor B................. 32  Figure 3.5  Steady-state torque-speed characteristic as predicted by various models for the  Motor A. .................................................................................................................................. 34 Figure 3.6  Steady-state torque-speed characteristic as predicted by various models .......... 34 vi  Figure 4.1  Commutation angle look-up table function for the Motor A. ............................ 41  Figure 4.2  Commutation angle look-up table function for the Motor B. ............................. 42  Figure 4.3  Block diagram of the AVM implementation. ..................................................... 43  Figure 4.4  Steady-state torque as predicted by various models for Motor A. ..................... 45  Figure 4.5  Steady-state torque as predicted by various models for Motor B. ..................... 45  Figure 4.6  Steady state torque-speed characteristic as predicted by various models for  Motor A. .................................................................................................................................. 46 Figure 4.7  Steady state torque-speed characteristic as predicted by various models for  Motor B. .................................................................................................................................. 46 Figure 4.8  System response to sudden load change for the Motor A. ................................. 47  Figure 4.9  System response to sudden load change for the Motor B. ................................. 48  Figure 4.10  Start-up transient of Motor A as predicted by various models. ........................ 49  Figure 4.11  Start-up transient of Motor B as predicted by various models. ........................ 49  Figure 4.12  Measured and simulated response to the input voltage change as predicted by  the detailed and proposed average-value models for Motor A. .............................................. 50 Figure 4.13  Measured and simulated response to the input voltage change as predicted by  the detailed and proposed average-value models for Motor B. .............................................. 51 Figure B.1  Motor controller schematic. ............................................................................... 59  Figure B.2  Controller PCB. ................................................................................................. 60  Figure B.3  BLDC motor controller box. .............................................................................. 61  vii  Acknowledgements First of all, I would like to express my deepest appreciations to my research supervisor, Dr. Juri Jatskevich, whose strong academic support and dedication to his students have been the most precious assets to my studies and research during the last two years at UBC. I am also very grateful for the Research Assistantship that has been made available to me through the NSERC Discovery Grant lead by Dr. Jatskevich. The availability of research equipment and resources in the Alpha technology Lab has also been an important asset for my research. In this regard, I would like to express my special thanks to our close colleague and collaborator, Dr. S. D. Pekarek at Purdue University, who has generously donated to our group the high-power BLDC machine with trapezoidal back EMF which has been extensively utilized in my research. I also like to thank Dr. Jose Marti and Dr. K.D. Srivastava, who have accepted to be the committee members and dedicated their time and effort for reading this thesis and providing their constructive and valuable comments. My special thanks go to my friends and colleagues in the UBC‟s Power Lab and Electrical Power and Energy Systems Group, particularly to Jaishankar Iyer, Mehrdad Chapariha, Hamid Atighechi, Sina Chiniforoosh, Milad Gougani, and all the members of research group who have always been a supportive and helpful friend to me. I also owe a debt of thanks to my loving parents and my brother who have supported me morally and financially throughout these intense years of my graduate studies.  viii  Dedication  To My Family  ix  1 1.1  Introduction Why Average-Value Modeling?  Modeling and simulation are indisputable steps in design and development of electromechanical systems with power electronic drives which are widely used in industrial automation, robotics, automotive products, ships, and aircraft. Figure 1.1 represents the block diagram of such an electro-mechanical system. The Electrical Subsystem often consists of an electrical power distribution that may also contain energy source and/or storage (i.e. battery in the case of vehicles). The Machine-Drive subsystem is basically composed of „Inverter‟, and „Electrical Motor/Generator‟ modules. The inverter controls the flow of energy from source to the electrical machine which is responsible for the electro-mechanical energy conversion. The „Mechanical Subsystem‟ may also represent a mechanical drive train (i.e. in vehicles) or an assembly actuating system (i.e. in industrial manufacturing/automation). In general, if the inverter can operate in all four quadrants, the energy may flow in either direction; from the Electrical Source to the Mechanical Subsystem, or from the Mechanical Subsystem back to the Electrical Source. In the former case, the machine operates as a motor where in the latter it functions as a generator. The output of the Mechanical Subsystem which can be position or speed of the rotor, or any other mechanical variables might be used directly or indirectly, to control the inverter. The control signal may also consist of external variables such as duty cycle, voltage amplitude and etc. Figure 1.1  A typical power-electronic-based electro-mechanical system  1  For the purposes of stability analysis and design of respective controllers, it is often desirable to investigate both the large-signal time-domain transients as well as the small-signal frequency-domain characteristics of such systems. Since the experimental tests with the hardware is not always possible and/or cost-effective, in actual industrial practice most of the studies are carried out using appropriate models, simulations, and mathematical apparatus. Such computer-bases studies are usually carried out many times for tuning the system and achieving the desired performance while satisfying the design specifications. This requires the simulation speed and accuracy to be as high as possible. In particular, modeling and simulation of Inverter Module is not often trivial, since this module includes switching components such as MOSFETs, IGBTs and diodes, which make the respective models discontinuous and time-variant. There are various simulation software packages such as [1]– [5], which can be used to develop and implement the models where the switching of all transistors and diodes is represented in full detail. On one hand, there is a need to run the simulation for a sufficiently long time in order to capture the electromechanical transients that may have relatively long time constants (on the order of several seconds); but on the other hand, the presence of power electronic components and fast switching requires using very small time-steps. Therefore, this type of detailed models requires excessively long CPU (computing) times, especially for large systems that include many switching components and consist of several subsystems. However, since the fast switching of the transistors and diodes has only an average effect on the system‟s slow dynamic behavior, it is advantageous to construct a simplified model that matches the original detailed switching model in the low-frequency range. The approach of establishing such simplified models is known as average-value modeling (AVM), wherein the effects of fast switching are neglected or averaged within a prototypical switching interval. Unlike the detailed models, average-value modes (AVMs) are continuous and the respective state variables are constant in steady states. Therefore AVMs can be linearized about a desired operating point; thereafter, obtaining a local transfer function and/or frequency-domain characteristics becomes a straightforward and almost instantaneous procedure. Many simulation programs offer linearization and frequency domain analysis tools [4], [5]. In addition, since there is no switching, the AVMs typically execute, by orders  2  of magnitude, faster than their corresponding detailed models, making them ideal for representing respective components in system-level time-domain transient studies. Such dynamic average models have been very successfully used for modeling of distributed DC power systems of spacecraft [6]–[8] and aircraft [9], [10], naval electrical systems [11], [12] and vehicular electric power systems [13]. Average-value modeling has also been often applied to variable speed wind energy systems [14]–[21], where the machines are typically interfaced with the grid using the power electronic converters.  1.2  Literature Review  Average-value modeling of electro-mechanical systems with power electronic drives has attracted attention of many researchers during the past few decades. R. Krishnan at the Virginia Polytechnic who developed a basis for modeling of the PMSM with power electronic drive [22], and P. C. Krause and S. D. Sudhoff at Purdue University who established the detailed and average-value modeling of the power electronic driven motors including the BLDC [23], [24] were among the pioneers of research in this area. Later on, P. L. Chapman at the University of Illinois at Urbana Champaign, K. A. Corzine at the University of Missouri-Rolla, and H. A. Toliyat at the Texas A& M University made contributions to this field too by developing the models for continuous current operation. Development of the current-regulated BLDC motor-drive system [25], [26] and analysis of the BLDC motor-drive system in hybrid sliding mode observer [27] is often noted as the outcome of their research in this field. In addition, most recently, there have been significant contributions to this area lead by J. Jatskevich and his graduate students at The University of British Columbia, which resulted in several state-of-the-art models including the numerical state-space AVM for the DC-DC converters [28], parametric AVM for the synchronous machine-rectifier system [29], and the most relevant AVM for the sinusoidal BLDC motorinverter system [30]. Prior to the work presented in this thesis, the best known to us average-value model of a BLDC motor with a 120-degree inverter remains the dynamic AVM presented in [31]. That work represents a significant contribution to the area but considers only the sinusoidal back EMF of the machine.  3  1.3  Brushless DC Motor-Inverter System  A brushless dc (BLDC) motor-inerter system consists of a permanent magnet synchronous machine (PMSM) that is driven by a voltage source inverter (VSI). Typically such motors provide good torque-speed characteristics, fast dynamic response, high efficiency, long life, etc., which make them favorable in wide range of applications including industrial automation, instrumentation, and many other equipment and servo applications. This paper considers typical voltage-source inverter driven (VSI-driven) BLDC motors wherein the inverter operates using 120 commutation method [32], [33]. In this switching logic, each phase is allowed to be open-circuited for a fraction of revolution, giving rise to complicated commutation-conduction patterns of the stator currents [32]. In general, derivation of dynamic average-value modeling of such BLDC systems requires careful averaging of the stator phase voltages and currents over a prototypical switching interval (SI) to find the corresponding average-value relationships for the state variables and electromagnetic torque. A pioneering step in this approach has been the AVM for the BLDC motor-inverter system with sinusoidal back EMF [31]. This approach has been extended to include both conduction and commutation sub-intervals [31]. Another average-value model was proposed in [34] for the non-sinusoidal back EMF PMSM driven by a three phase H-bridge inverter that can only operate in continuous-current voltage-control mode by adjusting the duty cycle. However, the average-value modeling of the 120 ° VSI driven BLDC motors with trapezoidal back EMF becomes more challenging due to the discontinuous current and harmonics in the voltage and torque equations; and to the best of our knowledge this has not been addressed in the literature.  4  1.4  Contributions  This thesis extends the previous work in this area and presents a new AVM for the 120 VSI-driven BLDC motor with non-sinusoidal back EMF. The contributions of this thesis and the property of the proposed model can be summarized as follows: 1) We show that there is a need for including the back EMF harmonics for modeling the BLDC motors with pronounced non-sinusoidal back EMF. The new AVM is proposed that simultaneously includes the back EMF harmonics and the commutation and conduction subintervals. 2) The multiple reference frame theory [34] is utilized to properly include the effect of back EMF harmonics into the average-value relationships of the AVM. 3) Since it is not practical to analytically derive a closed form solution for the commutation angle, the solution is obtained numerically using detailed simulation. This method reduces the complexity of analytical derivations and has been shown to provide accurate results [31], [28], [29], [35]. 4) The conducted studies are based on two typical industrial BLDC motors with various back EMF harmonics content and parameters summarized in the Appendix A. Figures 1.2, and 1.3 show the prototype motors‟ test set-up arranged for the measurement purposes in the laboratory. The details of the BLDC Motor Controller Box are shown in Appendix B. The results of studies are compared with the experimental measurements as well as previously established models [24], [31] whereas the new model is shown to provide appreciable improvement.  5  Figure 1.2  Prototype Motor A Set-up Including Drive Circuit and Mechanical Load.  Figure 1.3  Prototype Motor B Set-up Including Drive Circuit and Mechanical Load.  6  1.5  Thesis Composition  This thesis is comprised of the following chapters:   Chapter 2 presents an improved approach for detailed modeling of the BLDC motor-drive system where the effects of back EMF harmonics are properly incorporated into the model. The proposed model is then verified against the hardware measurement from the actual machine. It is also shown that including the back EMF harmonics often significantly increases the accuracy of the detailed model in predicting the behavior of the system and hence, it is necessary to take these harmonics into consideration for further studies on the inverter-driven BLDC systems.    Chapter 3 describes a new average-value model for the BLDC machine-drive system in which the multiple reference frame theory [34] is used to properly include the back EMF harmonics into the AVM. The developed model is then implemented in Matlab/Simulink [4] along with the previously developed AVMs in which the back EMF waveform is assumed to be sinusoidal, and it is shown that the proposed AVM is more accurate. However, since the commutation interval is neglected, the new AVM may still result in some error in prediction of the system performance.    Chapter 4 completes the presented AVM in Chapter 3 by simultaneously including the effects of the back EMF harmonics and the commutation and conduction subinterval. In particular, this process is very challenging due to presence of both the higher harmonics and the commutation angle in the voltage and torque relationships. The proposed AVM is then shown to be considerably more accurate in comparison with the previously developed AVMs and compensates the error arising due to neglecting the back EMF harmonics and/or the commutation subinterval.    Chapter 5 concludes the thesis by summarizing the conducted research.    The parameters of the motors and the motor controller are summarized in Appendix.  7  2  Detailed Modeling of Brushless DC Motors with Non-Sinusoidal  Back EMF The detailed modeling of the BLDC motor-inverter system has been described in literature quite well [22], [23], [32], [33], [36]–[38] and can be easily carried out using a number of simulation packages [1]–[5]. In many available detailed models, it is often assumed that the induced back EMF waveform of the machine is sinusoidal [23], [24], [31], [32]. However, the actual back EMF waveform might quite non-sinusoidal. Including the back EMF harmonics into the voltage and torque equations increases the accuracy of the model. In addition, to develop a detailed model that precisely predicts the performance of the BLDC motor-drive system with trapezoidal back EMF, an appropriate simulation package must be used such that the back EMF waveform can be modified to include the desired amount of harmonics [2], [3]. Herein the typical voltage-source-inverter-driven (VSI-driven) BLDC motors are considered where the inverter operates using the 120 commutation method [32]. The steady state analysis of such motors has been carried out by several researchers [22], [23], [36]. In this chapter an improved detailed model of the typical 120 BLDC motor-drive system is proposed in which the trapezoidal back EMF harmonics are properly included into the model. It is also shown that an accurate model may be only obtained using simulation packages that allow making proper changes in the model such that the effects of back-EMF harmonics are appropriately included in the current and torque relationships.  2.1  Model Description  A schematic of the considered BLDC motor-inverter system is shown in Figure 1.2, in which the logical signals from hall sensors are used to control the inverter switches-transistors S1 –  S 6 . Here, as previously described, the motor is driven according to the 120 switching logic. In this method, switching signals are of the sequence shown in Figure 2.1 [31]. As a result, each phase carries current for 120 two times during one electrical revolution which delays  8  the fundamental component of the voltage by 30 electrical degrees. To align the fundamental component of the voltage with the back EMF, the advance firing angle of   30 is applied [33], [37], [38].  Figure 2.1  Schematic diagram of a typical VSI-driven BLDC motor-drive system.  Although some BLDC machines are specifically designed to have low cogging torque and consequently close to a sinusoidal back EMF waveform [39], [40], in practice, BLDC motors often have trapezoidal back EMF. Including the back EMF harmonics into the voltage and torque equations increases the accuracy of the model. The presented model is expressed in physical variables and coordinates [38]. In particular, the electrical dynamics of stator shall be described by the well-known voltage equation  9  v abcs  rs i abcs   dλ abcs . dt  (1)  Here, the variable are represented in vectors such that f abcs   f as  f bs  f cs T , where  f may be voltage, current, or flux linkage. The stator resistance matrix is rs  diag rs , rs , rs .  (2)  The flux linkages are then given by  λ abcs  L s i abcs  λ m  (3)  where the inductance matrix is defined by   Lls  Lm L s    0.5Lm   0.5Lm   0.5Lm Lls  Lm  0.5Lm   0.5Lm   0.5Lm  Lls  Lm   (4)  in which Lls and Lm are the stator leakage and magnetizing inductances, and λ m is the vector of flux linkages. Assuming that stator windings are wye-connected, the three phase currents add up to zero. Thus, (3) may be simplified as  λ abcs  Ls i abcs  λ m  where Ls  Lls   (5)  3 Lm . 2  10  Figure 2.2  Switching sequence of the inverter according to the 120° switching logic.  Equations (1)–(5) hold true regardless of shape of the back EMF waveform. In general, the flux linkages vector can be expressed as   sin 2n  1 r      2  λ m   m  K 2 n 1 sin  2n  1 r  3  n 1   2    sin  2n  1 r  3                      (6)  where  r is the rotor‟s electrical position, and  m is the magnitude of the fundamental component of the permanent magnet flux linkage. The coefficient K n denotes the normalized magnitude of n th flux harmonic relative to the fundamental, i.e. K1  1 . Also, the index 2n  1 explains that only odd harmonics may be present since the rotor is assumed to be symmetrical. 11  The developed electromagnetic torque in presence of back EMF harmonics may then be presented as [34],   T cos 2n  1 r  i as      P 2  Te   m  2n  1K 2 n 1 ibs  cos 2n  1 r  2 3  n 1 i cs    2       cos 2 n  1    r   3            .          (7)  The phase back EMF voltages can be measured at the stator terminals when the machine is rotated by a prime mover and terminals are open-circuited. They can be also calculated based on (1)–(6) as  e abcs    T cos 2n  1 r  i as      2    r  m  2n  1K 2 n 1 ibs  cos 2n  1 r  3  n 1 i cs    2    cos 2n  1 r  3                     (8)  The mechanical subsystem is considered to be a single rigid body, for which the dynamics shall be modeled by  d  P  1    Te  Tm  dt  2  J  (9)  where  r is the rotor‟s electrical angular speed, J is the combined moment of inertia of the load and the rotor, P is the number of magnetic poles, and Tm denotes the combined mechanical torque. Herein, a fan type load is used for which  Tm  T1n  To  (10) 12  where n represents the mechanical speed in revolution per minute (rpm), and the terms T1n and To describe the dynamometer torque and the torque due to mechanical losses and friction respectively. Equations (1)–(10) form the detailed model of the BLDC motor driven by a 120 voltagesource inverter, where the back EMF waveform may be modified to possess the desired amount of harmonics.  2.2  Effect of Back EMF Harmonics in Detailed Model  To demonstrate the importance of properly including the back EMF harmonics, the detailed model described in previous sub-section is compared against the commonly-used model [22], [31], [38] that considers sinusoidal back EMF. The considered detailed switching models have been implemented in Matlab/Simulink using toolbox [3]. The conducted studies are based on two typical industrial BLDC motors whose parameters summarized in the Appendix A. As can be seen in the Appendix A, Motor A has a typical trapezoidal back EMF that includes significant amount of 3rd , 5th , and 7th harmonics; whereas the back EMF waveform of Motor B is much closer to sinusoidal.  2.2.1  Model Verification in Steady-State  The simulated steady state waveforms predicted by the models with sinusoidal and nonsinusoidal back EMF are superimposed in Figures 2.3 and 2.4 for Motor A and Motor B, respectively. These waveforms correspond to a steady state operation when the inverter is supplied with Vdc  26V , and a mechanical load of 330W at 2140 rpm applied for Motor A, and 90W at 1650 rpm is applied for Motor B, respectively. The first subplot in Figure 2.3 and 2.4 shows the back EMF with and without the harmonics. As can be seen in Figure 2.3 (first subplot), the Motor A has a strongly-pronounced trapezoidal back EMF, unlike the back EMF of the Motor B in Figure 2.4 (first subplot), which is visibly close to sinusoidal. Figures 2.3 and 2.4 (second subplot) show that the back EMF harmonics also have effect on the shape of the phase current during the conduction interval, which is also more pronounced for the Motor A than Motor B. The simulated electromagnetic torque waveforms are shown in  13  Figures 2.3 and 2.4 (third subplot), where the effect of back EMF harmonics is also clearly observed. According to (7), the electromagnetic torque is expected to have a larger average value in the presence of harmonics. As a result, the difference in the torque ripple and its average value when the harmonics are included or not for Motor A is more significant than for Motor B. Next, the detailed models that include the back EMF harmonics are compared to the actual Motor A and Motor B. The measured and simulated waveforms corresponding to the same steady state operating condition are superimposed in Figures 2.5 and 2.6. The first subplot shows the measured back EMFs that have been recorded under the open-circuit condition corresponding to the same speeds for the Motor A and Motor B, respectively. This measurement was also used to extract the back EMF harmonics for each of the motors (with the results summarized in Appendix A). Furthermore, Figures 2.5 and 2.6 (see second and third subplots) also show that by including the back EMF harmonics into the detailed models, an excellent match between the measured and simulated phase currents and voltages for both motors is achieved. Therefore, these models can be considered as the reference for the future studies.  14  Figure 2.3  Steady state waveforms of phase back EMF, phase current and electromagnetic torque  predicted by various models for Motor A.  Figure 2.4  Steady state waveforms of phase back EMF, phase current and electromagnetic torque  predicted by various models for Motor B.  15  Figure 2.5 Measured and simulated waveforms of phase back EMF, phase current, and phase voltage for Motor A.  Figure 2.6 Measured and simulated waveforms of phase back EMF, phase current, and phase voltage for Motor B.  16  2.2.2  Model Verification in Transient  The considered detailed model has been further verified in a transient study against the actual Motor A and Motor B. In the following study, the motors are initially supplied with a voltage  Vdc1  20V , driving a fan–type load (emulated by a dynamometer machine). The corresponding load characteristics for both machines (coefficients T1 and To ) are also summarized in Appendix A. Then, at t  1s , the input voltage is stepped to Vdc2  23V . For better comparison, the measured and simulated currents have been carefully aligned in time axis and superimposed in Figure 2.7 and 2.8 for the Motor A and Motor B, respectively. As can be seen in Figure 2.7 and 2.8, the two motors have different inertia and currents corresponding to their respective loading conditions. However, the predicted dc current idc (see Figure 2.1) and the phase current ibs are in good agreement with the experimental results obtained for both motors, which also supports the use of these detailed models as the reference in the future transient studies.  Figure 2.7 Measured and simulated source current and stator phase current waveforms for Motor A.  17  Figure 2.8 Measured and simulated source current and stator phase current waveforms for Motor B.  2.3  Case studies  The same industrial BLDC prototypes with parameters summarized in the Appendix A are used to show the improvement of the proposed model when implemented in Matlab/ASMG [3] against models for the BLDC motor that are implemented in other simulation packages such as SIMPOWERSYSTEMS [1].  2.3.1  ASMG vs. SIMPOWER Model  The BLDC motor-inverter system modeling may be carried out using various simulation packages [1]–[3]. Experimenting with these packages, it has been found that implementing the BLDC motor that considers only the sinusoidal back EMF will result in the same output waveforms of the currents and electromagnetic torque that are consistent among all the mentioned tools. However, when the model includes the back EMF harmonics, the simulation results of some packages might not be consistent anymore. To demonstrate this point, the prototype BLDC Motor A with trapezoidal back EMF (see Appendix A.1) has been also implemented in SIMPOWERSYSTEMS [1], wherein the user has a choice of using either sinusoidal or trapezoidal back EMF. For comparison, the simulated waveforms of the phase current, phase back EMF, and electromagnetic torque predicted by ASMG [3] and 18  SIMPOWERSYSTEMS [1] for the same operating point of Motor A are shown in Figure 2.8. As can be seen in this figure, SIMPOWERSYSTEMS uses an ideal trapezoid for representing the back EMF, which can be well matched with the measured/simulated waveform of the back EMF shown in Figure 2.3 with the specified 3rd, 5th and 7th harmonics. However, as it can be seen in Figure 2.9, when the trapezoid parameters are selected to match the back EMF waveform (see top subplot), quite noticeable error will appear in the phase current (see middle subplot) and electromagnetic torque (see bottom subplot).  Figure 2.9  Steady-state waveforms of phase current, back-EMF, and electromagnetic torque as  predicted by models with trapezoidal back-EMF implemented in different simulation packages.  2.3.2  Torque-Speed Characteristic  To further show the impact of accurately including the back EMF harmonics on the system performance, the steady-state torque-speed characteristic of the prototype Motor A predicted by various models is shown in Figure 2.10. As expected, the implemented model in ASMG [3] that includes the effect of back EMF harmonics represents an improvement and predicts higher torque that also confirms the results shown in Figure 2.3 and 2.9. Hence, it can be again implied that neither the model that considers sinusoidal back EMF nor the  19  implemented model in SIMPOWERSYSTEMS can be used as a reference detailed model for further investigation on the BLDC systems with trapezoidal back EMF. Figure 2.10  Steady-state torque-speed characteristic predicted by various models and simulation  packages.  20  3  Average-Value Modeling of Brushless DC Motors with Trapezoidal  Back EMF This chapter focuses on average-value modeling of typical voltage-source-inverter-driven (VSI-driven) BLDC motors, where the inverter operates using the 120o commutation method [24], [31], [32]. The AVM for the BLDC with sinusoidal back EMF has been proposed in the literature [31]. Another average-value model was also proposed in [34] for the non-sinusoidal back EMF PMSM with a 3-phase H-bridge inverter that can operate in continuous-current voltage-control mode only by adjusting the duty cycle. However, average-value modeling of the 120-degree BLDC with trapezoidal back EMF is more challenging due to the discontinuous current and harmonics in the voltage and torque equations [34] and, to the best of our knowledge, has not been addressed in the literature. In this chapter a new and improved AVM for the typical 120-degree BLDC motor-drive systems is proposed which makes a contribution by properly including the harmonics of the trapezoidal back EMF into the average-value relationships of the model. It is also shown that by utilizing the multiple reference frame theory and properly including the contributions of harmonics, a more accurate AVM can be derived.  3.1  Model Description  For the purpose of deriving the AVM, the back EMF waveform is assumed to include only 5th and 7th harmonics because: i) the higher harmonics are negligible due to their relatively smaller magnitude; and ii) the 3rd harmonic will have no effect in the averaging process since the stator windings are wye-connected. Furthermore, the AVM is derived in a reference frame in which the state variables are constant in steady-state. Therefore, the stator variables are transformed into the qd -rotor reference frame using Park‟s Transformation [38]  f qd 0 s  K rs f abc  (11)  where  21   cos  r  2 r Ks  sin  r 3  1   2  2  2    cos r   cos r   3  3    2  2     sin  r   sin  r   . 3  3     1 1  2 2   (12)  Applying transformation (11) to (1)–(9), the stator phase voltages in qd –rotor reference frame may be described by  r vqs  r vds  r diqs  r   r Ls ids   r  ' m 1  5K 5  7 K 7 cos6 r     r rs iqs   Ls    r rs ids  r dids r  Ls   r Ls ids   r  ' m 5K 5  7 K 7 sin 6 r  . dt  dt  (13) (14)  The electromagnetic torque in qd -rotor reference frame may also be found by applying (11) to (7) as       3  P  r r . Te    m  1  5K 5  7 K 7  cos6 r iqs  5K 5  7 K 7 sin 6 r ids  2  2   (15)  It is noticed that in (13)–(15), the addition of 5th and 7th harmonics into the model results in extra harmonic terms corresponding to 6 r . This can be explained considering that (13)–(15) are expressed in the reference frame rotating at rotor‟s electrical speed. In general, for a round rotor PMSM, the n th harmonic term of the induced voltage in physical coordinates can be expressed by an equivalent vector rotating at n times the rotor‟s electrical speed. Here, only 5th and 7th harmonics of the back EMF are considered where the 5th travels in the negative direction and 7th in the positive direction with respect to the fundamental harmonic resultant vector [41]. However, in the qd -rotor reference frame, the transformed voltage harmonics are equivalent to the vectors rotating at the relative speed of their respective vectors in physical coordinates with respect to the fundamental harmonic which rotates with  22  the rotor. Therefore, considering the direction, both the 5th and 7th voltage harmonics result in  6 r dependant terms in (13)–(15).  r r The state variables iqs and ids must now be averaged with respect to a prototypical switching  interval, Ts (see Figures 2.5 and 2.6, second subplot), using  f   1 Ts  t  t T  f ( )d  (16)  s  where f may represent voltages or currents, and the bar symbol is used to denote the corresponding average-value. For the six-pulse converter, Ts   / 3 r  [7]. However, (15) r r cannot be simply averaged using (16) since iqs and ids are both functions of rotor‟s electrical  position,  r . To establish correct average-value relationship for torque, the multiple reference frame theory is used [8], [9]. This step requires the state variables to be transformed from qd -rotor reference frame to another reference frame rotating at some multiple of the rotor‟s electrical speed. In particular, this transformation is described by xr r f qd  xr K rs f qd  (17)  where  xr  K rs     r  K sxr    1  cosx r   r   sin x r   r   .  sin x r   r  cosx r   r    (18)  Transformation (18) can be used to transform the variables from the rotor reference frame, ‗r‘, into the ‗xr‘ reference frame, rotating at ‗x‘ times the electrical speed of the rotor. After  23  algebraic manipulations, (15) can be expressed considering the multiple reference frame theory as [34]     3  P  5r 7 r r Te     m  i qs  5K 5 i qs  7 K 7 i qs 2 2       (19)  5 r r where, for example, iqs represents the transformation of iqs into a reference frame rotating  at ‗5‘ times the rotor electrical speed and should not be misunderstood with the power sign. Since there are no  r -dependent terms in (19), it can now be averaged using (16) resulting in the following:       3  P  Te     m  i qsr  5K 5 i qs5r  7 K 7 i qs7 r .  2  2   (20)  To make use of (20), it is necessary to establish the state equations where the averaged currents are the state variables. Since the state equations are derived from the voltage equations, the next step is to find the transformed voltages in '5r ' and '7r ' reference frames by applying (17) to (13)–(14), as  5 r 5 r vqs  rs iqs  Ls  5 r 5 r vds  rs ids  Ls 7 r  vqs  7 r   rs iqs  Ls  7 r 7 r vds  rs ids  Ls  5 r diqs  dt  5 r  5 r Ls ids  5K 5 r m   r  ' m cos6 r   7 K 7 cos12 r  (21)  5 r dids 5 r  5 r Ls iqs   r  ' m sin 6 r   7 K 7 sin 12 r  dt 7 r diqs  dt  (22)  7 r  7 r Ls ids  7 K 7 r m   r  ' m cos6 r   5K 5 cos12 r  (23)  7 r dids 7 r  7 r Ls iqs   r  ' m sin 6 r   5K 5 sin 12 r . dt  (24)  The transformed voltages are then averaged by applying (16) to (13)–(14) and (21)–(24), which results in the following:  24  v qsr v dsr  diqsr   Ls    rs idsr  didsr  Ls   r Ls iqsr . dt  dt  v qs5r  rs i qs5r  Ls 5 r  v ds  7 r  v qs    r Ls i dsr   r  ' m    rs iqsr  5 r   rs i ds  7 r   rs i qs   Ls   Ls  v ds7 r  rs i ds7 r  Ls  di qs5r dt  di ds5r dt  di qs7 r dt  di ds7 r dt  (25) (26)   5 r Ls i ds5r  5K 5 r  m  (27)   5 r Ls i qs5r  (28)   7 r Ls i ds7 r  7 K 7  r  m  (29)   7 r Ls i qs7 r .  (30)  To provide the input into the state model formed by (25)–(30), the averaged stator voltages have to be established from the instantaneous voltages over a prototypical switching interval,  Ts , which consists of commutation and conduction subintervals [31]. The commutation subinterval is often neglected [23] in order to simplify the averaging process. This assumption is basis of the dynamic AVM proposed in [31] and also in this paper. In particular, considering the switching interval II (see [24], [31], and Figures 2.5 and 2.6, second subplot), which starts when the phase b transistor S 5 is being switched “off”, the average voltages may be expressed as    3   xr vqsxr    2 vqs ,cond  r d r    6   (31)     3   xr vdsxr    2 vds ,cond  r d r    6   (32)  xr xr where vqs ,cond and v ds ,cond are the instantaneous voltages in the conduction subinterval  transformed into the ‗xr‘ reference frame. To obtain these voltages, the phase voltages in  25  direct abc coordinates should be known first. Based on analysis of inverter circuit [32], [38] we have  v abc ,cond  1  1   2 Vdc  2 V B    VB    1 Vdc  1 V B  2   2  (33)  where    2  10  VB  m  r cos r    5K 5 cos 5 r  3  3    14    7 K 7 cos 7 r   3        .  (34)  Applying (11) to (33) and then transforming the result using (17), the conduction voltages in the ' r ' , '5r ' and '7r ' reference frames are found as   1 2   r v qs  ,cond  Vdc cos  r  cos   r  3 3     2  5 4     'm r cos 2  r    K 5 cos 4 r   3  2 3      (35)  7 7 4  5     K 5  K 7  cos 6 r   K 7 cos 8 r   2 2 3  2    1 2   r v ds  ,cond  Vdc sin  r  sin   r  3 3      2   2   'm r cos r   sin  r   3   3      5 4   5 7 7 4    K 5 sin  4 r    K 5  K 7  sin 6 r   K 7 sin  8 r  2 3  2 2 2 3     (36)      26   1 2    5 r  v qs  ,cond  Vdc cos 6 r cos  r  cos   r  3 3     2  5 4     'm r cos 6 r cos 2  r    K 5 cos 4 r   3  2 3      7 7 4  1 2  5      K 5  K 7  cos 6 r   K 7 cos 8 r    Vdc sin 6 r sin  r  sin  r   2 2 3  3 3  2       2   2  5 4    'm r sin 6 r cos r   sin  r    K 5 sin  4 r   3   3  2 3     7 7 4 5     K 5  K 7  sin 6 r   K 7 sin  8 r  2 2 3 2       (37)   1 2    5 r  v ds  ,cond   Vdc sin 6 r cos  r  cos   r  3 3     2  5 4     'm r sin 6 r cos 2  r    K 5 cos 4 r   3  2 3     7 7 4 5     K 5  K 7  cos 6 r   K 7 cos 8 r  2 2 3 2     2  1    Vdc cos 6 r sin  r  sin  r  3  3          2   2  5 4    'm r cos 6 r cos r   sin  r    K 5 sin  4 r   3   3  2 3     7 7 4  5     K 5  K 7  sin 6 r   K 7 sin  8 r   2 2 3  2   (38)   1 2   7 r  v qs  ,cond  Vdc cos 6 r cos  r  cos   r  3 3     2   'm r cos 6 r cos 2  r  3    4   5    K 5 cos 4 r   3   2    7 7 4  1 2  5      K 5  K 7  cos 6 r   K 7 cos 8 r    Vdc sin 6 r sin  r  sin  r   2 2 3  3 3  2       2   2  5 4    'm r sin 6 r cos r   sin  r    K 5 sin  4 r   3   3  2 3     7 7 4  5     K 5  K 7  sin 6 r   K 7 sin  8 r   2 2 3  2   (39)  27   1 2   7 r  v ds  ,cond   Vdc sin 6 r cos  r  cos   r  3 3     2  5 4     'm r sin 6 r cos 2  r    K 5 cos 4 r   3  2 3      7 7 4  1 2  5      K 5  K 7  cos 6 r   K 7 cos 8 r    Vdc cos 6 r sin  r  sin  r   2 2 3  3 3  2       2   2  5 4    'm r cos 6 r cos r   sin  r    K 5 sin  4 r   3   3  2 3     7 7 4 5     K 5  K 7  sin 6 r   K 7 sin  8 r  2 2 3 2       (40)  The averaged voltages over the conduction subintervals may now be obtained by substituting (35)–(40) into the equations (31)–(32), respectively. The results in the ' r ' , '5r ' and '7r ' reference frames can be expressed as  v qsr  A r  Vdc  B r  m  r  (41)  vdsr  C r  Vdc  D r  m  r  (42)  v qs5r  A 5r  Vdc  B 5r  m  r  (43)  v ds5r  C 5r  Vdc  D 5r  m  r  (44)  v qs7 r  A 7 r  Vdc  B 7 r  m  r  (45)  v ds7 r  C 7 r  Vdc  D 7 r  m  r  (46)  where the coefficients A r   , B r   , C r   , D r   , A5r   , B 5r   , C 5r   , D 5r   , and A7 r   , B 7 r   , C 7 r   , D 7 r   are  A r       cos     6   (47)  B r     1 3 3   15 3 2  21 3      cos 2    K 5 cos 4  K 7 cos 8    2 4 3  8 3  16 3     (48)  3  28  C r      2   cos     3    (49)  D r      3 3 2  15 3   21 3     cos 2  K 5 cos 4    K 7 cos 8    4 6  8 6  16 6     (50)  3   2  cos5   cos 5  3        A 5r     1 5  B 5r     5 21 3 2  3 3 2  15 2     K5  K 7 cos 2  cos 4  K 5 cos10     2 4 3  8 3  20 3     (52)  C 5r     1 5  D 5r     A 7 r       2  sin 5   sin 5  3    (51)      (53)  21 3 2  3 3  2  15 2    K 7 sin 2  sin 4  K 5 sin10     4 3  8 3  20 3      1     cos7   cos 7    7  3    (54) (55)  B 7 r     7 15 3 2  3 3 2  21 3 2     K7  K 5 cos 2  cos 8  K 7 cos14     2 4 3  16 3  28 3     (56)  C 7 r     1 7  D 7 r      3.2      sin 7   sin 7  3      15 3 2  K 5 sin 2  4 3   2  3 3  sin 8   3  16   (57)    21 3  K 7 sin14    3  28   (58)  Case Studies  The proposed AVM has been implemented in Matlab/Simulink together with the detailed model. The same prototype motors with parameters summarized in the Appendix A are used here. It is important to recall that since we have neglected the commutation time, the model accuracy depends on how large or small the commutation interval is, which in turn depends on the motor parameters.  29  3.2.1  Start-Up Transient  Figures 3.1 and 3.2 depict the typical start-up transients of the prototype motors as predicted by the detailed model, the average-value model with sinusoidal back EMF, and the proposed AVM that takes the back EMF harmonics into account. As can be observed, the developed AVM provides more accurate results in prediction of the transient response. However, neglecting the commutation interval affects the accuracy of the proposed AVM despite the improvement against the previous AVMs which consider sinusoidal back EMF [31]. Figure 3.1  Start-up transient response as predicted by various models for the Motor A.  30  Figure 3.2  3.2.2  Start-up transient response as predicted by various models for the Motor B.  Steady-State  To further explore the improvement of the proposed model against the previously established AVM that considers a sinusoidal back EMF [31], all three models have been implemented and compared. Without loss of generality, and to have consistent studies with chapter 2, the same steady-state operating point which were defined by 330W mechanical load at 2140 rpm for motor A, and 90W mechanical load at 1650 rpm for motor B, supplied with Vdc  26V , are used here again. The electromagnetic torque predicted by the three models is superimposed in Figure 3.3 and Figure 3.4 corresponding to Motor A and Motor B respectively. As shown in these figures, including the effect of back EMF harmonics appreciably improves the accuracy of the new AVM, especially in the case of Motor A which possess more significant back EMF harmonics in comparison with Motor B. As expected, since the commutation angle is ignored, the proposed AVM which is noticeably more accurate than the sinusoidal AVMs, might still result in some error if commutation time is not negligible compared to the length of conduction period.  31  Figure 3.3  Steady-state torque as predicted by various models for the Motor A.  Figure 3.4  Steady-state torque as predicted by various models for the Motor B.  32  To examine the model performance in a wider operating range, the calculated steady state torque-speed characteristics for all considered models are plotted in Figure 3.5 and 3.6 for the prototype motors A and B, respectively. As expected, the AVM that includes the effect of back EMF harmonics represents an improvement and overall predicts higher average torque that is also closer to the torque predicted by the detailed model. This improvement is more pronounced at light loads, which is also expected, because the commutation interval is smaller in this operating region. In addition, it may be again noticed that neglecting back EMF harmonics results in more significant error in the case of Motor A which has trapezoidal back EMF.  33  Figure 3.5  Steady-state torque-speed characteristic as predicted by various models for the Motor A.  Figure 3.6  Steady-state torque-speed characteristic as predicted by various models  34  4  Dynamic Average-Value Modeling of 120° VSI-Commutated  Brushless DC Motors with Trapezoidal Back EMF Dynamic Average-value modeling of 120° VSI-Commutated brushless DC motors with sinusoidal back EMF has been well investigated in [31] where the challenge of including the commutation interval into the voltage equations, was conventionally manipulated by utilizing the data from detailed simulation in the form of a numerical look-up table. However, as discussed in the previous section, the error arising due to neglecting the back EMF harmonics might be quite significant in some cases. The proposed AVM in this chapter complements the presented model in the previous chapter such that the effect of back EMF harmonics and the commutation subinterval are taken into consideration simultaneously.  4.1  Model Description  To develop the average-value model of the BLDC motor-inverter system with trapezoidal back EMF which properly considers the commutation time, the proposed procedure in chapter 3 can be followed where equation (11)–(30) still hold true. However, the commutation time is not ignored anymore meaning that the switching interval Ts , consists of commutation and conduction subintervals. Therefore, the averaged instantaneous voltages can be represented as [23], [32]  vqsxr  vqsxr,com  vqsxr,cond  (59)  vdsxr  vdsxr,com  vdsxr,cond .  (60)  xr xr where vqds ,com and vqds ,cond are the instantaneous voltages in the commutation and  conduction subintervals respectively, in the ‗xr‘ reference frame. Considering the switching interval II (see [32]) which starts when the phase b transistor S 5 is being switched “OFF”, the averaged commutation and conduction voltages in the ‗xr‘ reference frame are  35     3     xr vqsxr,com    6 vqs ,com  r d r    6   (61)     3     xr vdsxr,com    6 vds ,com  r d r    6   (62)     3   xr vqsxr,cond    2 vqs ,cond  r d r    6     (63)     3   xr vdsxr,cond    2 vds ,cond  r d r    6     (64)  where  is the commutation angle, in electrical degrees, and  , is the advance in firing angle that is assumed to be 30 . Equations (61)–(64) require finding the instantaneous voltages during the commutation and conduction subintervals prior to the averaging process. The commutation time (and angle) depends on the stator winding electrical time constant and operating conditions, but in general cannot be zero since the current in the inductor cannot be switched “OFF” instantaneously. In the commutation subinterval of the SI II, the phase current ibs is negative and going to zero through the upper diode. The stator phase voltages in direct abc coordinates can then be readily established based on the inverter topology. In particular, after some algebraic manipulation [23], the phase voltages during the commutation time may be expressed as  v abc,com   Vdc 1 1  2T . 3  (65)  Hence, (65) must also be transformed to the ' r ' , '5r ' and '7r ' reference frames according to the multiple reference frame theory [34]. This is achieved by first applying the transformation (11) to (65) and then transforming the result using (17), which results in the following  36  2 2   r v qs  ,com   Vdc cos r  3 3    (66)  2 2   r v ds . ,com   Vdc sin  r  3 3    (67)     2 2  2   5r  vqs  cos6 r  sin r   sin 6 r  ,com   Vdc cos r  3 3  3       (68)     2 2  2   5r  vds  sin 6 r  sin r   cos6 r  . ,com   Vdc cos r  3 3  3       (69)     2 2  2   7 r  vqs  cos6 r  sin r   sin 6 r  ,com   Vdc cos r  3 3  3       (70)   2 2  7 r  vds ,com   Vdc  cos r  3 3    (71)  2    sin 6 r  sin r  3       cos6 r  .    However, the instantaneous voltage harmonics during the conduction subinterval remain the same as (35)–(40). The total averaged voltages over commutation and conduction subintervals can now be obtained by substituting (66)–(71) and (35)–(40) into (61)–(64), respectively. The results in the 'r ' , '5r ' and '7r ' reference frames are:  vqsr  A r  ,  Vdc  B r  ,  m  r  (72)  vdsr  C r  ,  Vdc  D r  ,  m  r  (73)  vqs5r  A5r  ,  Vdc  B 5r  ,  m  r  (74)  vds5r  C 5r  ,  Vdc  D 5r  ,  m  r  (75)  vqs7 r  A7 r  ,  Vdc  B 7 r  ,  m  r  (76)  vds7 r  C 7 r  ,  Vdc  D 7 r  ,  m  r  (77)  37  where the coefficients A r  ,   , B r  ,   , C r  ,   , D r  ,   , A5r  ,   , B 5r  ,   ,  C 5r  ,   , D 5r  ,   , A7 r  ,   , B 7 r  ,   , C 7 r  ,   , D 7 r  ,   are  A r     2        cos     sin       3 2 6 2       5         cos    sin     2 cos     sin   2 6 2 6 2   2     (78)  3   2     15    2     cos 2     sin      K 5 cos 4   2  sin   2   2  3 3 3   3  4    3  1 5K 5  7 K 7 cos6  3 sin 3   21 K 7 cos 8    4  sin 2  4   2 8 3    3  (79) B r     C r     2             5        sin    sin   sin    sin    2 sin    sin     3 2 6 2 2 6 2 6 2   2    (80)  3 2 2     15    2  sin  2     sin      K 5 sin  4   2  sin   2  2 3 3   3  4    3  1 5K 5  7 K 7 sin 6  3 sin 3   21 K 7 sin 8  2  4  sin 2  4   2 8 3    3  D r      A5 r     2 5    5    5  cos 5  2  sin 6  2        2 5   5 5   5   5      cos 5     sin  2 cos 5    sin  3 2   6 2  6 2   2    1 3 2    2  cos 6  3 sin 3   cos 4   2   sin   2  2 4 3    3  3 2     3    K 5 cos10   5  sin   5      5K 5  2 3   3  2  3   (81)  (82)  B 5 r       (83)  7 21 2     K 7 cos 12  6 sin 6    K 7 cos 2     sin     4 2 3   3   38  C 5 r       5   5 5  sin 5  2  sin 6  2        2 5  2 5   5 5   5   5      sin 5     sin  2 sin 5    sin  3 2   6 2  6 2   2    1 3 2    2  sin 6  3 sin 3   sin  4   2  sin   2  2 4 3    3  2    2  K 5 sin 10   5  sin   5  3    3  21 2     K 7 sin 12  6  sin 6    K 7 sin  2     sin     2 3   3   (84)  D 5 r      3 2    7 4  A7 r      2 7    7    7  cos 7  2  sin 6  2         7    7   7    7      cos 7    sin   2 cos 7    sin  3 2  6 2  6 2   2    1 3 2    2  cos 6  3 sin 3   cos 8   4  sin   4  2 8 3    3  15 2     3    K 5 cos 2     sin         7 K 7  2 3   3  2  3   (85)  (86)  B 7 r       (87)  5 3 2     K 5 cos 12  6  sin 6    K 7 cos14   7   sin   7   4 2 3   3   C 7 r     2 7    7    7  sin  7  2  sin  6  2         7    7   7    7      sin  7    sin    2 sin  7    sin   3 2  6 2  6 2   2     (88)  39  1 3 2    2  sin 6  3 sin 3   sin  4   2   sin   2  2 4 3    3  2    2  K 5 sin 10   5  sin   5  3    3  21 2     K 7 sin 12  6  sin 6    K 7 sin  2     sin     2 3   3   D 7 r      3 2    7 4  (89)  To complete the AVM, the commutation angle  has to be defined. An implicit transcendental equation was proposed in [28] that considers steady-state operation and sinusoidal back EMF, and requires iterative numerical solution. However, due to complexity of equations in our case, it is impractical (and even impossible) to derive a closed-form explicit analytical expression for the commutation time/angle.  4.2  Model Implementation  r In general, the commutation angle depends on the motor speed  r , the phase currents iqs  r and ids , the supply voltage Vdc , and the machine parameters. Following the approach  established in [31],  may be expressed as an algebraic function of the state, and input r variables  r , iqds , and Vdc , respectively, and can be numerically established based on the  results from the detailed simulation. Similar approach was also used for establishing the duty ratio constraint for the analysis of dc-dc converters [28]–[29]. Thus, a function      r  r ,Vdc , iqds may be established by running the detailed simulation in a loop spanning a  desired range of operating conditions, where the currents are averaged according to (16), and   r , Vdc , and  are recorded in a look-up table. The results can be further simplified by defining the dynamic impedance of the inverter switching cell as  z  Vdc r iqds  (90)  which combines two parameters into one, and consequently reduces the dimension of the developed look-up table. As a result, the commutation angle can be defined by a two40  dimensional look-up table   r , z  . The corresponding results for Motor A and Motor B are plotted in Figures 4.1 and 4.2, respectively. Here, it may be noticed that Motor B, has slightly higher commutation angle compared to Motor A, which is also consistent with their respective stator electrical time constants. The considered range for each motor has been chosen based on the rated values for each motor and the available measuring equipment. Figure 4.1  Commutation angle look-up table function for the Motor A.  41  Figure 4.2  Commutation angle look-up table function for the Motor B.  Finally, the implementation of the proposed AVM is established according to the block diagram depicted in Figure 4.3. The input into model is the inverter instantaneous dc voltage  v dc . Based on the commutation angle   r , z  , the combined average voltages in each reference frame are calculated using (72)–(77) which provides the input into the state equations of the electrical subsystem defined by (25)–(30). The outputs of the six-order state model are the averaged currents (for each reference frame) that are used to calculate the torque according to (20). The mechanical subsystem is defined by (9)–(10), and it calculates the (mechanical and electrical) speed of the motor.  42  Figure 4.3  4.3  Block diagram of the AVM implementation.  Case Studies  To demonstrate the improvement introduced by the proposed average-value model the proposed model has been implemented in Matlab/Simulink [4] together with the previously established models and the detailed switching model (which is also used as the reference). To demonstrate the generality of the new model, all studies are carried out for the two considered motors with different back EMF shapes.  43  4.3.1  Steady-State  Without loss of generality, the same steady-state operating points described in Chapter 3, – subsection 3.2.2– are considered here. The instantaneous and averaged electromagnetic torque predicted by different models, are superimposed in Figures 4.4 and 4.5 for Motor A and Motor B, respectively. It may be noticed from the figures that as expected, the relative error due to neglecting back EMF harmonics is more significant for Motor A, where ignoring the commutation angle has more pronounced impact on the performance of Motor B. The reason can be explored considering that Motor A possesses more considerable amount of back EMF harmonics in comparison with Motor B in which the commutation angle is relatively larger. To examine the model performance in a broader operating range, the torque-speed characteristics predicted by all considered models are superimposed in Figures 4.6 and 4.7 for Motor A and Motor B, respectively. For convenience of comparing the two motors, here the characteristics are plotted in terms of mechanical speed,  m  30  2 P  r , that has units of rpm. As can be seen in Figure 4.6, the AVM that assumes sinusoidal back EMF and neglect the commutation predicts the lowers torque. Including the back EMF harmonics increases the torque and improves the AVM accuracy. The most accurate AVM is the one that includes both the back EMF harmonics and the commutation. The characteristic predicted by this AVM matches the detailed model very well. Similar observations can be made in Figure 4.7 with regard to Motor B. However, since this motor has close to sinusoidal back EMF, the impact of including the harmonics is not very significant. At the same time, this motor has generally larger commutation angle (see Figure 4.2). Therefore, including the commutation subinterval in this case has more pronounced improvement, which is also achieved by the proposed AVM.  44  Figure 4.4  Steady-state torque as predicted by various models for Motor A.  Figure 4.5  Steady-state torque as predicted by various models for Motor B.  45  Figure 4.6  Steady state torque-speed characteristic as predicted by various models for Motor A.  Figure 4.7  Steady state torque-speed characteristic as predicted by various models for Motor B.  46  4.3.2  Transient Response to Mechanical Load Change  To further verify the proposed AVM, the following transient study is considered next. The motor is assumed to initially operate in steady state defined by a certain mechanical load (Motor A: 150 W at 1820 rpm; and Motor B: 90 W at 1650 rpm). The corresponding mechanical torque is defined by (10) for each motor. At t  1s , the load torque To is changed from 0.2 to 1N.m . The system response predicted by various models is plotted in Figures 4.8 and 4.9 corresponding to Motor A and Motor B, respectively. As expected, the increase in load is followed by the decrease in motor speed  r and increase of the dc current idc . It can be further seen in Figures 4.8 and 4.9, that the AVMs that do not include the commutation effect are noticeably off in predicting the qd -axis current, ids . This is more noticeable for Motor B, which has larger commutation angle (see Figure 4.9). At the same time, including the commutation has less of an effect for Motor A (see Figure 4.8), wherein including the back EMF harmonics has more pronounced improvement in accuracy. Figure 4.8  System response to sudden load change for the Motor A.  47  Figure 4.9  4.3.3  System response to sudden load change for the Motor B.  Start-Up Transient  Next, the proposed AVM is next against the detailed model based on the prototypical start-up transients of the prototype motors as shown in Figures 4.10 and 4.11, corresponding to Motor A and Motor B, respectively. As can be observed in these figures, the developed AVM precisely predicts the start-up transient of the motors, regardless of the back EMF harmonics content and the length of the commutation interval.  48  Figure 4.10  Start-up transient of Motor A as predicted by various models.  Figure 4.11  Start-up transient of Motor B as predicted by various models.  49  4.3.4  Transient Response to Input Voltage Change  The accuracy of the detailed switching model has been established in Section II. For consistency, the same transient study of Chapter 2, subsection 2.2.2 is considered again; wherein the motors are subjected to the dc supply voltage step increase from 20V to 23V. The resulted waveforms of idc (measured and simulated) and Te predicted by the AVM and the detailed model are superimposed in Figures 4.12 and 4.13, for Motor A and Motor B, respectively. Due to the limited space and for better clarity, only the final proposed AVM that includes back EMF harmonics and commutation subinterval is considered in this study. As can be seen in Figures 4.12 and 4.13, the increase in the applied voltage causes the respective increase in the electromagnetic torque Te and the drawn dc current idc , and the motors go through the transient that is determined by their respective electromechanical parameters and inertia. Furthermore, the AVM predicts the transient responses for both Motor A and Motor B with very good agreement with the measurement and the detailed switching model.  Figure 4.12 Measured and simulated response to the input voltage change as predicted by the detailed and proposed average-value models for Motor A.  50  Figure 4.13 Measured and simulated response to the input voltage change as predicted by the detailed and proposed average-value models for Motor B.  51  5  Conclusion  Average-value modeling is indisputable for analysis of power-electronic-based electromechanical systems where the simulation speed and accuracy must be as high as possible. Nevertheless, the AVM is particularly valuable for analysis of the so-called voltage-source inverter driven brushless dc motors which are widely used in various industrial applications. Several AVMs were previously proposed in the literature for the BLDC motor-drive system in which the effects of the back EMF harmonics and/or the commutation subinterval were ignored resulting in significant amount of error in predicting the machine‟s performance. However, this thesis presents a new and improved AVM for the 120° VSI-driven BLDC which is capable of precisely predicting the actual motor‟s behavior, regardless of the shape of the back EMF waveform and/or the length of the commutation subinterval.  5.1  Summary  This thesis presented a new and improved average-value model for the commonly used 120degree VSI-controlled BLDC motors. The challenges in establishing dynamic average models for such motor-drive systems include the commutation/conduction pattern of the stator phase currents as well as the back EMF harmonics which may be quite significant in trapezoidal BLDC machines. The approach considered in this paper is based on utilizing the multiple reference frame theory for including the contribution from each significant harmonic over conduction and commutation subintervals into the average-value relationships of the voltages, currents, and developed electromagnetic torque, which are all combined into the final state model. The presented studies are based on two typical industrial BLDC motors with different back EMF waveforms and electrical time constants. The results are compared with experimental measurements as well as previously established reference models, whereas the proposed average-value model is shown to provide appreciable improvement for either trapezoidal or sinusoidal motors.  52  5.2  Future Research Topics  In this thesis, only one operating mode of the BLDC motor-inverter system, the so-called negative-zero (NZ), is considered in which the overall switching interval is divided into two subintervals. Average-value modeling of the BLDC motors with non-sinusoidal back EMF in other operating modes has not been done can be further pursued by the UBC research group. It has also been assumed that the BLDC operates with advance firing angle fixed at 30 degrees. However, in some applications, the BLDC machine can operate with different firing angles and more complicated switching patterns of the stator currents with up to three subintervals within a single switching interval may be taken into consideration for future studies. In addition, the hall sensors are considered to be exactly 120° apart, which may not be true in some cases. Including the effect of misalignment of the hall sensors on the presented AVM can be another potential topic for the future research. Parameter identification, which provides the model with the online dynamic changes of the system parameters, is another potential subject for extending this research on the BLDC motorinverter system with the non-sinusoidal back EMF. These and other topics will be considered by other graduate students who are joining the UBC power group.  53  References [1]  SimPowerSystems: Model and simulate electrical power systems User‘s Guide, The MathWorks Inc., 2006 (www.mathworks.com).  [2]  Piecewise Linear Electrical Circuit Simulation (PLECS) User Manual, Version. 1.4, Plexim GmbH (www.plexim.com).  [3]  Automated State Model Generator (ASMG) Reference Manual, Version 2, P C Krause & Associates, Inc. 2003 (www.pcka.com).  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Wood, Power System Harmonic  Analysis, John Wiley and Sons, 1997.  57  Appendices Appendix A : Prototype Parameters  A.1 Motor-A Parameters ECycle Inc., Model MGA-1-13, 4.5kW, 12poles, rs  0.2, Ls  0.025mH,  m  10.9mV.s , back-EMF harmonics: K1  1, 3K 3  0.20, 5K 5  0.047, 7 K 7  0.0067, J  5  10 3 kg.m 2 , J  5 10 3 kg.m 2 , T1  2.7 10 3 N.m  rpm  ,  To  0.2N.m A.2 Motor-B Parameters Arrow Precision Motor Co., LTD., Model 86EMB3S98F-B1, 210W, 8poles, rs  0.125,  Ls  0.4mH,  m  21.8mV.s , back-EMF harmonics: K1  1, 3K 3  0.0035, 5K 5  0.039, 7 K 7  0.017, J  5 10 4 kg.m 2 , T1  7 10 4 N.m  rpm  , To  0.1N.m  58  Appendix B : BLDC Motor Controller  Figure B.1 Motor controller schematic.  59  Figure B.2  Controller PCB.  60  Figure B.3  BLDC motor controller box.  61  

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