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Average-value modeling of brushless dc motors with 120-degree voltage source inverter Han, Qiang 2007

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AVERAGE-VALUE MODELING OF BRUSHLESS DC MOTORS WITH 120-DEGREE VOLTAGE SOURCE INVERTER by Qiang Han B. Eng., Tsinghua University, 2004 A THESIS SUBMITTED IN PARTIAL FULFILMENT 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 September 2007 © Qiang Han, 2007 / ABSTRACT Average-value modeling is indispensable for large and small-signal analysis of electromechanical systems with power electronic drives. Analytical development of accurate average-value models for the brushless dc motor with 120-degree inverter systems is particularly challenging due to the complicated commutation-conduction patterns of the stator currents. This thesis is comprised of several manuscripts that present the research results in this area. First manuscript compares the brushless dc motors with 180 and 120-degree inverters, and the average-value modeling of these systems. In 180-degree inverters, each phase is always connected to the dc source and the average voltages are readily known. In 120-degree inverters, each phase is open-circuited for a fraction of a revolution resulting in discontinuous phase current and different commutation-conduction patterns, which makes it challenging to obtain an accurate average-value model. The second manuscript extends a recently proposed parametric approach to numerical averaging of power electronic converters to construct an average-value model for a brushless dc motor with a 120-degree inverter system. In the proposed model, a proper qd model of the permanent magnet synchronous machine is used, and the inverter dynamics are represented by nonlinear algebraic functions that relate the averaged terminal voltages and currents of the inverter switching-cell. These parametric functions are obtained numerically using the detailed model. The third manuscript presents a different parametric approach to construct a new average-value model that possesses greater accuracy. This new model also uses proper qd model of the permanent magnet synchronous machine but is based on appropriately averaging the transformed stator voltages in both commutation and conduction subintervals. In this approach, the required commutation angle is represented as a nonlinear algebraic function, which is obtained from the detailed simulation. The conducted case studies are based on two typical industrial motors with different electrical time constants, and include hardware measurements, detailed simulation, and comparison with the best-knbwn-to-day published average-value model. The proposed models are shown to be more accurate in a large- and small-signal sense than the previous published model and applicable to motors with a large range of parameters including electrical time constant. TABLE OF CONTENTS Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgements viii Co-Authorship Statement ix 1 Introduction 1 1.1 Motivation for Average-Value Modeling of Machine-Drive Systems 1 1.2 Brushless DC Motors 3 1.3 Contributions 6 1.4 Composition of Thesis 7 1.5 Reference.1. 8 2 Comparison Of Brushless DC Motor Drives With 180/120-Degree Inverter Systems 10 2.1 Introduction : 10 2.2 System Operation and Detailed Modeling 12 2.3 180-Degree B L D M Average Value Modeling 14 2.4 120-Degree B L D M Average Value Modeling 15 2.5 Conclusion 17 2.6 References 18 3 Numerical Average-Value Modeling Of The Brushless DC Motor 120-Degree Inverter System 19 3.1 Introduction 19 " 3.2 Case System Detailed Model 21 3.3 Average Value Modeling 23 3.4 Computer Study 25 3.5 Conclusion 26 3.6 References 26 4 Average-Value Modeling Of Brushless DC Motors With iii 120-Degree Voltage Source Inverter 28 4.1 Introduction 28 4.2 System Operation And Detailed Modeling 30 4.2.1S ystem Operation 30 4.2.2De tailed Modeling 32 4.3 Average-Value Modeling 34 4.3.1A veraging Stage 34 4.3.2A verage-Value Modeling of 180-Degree System 35 4.3.3A verage-Value Modeling of 120-Degree System 36 4.4 Proposed Average-Value Model 40 4.5 Case Studies ^ 43 4.5.1T ime Domain Model Verification 44 4.5.2F requency Domain 47 4.6 Conclusion 48 4.7 References 49 5 Summary 51 5.1 Conclusions : 51 5.2 Future Work 52 Appendices 53 iv LIST OF TABLES TABLE 1.1 Comparison of simulation times between the detailed model and the proposed average-value model 5 TABLE 3.1 Switching Logic For 120° Inverter 20 i v LIST OF FIGURES Figure 1.1 Block diagram of a typical electro-mechanical system 1 Figure 1.2 Diagram of a typical BLDC motor-inverter system 3 Figure 1.3 Maxon brushless dc motor. 4 Figure 1.4 Arrow precision motor. 4 Figure 1.5 Speed and torque transient of the Maxon motor. 6 Figure 2.1 PMSM with Hall sensors 10 Figure 2.2 B L D M inverter circuit 11 Figure 2.3 Switching signals 12 Figure 2.4 Phase voltage and current waveforms for the 180-degree B L D M 14 Figure 2.5 Start-up transient of the 180-degree B L D M 15 Figure 2.6 Typical voltage and current waveforms for the 120-degree B L D M 15 Figure 2.7 Start-up transient of 120-degree BLDM: small stator resistance 16 Figure 2.8 Start-up transient of 120-degree BLDM: large stator resistance 16 Figure 3.1 B L D M inverter circuit 19 Figure 3.2 Measured and simulated phase current for mode 1 22 Figure 3.3 Phase current response to the change in mechanical torque 23 Figure 3.4 Phase current operating modes: (a) NZ; (b) PZ; and (c) PZN 23 x Figure 3.5 Steady-state torque-speed characteristic and operating modes 23 Figure 3.6 Average-value model block diagram ; 24 Figure 3.7 Numerically extracted parametric functions a, B, and S 25 Figure 3.8 Start up transient response 25 Figure 4.1 PMSM with Hall sensors 30 Figure 4.2 BLDC motor inverter circuit 31 Figure 4.3 Switching signals 31 Figure 4.4 Typical phase voltage and current waveforms for the 180-degree inverter operation. ; .36 Figure 4.5 Start-up transient of the BLDC motor driven by 180-degree inverter. 36 Figure 4.6 Typical voltage and current waveforms for the 120-degree B L D C motor-inverter system ; 37 Figure 4.7 Start-up transient of the Motor A driven by 120-degree inverter. 39 Figure 4.8 Start-up transient of the Motor B driven by 120-degree inverter. 39 Figure 4.9 Commutation angle j3 for the Motor A 43 vi J Figure 4.10 Commutation angle p for the Motor B 43 Figure 4.11 Average-value model block diagram 43 Figure 4.12 Measured and simulated phase current of the Motor A 45 Figure 4.13 Measured and simulated phase current of the Motor B 45 Figure 4.14 Measuren and simulated esponse in current idc: (a)-Motor A; (b) - Motor B. 45 Figure 4.15 Speed, torque, ids response of Motor A 46 Figure 4.16 Speed, torque, ids response of Motor B 47 Figure 4.17 Voltage-to-torque transfer function for the Motor A. 48 Figure 4.18 Voltage-to-torque transfer function for the Motor B 49 vii ACKNOWLEDGEMENTS My foremost thank goes to my research supervisor Dr. Juri Jatskevich, for introducing me to this very fascinating area of research. I greatly appreciate his guidance and financial support during my research and study at the University of British Columbia. His valuable feedback and suggestions greatly contributed to this thesis. I would also like to thank my thesis committee members: Dr. Hermann Dommel and Dr. William Dunford, who have dedicated their time and effort to this thesis. Nikolay Samoylenko deserve a special thank as the co-author of the manuscripts in this thesis. He helped and assisted me in various aspects of the.research, especially in many experiments that we conducted in the lab. His valuable feedback helped me improve my thesis. A l l the students at the UBC Power Systems Lab made it a very convivial and intellectually stimulating place to work. They have also helped me live through difficult times by encouraging me and sharing interesting and insightful life stories with me. My special thanks go to Liwei Wang, Michael Wrinch, Yong Zhang, Nathan Ozog, Lucy Liu, Weidong Xiao, Yan L i , Tom de Rybel, Marcelo Tomim, and many other students in the group. I also owe a debt of thanks to my loving parents Linlong Han and Xiuhua Wang, and my sister Lanyan Han, who have supported me morally and financially throughout these wonderful and stressful years of my graduate education. CO-AUTHORSHIP STATEMENT The publications included in my thesis are co-authored by me, Nikolay Samoylenko, and Dr. Juri Jatskevich. Nikolay and I collaborated on some common aspects of our research including conducting the laboratory experiments, parameter estimation, and the detailed modeling, since his master's project is also related to the brushless dc motors. While my project and contributions are focused on the average-value modeling of brushless dc motors, Nikolay's research is on improving the operation of brushless dc motors with misaligned Hall sensors. I have designed and implemented modeling techniques, conducted simulations and experiments, and prepared the manuscripts, which was then iteratively edited by the authors. Dr. Juri Jatskevich has supervised and directed my work. ix 1 Introduction 1.1 Motivation for Average-Value Modeling of Machine-Drive Systems Modeling and simulation of power-electronic-based systems are essential steps that are required during the design of electro-mechanical systems such as those in modern industrial automation, robotics, automotive products, ships, and aircraft. For the discussion in this section, a block diagram of an electro-mechanical system is depicted in Fig. 1.1. The External Electrical Subsystem often represents an electrical power distribution that may also contain energy source and/or storage (e.g., battery in the case of a vehicle). The Machine-Drive Subsystem is comprised of two components - Inverter Module, and Electrical Machine. The Inverter Module conditions and regulates the energy flow from the source to the Electrical Machine which actually performs the electromechanical energy conversion. The External Mechanical Subsystem may represent a mechanical drive train (e.g., in case of vehicles) or an assembly positioning system (e.g. in case of industrial manufacturing/automation). In general, if the Inverter Module can operate in all four quadrants, the energy may flow in either direction - from electrical source to External Mechanical Subsystem, or the other way around - from External Mechanical Subsystem back to the electrical source. The operation of Inverter Module is controlled by a control signal denoted in Fig. 1.1 by input variable u(t) (which can be duty cycle, voltage amplitude, etc.). The, output of the Mechanical Subsystem is denoted by y(t) (which can be position or speed of the rotor, or other mechanical variables). External Electrical Subsystem 1 1 —h* I Inverter Module Electrical Machine 1 ! 1 External Mechanical Subsystem y(t) \ Machine-Drive u(t) Subsystem Figure 1.1 Block diagram of a typical electro-mechanical system. 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 for establishing 1 \ the time- and/or frequency-domain relationship between input/control variable u(t) and output variable y(t). Such computer-aided studies are carried out many times with the overall goal of tuning the system and achieving the desired performance while satisfying the design specifications. Because of the very extensive use of computer models and the reliance on their accuracy, both factors - (i) simulation speed and (ii) simulation accuracy, are very important. Modeling and simulation of power-electronic-based systems (Inverter Module) is not trivial in general, since these systems include 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 build and implement the models where the switching of all transistors and diodes is represented in full detail. On the 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). 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. In addition to that, since the detailed models are discontinuous and time-variant, and the state variables are not constant even in steady states due to the switching behaviour, these models cannot be linearized and used for the small-signal analysis. The frequency-domain characteristics can be extracted from a detailed model using, for example, small-signal frequency sweep techniques [6], and simultaneous injection of sinusoidal and possibly spike-like signals [7]. However, since the detailed models are computationally intensive, determining the frequency-domain characteristics using the afore-mentioned techniques over a wide range' of frequencies is a very time consuming procedure, particularly if data points at very low frequencies (/ < 10Hz) are required. At the same time, the very fast switching of the transistors and diodes has only an average effect on the system's slow dynamic behavior [8]. Therefore, 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 2 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 of magnitude, faster than their corresponding detailed models, making them ideal for representing respective components in system-level time-domain transient studies. 1.2 Brushless DC Motors This thesis focuses on brushless dc (BLDC) motor-inverter-systems, which usually consists of two smaller subsystems: Inverter Module, and Permanent Magnet Synchronous Machine (PMSM), as shown in Fig. 1.2. The BLDC motors generally have good torque-speed characteristics, fast dynamic response, high efficiency and long life. The name "brushless dc" is partly attributed to the fact that their external torque-speed and control characteristics resemble those of the conventional brushed DC Motors, but all the commutations are performed using electronic switches - transistors instead of the brushes. In this thesis, we have carried out simulation and experimental studies using the two typical industrial BLDC motors shown in Fig. 1.3 and Fig. 1.4. Similar BLDC motors are increasingly used in industrial automation, instrumentation, robotics, electric vehicles, and many other equipment and servo applications. Figure 1.2 Diagram of a typical BLDC motor-inverter system. 3 Figure 1.3 Maxon brushless dc motor. Figure 1.4 Arrow precision motor. A special characteristic of a BLDC motor is that the inverter transistors are switched depending on the instantaneous position of the rotor, therefore converting a dc supply voltage into non-sinusoidal 3-phase voltages with the frequency directly corresponding to the rotor speed. Such self-commutation is commonly achieved by using the Hall-effect sensors that are mounted inside the PMSM and provide the logical signals of the rotor position. Although use of optical encoders is also possible in more advanced drives and applications, this thesis focuses on typical Hall-sensor-based voltage-source-inverter-driven (VSI-driven) BLDC motors because such motors are less expensive and cover a much wider range of applications. In a typical arrangement, three Hall sensors are mounted 120 electrical degrees apart from each other providing 180 degree square-wave "on/off' signals depending on the rotor flux. This arrangement results in knowing the rotor position within six sectors of a circle (60 electrical degrees each). Using the Hall-sensor signals, the inverter phases may be turned "on" for 180 or 120 electrical degrees corresponding to continuous or discontinuous operation, respectively. In the 120-degree BLDC motor-inverter 4 systems, which are actually very common and are the main focus of this thesis, each phase is allowed to be open-circuited for a fraction of a revolution, giving rise to complicated commutation-conduction patterns of the stator currents. To give the reader a better idea of the difference in simulation speed of the detailed models and the AVMs, we conducted a transient study using one of the BLDC motors used in this thesis. In the following study, the Maxon motor (see Fig. 1.3) starts from a stall, and at t = 0.6s, a mechanical load of 1 Nm is applied after which the model is continued to run until t = 1 .Os. The details of the models used in this study are presented in Chapter 4. The corresponding time-domain transient responses of the rotor speed 0)r and the electromagnetic torque Te computed by each model are shown in Fig. 1.5. The CPU times required by both models to simulate this ls-study are shown in Table 1.1. As can be observed in Fig. 1.5, the responses predicted by the A V M are in excellent agreement with those predicted by the detailed model. However, using the same relative and absolute error tolerances, the A V M took only 128 time steps and 0.016s to compute, which amounts to the simulation speed-up of over 320 times as compared to the detailed switching model. To accurately capture all the switching information present in the detailed model, the time-step must not exceed the commutation time of the given BLDC motor. This results in a time-step limit of le-4s used with the detailed model. At the same time, the A V M has no switching and may use much larger time-steps to satisfy the same error tolerance. Other advantages are explained in detail in Chapters 2 - 4 . TABLE 1.1 Comparison of simulation times between the detailed model and the proposed average-value mod_el. Solver used in both models: Odel5s Simulation Time Max Step Size Relative Tolerance Absolute Tolerance Number of Time Steps CPU Time Detailed Model Is le-4 le-4 le-4 23874 5.157s Average-Value Model Is auto le-4 le-4 128 0.016s 5 300 0 0.2 0.4 • 0.6 0.8 1 Time, s Figure 1.5 Speed and torque transient of the Maxon motor. 1.3 Con t r i bu t i on s i Many averaging techniques have been previously developed in the literature including the popular state-space averaging method. The state-space averaging is a very useful tool primarily used for small-signal modeling of switching converters. It involves averaging the sets of state-space equations of the system in several possible topological configurations of the converter circuit and requires calculation of the times spent in each encountered topology. Although it may appear straightforward to use this technique with the basic D C - D C converters, the application of this approach to deriving AVMs of the B L D C motor-inverter system is very challenging, particularly when the inverter operates using the 120-degree switching logic and the phase currents are discontinuous. Over the past several decades, modeling of B L D C motor-drive systems has attracted attention of many researchers including R. Krishnan, P. C Krause, S. D. Sudhoff, K. A . Corzine, and H. A. Toliyat [9] - [15]. However, prior to the work reported in this thesis, the best known to us averagervalue model of a B L D C motor with a 120-degree inverter remain the dynamic A V M presented in [11], which represents a significant contribution to the area but does not consider the commutation time and current, which significantly simplifies the analytical derivations. This thesis recognizes that the presence of operating-point-dependent commutation and conduction subintervals in the motor phase current significantly complicates the classical averaging procedure making the analytical approaches not practical. This thesis is inspired by the previous research in this area. The motivation is also driven by the presently increasing range of applications of B L D C motors, on the one hand, and the need for more accurate and fast models, 6 on the other hand. In particular, the thesis analyzes the BLDC motors with 180- and 120-degree switching logic and makes the following contributions: 7 • We show that only for the motors with large stator resistance (small electrical time constant) the commutation subinterval may be neglected. For the motors with small stator resistance, neglecting the commutation subinterval introduces significant error. • This thesis proposes two new explicit AVMs thattake into account both commutation and conduction subintervals. The first A V M represents the inverter as a composite switching cell which is replaced by three algebraic functions. The second A V M requires only one parametric function to represent the commutation angle. The new AVMs are more accurate in predicting both large-signal time-domain transients and small-signal frequency-domain characteristics than the best known to us previously published model [11] which neglects the commutation subinterval. We also show that the proposed AVMs are accurate for motors with both large and small stator resistances (electrical time constants). • Since it is not practical to analytically derive a closed-form solution for the algebraic function(s) used in the proposed AVMs, these solutions have been obtained numerically using the detailed simulation. This numerical-parametric approach reduces laborious analytical derivations and has been shown to give accurate and practical results for DC-DC converters [16], and machine-rectifier systems [17]. 1.4 Composition of Thesis This thesis is written in the manuscript format and is comprised of several publications that describe research on average-value modeling of brushless dc motor-inverter systems. Chapter 2 compares the operational characteristics of brushless dc motors with 180-degree switching logic and 120-degree switching logic. It demonstrates that it is easy to derive an accurate average-value model for the 180-degree logic, while it is very challenging to do that for the 120-degree logic due to the complicated commutation-conduction pattern of the stator currents. Neglecting the commutation interval does not always give accurate results, especially when the electrical time constant of the brushless dc motor is large and the commutation interval is not small. Chapter 3 proposes a reduced-order average-value model. In this model, the dynamics of inverter are represented by a set of nonlinear algebraic functions, while the PMSM is expressed in the 7 qd -rotor reference frame. The new model is constructed (without laborious analytical derivations) using the detailed simulation and readily takes into account commutation and conduction intervals. The model is verified by simulation as well as experiments. Chapter 4 proposes a full-order average-value model. In this AVM, only the commutation angle is represented by a nonlinear algebraic function which is an advantage over the AVM proposed in Chapter 3. The PMSM is again expressed in the qd-xoXor reference frame. The algebraic function of the commutation angle is constructed using the detailed simulation. The inverter voltages are averaged over the commutation and conduction subintervals separately. Computer studies as well as experiments are conducted based on two typical industrial BLDC motors (see Fig. 1.3 and 1.4) with large/small electrical time constants. It is shown that the new AVM is applicable to both motors, while the previously published model [11] is only applicable to the motor with small electrical time constant. Chapter 5 concludes the thesis by summarizing the overall results. 1.5 Reference [1 ] SimPowerSys terns: Model and simulate electrical power systems User's Guide, The Math Works 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). [4] Simulink: Dynamic System Simulation for Matlab, using Simulink, Version 6, The MathWorks Inc., 2006 (www.mathworks.com). [5] Advanced Continuous Simulation Language (acslXtreme) Language Reference Guide, Version 2.3, The AEgis Technologies Group, Inc., Huntsville, AL, 2006. [6] M.B. Harris, A.W. Kelley, J.P. Rhode, M.E. Baran, "Instrumentation for measurement of line impedance," Conf. Proc. of Applied Power Electronics Conference (APEC'94), Vol.2, p.87-893, 1994 [7] B. Palethorpe, M. Sumner, D.W. Thomas, "Power system impedance measurement using a power electronic converter," Proc. Ninth International Conference on Harmonics and Quality of Power, Vol. 1, p. 208-213, 2000. [8] I. Jadric, D. Borojevic, M. Jadric, "Modeling and control of a synchronous generator with an active DC load," IEEE Transactions on Power Electronics, Vol.15, no. 2, pp. 303-311, Mar. 2000. 8 [9] P. Pillay, R. Krishnan, "Modeling, simulation, and analysis of permanent-magnet motor drives. Part II. The brushless DC motor drive," IEEE Trans. Industry Applications, vol. 25, pp. 274-279, March-April 1989. [10] R. R. Nucera, S. D. Sudhoff, and P. C. Krause, "Computation of Steady-state Performance of an Electronically Commutated Motor", IEEE Transactions on Industrial Applications, Vol. 25, pp. 1110-1117,Nov.-Dec. 1989. [11] S. D. Sudhoff, P. C. Krause, "Average-value model of the brushless DC 120° inverter system," IEEE Trans. Energy Conversion, vol. 5, pp 553-557, 1990. [12] K. A. Corzine, S. D. Sudhoff, H. J. Hegner, "Analysis of a current-regulated brushless DC drive," IEEE Trans. Energy Conversion, vol. 10, pp. 438^ 145, Sep. 1995. -[13]K. A. Corzine, S. D. Sudhoff, "A hybrid observer for high performance brushless DC motor drives," IEEE Trans. Energy Conversion, vol. 11, pp. 318-323, lune 1996. [14] P. L. Chapman, S. D. Sudhoff, and C. A. Whitcomb, "Multiple reference frame analysis of non-sinusoidal brushless DC drives," IEEE Trans. Energy Conv., vol. 14, pp. 440-446, Sep. 1999. [15] Lei Hao, H. A.. Toliyat, "BLDC motor full-speed operation using hybrid sliding mode observer," In Proc. 18th IEEE Applied Power Electronics Conference and Exposition (APEC'03), 9—13 Feb. 2003, vol. 1, pp. 286-293. [16] A. Davoudi, J. Jatskevich, and T. DeRybel, "Numerical State-Space Average-Value Modeling of PWM DC-DC Converters Operating in DCM and CCM," IEEE Trans. Power Electronics, vol. 21, pp. 1002-1012, Jul. 2006. [17] J. Jatskevich, S. D. Pekarek, and A. Davoudi, "Parametric average-value model of a synchronous machine-rectifier system," IEEE Trans. Energy Conversion., vol. 21, pp. 9-18, Mar. 2006. 9 2 Comparison Of Brushless DC Motor Drives With 180/120-Degree Inverter Systems1 2.1 Introduction Three phase brushless dc motor (BLDM) inverters may operate using 180- or 120-degree commutation methods, whereas the latter method is very common for Hall-sensor-driven machines. This paper considers an inverter circuit with a permanent magnet synchronous machine (PMSM) as shown in Figs. 2.1 and 2.2. A detailed model of this system, in which the switching of each transistor and diode is represented, can be readily constructed using various simulation tools [l]-[3]. However, for the purpose of extracting small-signal transfer-function characteristics and/or large-signal system-level transient studies, it is advantageous to use the so-called average-value models (AVMs), where the effect of fast switching is averaged within a prototypical switching interval (60-degfee for 3-phase machines). Unlike detailed switch-level models, AVMs are continuous and can therefore be linearized about a desired operating point. Thereafter, obtaining any local transfer function and/or frequency-domain characteristics becomes a straightforward procedure. This feature makes AVMs indispensable for modeling and analysis of motor-drive systems. Figure 2.1 PMSM with Hall sensors. Derivation of AVMs requires careful averaging of the stator phase voltages and currents over a prototypical switching interval to obtain the corresponding average torque. In the 180-degree inverter system, each phase is always connected to either positive or negative terminal of the dc 1 A version of this chapter has been published. Q. Han, N. Samoylenko and J. Jatskevich, 'Comparison of Brushless DC Motor Drives with 180/120-Degree Inverter System', In proc: IEEE Canadian Conference on Electrical and Computer Engineering, April 2007, Vancouver, Canada. 10 source and the phase currents are always continuous. This is utilized in several models proposed in the literature for the machines with sinusoidal [4] and non-sinusoidal back emf [5], whereas for the latter case the use of multiple reference frames has been proposed. In the 120-degree inverter system, each phase is allowed to be open-circuited for a fraction of revolution. Depending on the firing offset angle and operating conditions, the commutation-conduction pattern of currents may change within a single switching interval giving rise to several distinct operational modes documented in the literature [6]. Multiple operational modes coupled with the difficulty in finding closed-form analytical solutions for the averaged currents in conduction and commutation sub-intervals make it challenging to obtain an accurate AVM. In [7], the authors derived an average-value model for the 120-degree B L D M inverter system considering one dominant operating mode, neglecting the commutation time and current. This paper compares the characteristics of B L D M with 180/120-degree inverters, and describes the procedure for developing AVMs. The A V M for 180-degree inverter is straightforward and the model accuracy is verified by simulation results. The challenges of developing an accurate A V M for the 120-degree inverter are investigated. This paper investigates the A V M proposed in [7] with different values of stator resistance and shows that neglecting commutation interval may give satisfactory results when the stator resistance is relatively large, whereas the model accuracy significantly degrades otherwise. 'dc s Figure 2.2 BLDM inverter circuit. 11 \S6 H, (a) H2 (b) 9r+<|) switching I | II | III | IV | V | VI | I interval ^ _n 5_7t In 3TC 1_1TC 6 2 6 6 2 6 Figure 2.3 Switching signals. 2.2 System Operation and Detailed Modeling The Hall-sensor-driven B L D M inverter system is a self-commutating device. The Hall sensors are used to detect the rotor position. Depending on the "construction of a particular PMSM, the Hall , sensors may be mounted inside of the motor case and interact directly with the rotor magnetic poles, or mounted outside and interact with magnetic poles of an auxiliary magnet-tablet. The logical output signals from the three Hall sensors //,, H2, H3 (see Fig. 2.1) are shifted in space by 120 electrical degrees and produce signals as depicted in Fig. 2.3 (a). Here, the rotor position and the advance firing angle are denoted by 6r and </>, respectively. If the motor is driven according to the 180-degree logic, the transistor signals S{ S2, S3 coincide with the Hall sensor outputs Hx, H2, H3, and the lower transistors signals are complementary S4 , S5 , S6 , as shown in Fig. 2.3 (a). When the motor is driven according to the 120-degree logic, each transistor is turned on 60 degrees later using the Hall sensor signal of the preceding phase but is turned off as before according to the sensor in its phase. Therefore, according to this logic, each phase is gated-on for 120 degrees two times during a complete revolution. This paper considers the B L D M prototype with parameters summarized in the Appendix A. The classical model in physical coordinates uses the following assumptions [4]: (i) saturation is negligible; (ii) back emf is sinusoidal; (iii) eddy currents and hysteresis losses are negligible. The stator electrical dynamics are described by the following voltage equation 12 ^abcs ^ s^abcs " dt (1) where fabcs = [fas fbs fcs J; and / may represent voltage, current or flux linkage. The stator resistance matrix is The flux linkage equation is r.v =diag[rs,rs,rs]. ^•abcs ~ ^ s^abcs ~*~ where the inductance matrix and the rotor flux are L , = Lls+Lm -0.5Lm -0.5Lm -0.5Lm Lls+Lm -0.5Lm -0.5Lm -0.5Lm Lls+L„ 1 ' — J ' Km ~ Am sin(9r sin(<9 r-120^ sinff?,. +120° The developed electromagnetic torque is given by |ias cos dr + ^ -(ibs - ics )sin dr (2) (3) (4) (5) (6) The mechanical subsystem is assumed here as a single rigid body for which the dynamics may be expressed as pcor P T -T 1 ±e ±m (7) 2 J where wr is the rotor angular speed, Tm is the mechanical load torque, and J is the combined moment of inertia of the load and the rotor. In AVM, the state variables should be constant in steady state. Therefore, (1) is transformed to the qd -rotor reference frame using Park's Transformation, which results in Vqs = (rs + PLs)'qs + <°rL/ds + <OrX'» Vds = {rs+pLsyds-corL/qs (8) 13 Replacing the variables in (8) with their fast averages over a switching interval (denoted by bar above) and rearranging the terms yields PlQ vqs ~ rJqs ~ MrLJds ~ Wr^'m L s (9) -r vds ~ rs'ds ~ cor^siqs Plds= : — The averaged developed electromagnetic torque is given by L ° - \ - ^ y m l q S (10) Eqs. (9H10) f o r m m e s t a t e model of the PMSM with and as inputs. These voltages are determined by the inverter and depend on the switching logic. 2.3 180-Degree BLDM Average Value Modeling Here, the switching transistors in the same inverter leg are complementary and the stator phase voltages are readily known [4]. Typical phase voltage and current waveforms for this type of inverter operation are shown in Fig. 2.4. The fast average of q- and d-axes voltages may be expressed as [4] vqs =-V r f cCOS^ -r 2 • A 71 "I Figure 2.4 Phase voltage and current waveforms for the 180-degree BLDM. 14 0 0.05 0.1 0.15 0.2 Time (s) Figure 2.5 Start-up transient of the 180-degree BLDM. The A V M is formed by (9)-(l 1) with mechanical subsystem represented by (7). To demonstrate the performance and accuracy of this model, a start-up transient was simulated by both the detailed and the averaged models. In this case study, the motor accelerates from stall with a friction torque Tj-ric. As can be observed in Fig. 2.5, the A V M predicts the dynamic response very well. 2.4 120-Degree BLDM Average Value Modeling In 120-degree BLDM, the switching of the two transistors in the same phase are not complementary, giving rise to two 60-degree intervals per revolution when both transistors are gated off. Typical phase voltage and current waveforms are shown in Fig. 2.6(a). Figure 2.6 Typical voltage and current waveforms for the 120-degree BLDM. 0 0.05 0.1 0.15 0.2 Time (s) Figure 2.7 Start-up transient of 120-degree BLDM: small stator resistance. 0 0.05 0.1 0.15 0.2 Time (s) Figure 2.8 Start-up transient of 120-degree BLDM: large stator resistance. The commutation-conduction pattern defines the operational mode [6]. A simplified fragment of the phase current ibs is depicted in Fig. 2.6(b) wherein the switching interval Ts is also shown. Since ibs is negative at the beginning of the switching interval Ts and then goes to zero and remains zero until the end of that interval, this mode is referred to as negative-zero (NZ). Here, the commutation time is denoted by tcom , whereas the conduction time is Ts - tcom during which the other two phases conduct the current. The commutation time tcom depends on the motor parameters (phase winding electrical time constant) and operating conditions, but in general cannot be zero. Presence of this operating-point-dependant factor makes it very difficult (if not impossible) to derive a closed-form explicit analytical expression for the average phase voltages and currents. The authors of [8] proposed an implicit transcendental equation for tcom , which requires iterative numerical solution. If,- however, the commutation time tcom is neglected - assuming that gated-off phase current 16 goes to zero instantaneously at the beginning of each switching interval Ts, the average voltages in qd -rotor reference frame may be expressed as [7] Thereafter, the A V M is formed by (7), (9), (10) and (12). The details of derivation of (12) can be found in [7]. The same start-up transient study has been implemented with the 120-degree inverter assuming original motor (rs = 0.15Q). The corresponding responses produced by detailed and averaged models are compared in Fig. 2.7. As can be observed, the A V M does not accurately predict the dynamic response. A second study, in which the stator resistance was increased to 0.8Q, is shown in Fig. 2.8. This results in significant reduction of the stator current and torque during the initial part of the transient, as well as the commutation interval tcom . As can be noted in Fig. 2.8, the A V M now predicts the transient trajectory quite well compared to the detailed simulation. This limits the application of this A V M to the B L D M with large stator resistance only. 2.5 Conclusion Modeling the average behavior of B L D M inverter system is of particular importance for analysis and control of electromechanical systems with B L D M drives. The A V M for the 180-degree B L D M is readily available. However, deriving an accurate A V M for the 120-degree B L D M is challenging due to the discontinuous mode of the phase current. The best existing A V M [7] (as presently known to the authors) neglects the commutation interval, and is therefore may not be sufficiently accurate for many practical BLDMs. An alternative A V M that completely overcomes these limitations is proposed by the authors in the companion paper [9]. (12) 17 2.6 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 Ver. 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). [4] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, IEEE Press, Piscataway, NJ, 2002. [5] P. L. Chapman, S. D. Sudhoff, and C. A. Whitcomb, "Multiple Reference Frame Analysis of Non-sinusoidal Brushless DC Drives," IEEE Trans. Energy Conv., vol. 14, pp. 440-446, Sept. 1999 [6] 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. [7] 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. [8] R. R. Nucera, S. D. Sudhoff, and P. C. Krause, "Computation of Steady-state Performance of an Electronically Commutated Motor", IEEE Transactions on Industrial Applications, Vol. 25, pp. 1110-1117, Nov.-Dec. 1989 [9] Q. Han, N. Samoylenko, and J. Jatskevich, "Numerical Average-Value Modeling of the Brushless DC Motor 120-Degree Inverter System," IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), April 22-26, 2007, Vancouver, BC, Canada. 18 3 Numerical Average-Value Modeling Of The Brushless DC Motor 120-Degree Inverter System2 3.1 Introduction Three-phase brushless dc motor (BLDM) inverters may operate using 180/120-degree commutation methods, whereas the latter method is very common for Hall-sensor-driven machines. This paper considers an inverter circuit with a permanent magnet synchronous machine (PMSM) as shown in Fig. 3.1. A detailed model of this system, in which the switching of each transistor and diode is represented, can be readily constructed using various simulation tools [l]-[3]. However, for the purpose of extracting small-signal transfer-function characteristics and/or large-signal system-level transient studies, it is advantageous to use the so-called average-value models (AVMs), which neglect or average the effects of fast switching within a prototypical switching interval. Unlike the detailed switch-level models, AVMs are continuous and can therefore be linearized about a desired operating point. Thereafter, obtaining any local transfer function and/or frequency-domain characteristics about a desired operating point becomes a straightforward procedure. Simulation times for AVMs are typically orders of magnitude faster than that of corresponding detailed models. This property makes AVMs indispensable for representing the respective components for design of controls and system-level studies with large number of modules and subsystems. 'dc permanent magnet synchronous machine Figure 3.1 BLDM inverter circuit. The derivation of AVMs for the 180-degree B L D M is straightforward [4], while analytical 2 A version of this chapter has been published. Q. Han, N. Samoylenko and J. Jatskevich, 'Numerical Average-Value Modelling of Brushless DC Motor 120-Degree Inverter System', In proc. IEEE Canadian Conference on Electrical and Computer Engineering, 2007, April 2007, Vancouver, Canada. 19 J derivation of accurate AVMs for the 120-degree B L D M is very challenging due to the complex switching pattern of stator currents and has not been sufficiently addressed in the literature. In the 120-degree Hall-sensor driven machine, the transistors are gated depending on the rotor position as defined in Table I, which splits one electrical cycle (revolution) of the rotor into six 60-degree switching intervals. Here, 6r is the rotor angle and A represents a possible shift in firing [4]. Within a 60-degree interval, the system's topology changes and may follow different conduction-commutation patterns depending on the operating conditions. These so-called operating modes have been identified and discussed in detail [5]. In [6], the authors derived an A V M that considers only one operating mode,, and even there, commutation current and time still had to be neglected to further simplify the derivation. This significantly reduces accuracy and range of application of the final model as has been shown in [7]. In general, both conduction and commutation intervals should be considered for deriving the correct averages of phase currents over the switching interval. This leads to significant complications and implicit transcendental equations that require iterative (expensive) solution. TABLE 3.1 Switching Logic For 120° Inverter Switching Interval Rotor Angle Gated Transistors I - 30° < <9, + <j> < 30° 1,5 II 30° < er + (f> < 90° 1,6 III 90° < 6"r + < 150° 2,6 IV 150° <6»r + (*< 210° 2,4 V 210° <(9r + ^<270° 3,4 VI 270° < 6r + <j> < 330° 3,5 This paper recognizes that the presence of commutation currents within the 60-degree switching interval is similar to the discontinuous conduction mode in DC-DC converters [8]. Since the B L D M with a 120-degree inverter is self-commutated, it is also similar to a synchronous machine with line-commutated converter system [9]. Moreover, since the accurate analytical formulation for machine-converter systems leads to implicit equations which are challenging and expensive to solve, the approach presented in this paper extends the numerical/parametric method of [8] and [9] to the B L D M with a 120-degree inverter system. In the resulting AVM, the dynamics of the inverter are represented by a set of nonlinear algebraic functions, while the PMSM is expressed in the qd-rotor reference frame. The new model is constructed (without laborious analytical 20 / derivations) using the detailed simulation and readily takes into account commutation and conduction intervals. 3.2 Case System Detailed Model This paper considers the B L D M prototype with parameters summarized in the Appendix A. As in [9], the detailed model of the system has to be implemented first. The classical model in physical coordinates is given by dk 'abcs 's'abcs abcs dt 0) where fabcs = [fas fbs / C i ] r , a n d / represents voltage, current or flux linkage. The stator resistance matrix is rs=diag[rs,rs,rs]. (2) The flux linkage equation is ^•abcs ~ Li'aicv + km (3) where the inductance matrix and the rotor flux are (4) Lis+Lm -0.5Lm -0.5Lm -0.54, Lis+Lm -0.54 -0.54 -o.5im 4 s in^ r sin(^r -120°) sin(f?r +120°) (5) The developed electromagnetic torque is given by p , T e = 2A" |ias cos 6r + - y (ibs - ics )sin 6r (6) In the detailed model, the inverter logic is implemented according to Table I using [3]. Although various firing shifts are possible [5], here we assume </> = 30°. To verify the detailed model, the measured and simulated waveforms of the phase current are 21 superimposed in Fig. 3.2. These currents correspond to an operating point defined by 200W mechanical load at 2386 rpm when the inverter is supplied with vdc = 40V . An excellent match of the detailed model with the hardware confirms that the model accuracy is acceptable. This operation corresponds to mode 1, which, without loss of generality, is defined here with respect to phase current ibs during the switching interval II. As shown in Fig. 3.2, at the beginning of switching interval, the current is negative and then it goes to zero and remains zero until the end of that interval. Therefore, this mode is referred to as negative-zero (NZ) mode [5]. The operating mode and the current waveform may change quite significantly depending on the operating conditions. An example study, in which the mechanical torque is reversed, is shown in Fig. 3.3. To better understand the definition of possible modes, a simplified fragment of the phase current is depicted in Fig. 3.4, where the NZ mode is shown in Fig. 3.4(a). Here, the period of one switching interval is denoted by Ts, and the commutation time of current ibs going to zero is denoted by t°£m . If the current is reversed as shown in Fig. 3.4(b), the corresponding mode is referred to as positive-zero (PZ). However, if the machine's back emf becomes sufficiently larger than the applied voltage, the phase current may start to conduct in the opposite direction even before the beginning of the next switching interval. This mode is referred to as positive-zero-negative (PZN), and it splits the switching interval into three sub-intervals as depicted in Fig. 3.4(c). Figure 3.2 Measured and simulated phase current for mode 1. 0.04 0.05 Time, s 0.06 0.07 22 Figure 3.3 Phase current response to the change in mechanical torque. t°lf . 'com / • / / NZ ''com (a) PZ (b) toff 'com PZN (c) Figure 3.4 Phase current operating modes: (a) NZ; (b) PZ; and (c) PZN 20 i |10 . Mode 1 \ (NZ) Mode 2 (PZ) Mode3 (PZN) 0 5 0 0 CDr,rad/s 1 0 0 0 1 5 0 0 Figure 3.5 Steady-state torque-speed characteristic and operating modes. Each of these modes includes a time when phase current ibs is zero (discontinuous modes). The current transition from NZ to PZN is depicted in Fig. 3.3. The steady state averaged-torque-speed characteristic for the given system is shown in Fig. 3.5, wherein the regions of possible operating modes are also depicted. 3.3 Average Value Modeling In general, obtaining closed-form analytical expressions for t°£m and t°"m , which would be required for average-value modeling, is challenging. The authors of [10] proposed an implicit transcendental equation for calculating commutation time f£m , which requires iterative numerical solution. Instead, the methodology presented in this paper is based on the explicit model structure shown in Fig. 3.6. Here, the PMSM is represented using standard qd state model. The corresponding voltage equations in transformed variables are vqs = (rs '+ PLsYqs + arL/dS + » A 'qs vds = (rs + pLs)irds - 0)rL/ (7) •qs 23 DC Voltage Source Vdc . Inverter AVM (Algebraic block) Vqds PM Syn. Machine qd Model Qc Iqds PM Syn. Machine qd Model PM Syn. Machine qd Model Figure 3.6 Average-value model block diagram. The developed electromagnetic torque and mechanical subsystem dynamics can be expressed as T» -1 — \^m'qs par = P T -T (8) (9) 2 J where cor is the rotor angular speed, Tm is the mechanical load torque, and J is the combined moment of inertia of the load and the rotor Since the inverter is assumed to have no energy-storing components, it is represented in Fig. 3.6 as an algebraic block between the dc voltage source and the machine qd model. Similar to [9], the averaged dc-link variables and the averaged qd variables on the ac side are related as VqS =a(-)vdccos{Sr) vds =a(-)vdcsm(Sr) ids] (10) (11) where a() and /?(•) are algebraic parametric functions that depend on the operating conditions, and Sr = <?(•) is the rotor angle. The operating condition may be specified in terms of rotor speed cor and the dynamic impedance of the inverter switching cell defined as z = Vdc ]r lqds (12) Deriving a(z,u)r) , fi{z,(or) and S(z,cor) analytically for all possible modes is impractical. Instead, these functions are established numerically using the detailed simulation and procedure similar to that described in [9]. These functions are plotted in Fig. 3.7 for the considered operating region. Once these functions are established, the proposed AVM is implemented according to the 24 small-signal frequency-domain characteristics is similar to that described in [8] and [9], and can be rapidly implemented using numerical linearization of the continuous AVM (not included here due to space limitation). 3.5 Conclusion In this paper, a numerically-constructed average-value model of the brushless dc motor 120-degree inverter system has been presented. The presented approach overcomes the challenges associated with existing analytically-derived average-value models that require solutions for commutation and conduction angles. Instead, in the proposed AVM, the relationships between the averaged dc-link variables and the averaged machine variables are established as nonlinear algebraic functions. Although establishing an AVM requires running the detailed model in a range of operating conditions, once established, the resulting model is continuous and valid for large-signal time-domain transients as well as for linearization and subsequent small-signal frequency-domain analysis. The proposed model is verified in time domain against the detailed model and measured data. 3.6 References [1] "SimPowerSystems: Model and simulate electrical power systems", User's Guide, The Math Works Inc., 2006 (www.mathworks.com). [2] Piecewise Linear Electrical Circuit Simulation (PLECS), User Manual Ver. 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). [4] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, IEEE Press, Piscataway, NJ, 2002. [5] 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. [6] 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. [7] Q. Han, N. Samoylenko, and J. Jatskevich, "Comparison of Brushless DC Motor Drives with 180/120-degree Inverter Systems," IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), April 22-26, 2007, Vancouver, BC, Canada. 26 [8] A. Davoudi, J. Jatskevich, and T. DeRybel, "Numerical State-Space Average-Value Modeling of PWM DC-DC Converters Operating in DCM and CCM," IEEE Transactions on Power Electronics, Vol. 21, No. 4, Jul. 2006, pp. 1002-1012. [9] J. Jatskevich, S. D. Pekarek, and A. Davoudi, "Parametric average-value model of a synchronous machine-rectifier system," IEEE Trans. Energy Convers., vol. 21, no. 1, Mar. 2006, pp. 9-18. [10] R. R. Nucera, S. D. Sudhoff, and P. C. Krause, "Computation of Steady-state Performance of an Electronically Commutated Motor", IEEE Transactions on Industrial Applications, Vol. 25, pp. 1110-1117, Nov.-Dec. 1989. [11] "Simulink: Dynamic System Simulation for Maflab", using Simulink Version 6, The MathWorks Inc., 2006 (www.mathworks.com) 27 4 Average-Value Modeling Of Brushless DC Motors With 120-Degree Voltage Source Inverter3 4.1 Introduction Brushless dc (BLDC) motors generally have good torque-speed characteristics, fast dynamic response, high efficiency, long life, etc., and are therefore increasingly used in industrial automation, instrumentation, and many other equipment and servo applications. A BLDC motor consists of a permanent magnet synchronous machine (PMSM) (with either sinusoidal or trapezoidal back emf) driven by an inverter. The detailed modeling of such machine-inverter systems, wherein the switching of each transistor and diode is represented, has been investigated in the literature quite well [l]-[6] and may be readily carried out using a number of available simulation packages [7]-[ 11]. However, for the large- and small-signal analysis of electromechanical and motor-drive systems, the so-called average-value models (AVMs), where the effect of fast switching is neglected or averaged with respect to a prototypical switching interval, are indispensable. Derivation of AVMs requires careful averaging of the stator phase voltages and/or currents over a prototypical switching interval to obtain the corresponding average-value relationships for the state variables and electromagnetic torque. However, unlike the switch-level models, AVMs are continuous, and can therefore be linearized about a desired operating point. Thereafter, obtaining local transfer functions and/or frequency-domain characteristics becomes a straightforward and almost instantaneous procedure. Many simulation programs offer automatic linearization and frequency domain analysis tools [10], [11]. In addition, AVMs typically execute orders of magnitude faster than the corresponding detailed models, making them ideal for representing respective motor-drive-components in system-level studies. This paper focuses on typical Hall-sensor-based voltage-source-inverter-driven (VSI-driven) BLDC motors, where the inverter may operate using 180- or 120-degree commutation methods [4]. In the 180-degree inverter system, the switching of transistors is complementary and each phase is always connected to either the positive or the negative rail and the phase currents are always continuous. Having continuous phase current simplifies the average-value modeling of BLDC machines with sinusoidal back emf (which may operate in a voltage-control mode [5, 3 A version of this chapter has been submitted for publication. Q. Han, N. Samoylenko and J. Jatskevich, 'Average-Value Modeling of Brushless DC Motor with 120-Degree Voltage Source Inverter', Manuscript No. TEC-00235-2007 28 Chaps. 3, 15] or current-control mode [5, Chap. 15], [12], [13]) or non-sinusoidal back emf where the use of multiple reference frames has been proposed in [14] and later used in [15]. In the 120-degree BLDC motor-inverter systems, which are actually very common primarily due to their simplicity [16], each phase is allowed to be open-circuited for a fraction of revolution giving rise to complicated commutation-conduction patterns of the stator currents [17]. Steady state analysis of such motors has been analyzed by several researchers [l]-[3]. The average-value modeling, of such BLDC motors also becomes more challenging and has not been sufficiently covered in the literature. The best presently known to us average-value model of BLDC motor with 120-degree inverter has been proposed in [18]. Another average-value model was proposed for the non-sinusoidal back emf PMSM with the 3-phase H-bridge inverter that can operate in voltage-control mode by adjusting the duty cycle [14], wherein the stator currents were continuous. Both models [18] and [14] were developed for the machines with large stator resistance and did not consider the commutation time and current, which significantly simplified the average-value modeling. This paper analyzes the BLDC motors with 120-degree switching logic and makes the following contributions: • It explains the challenges of the average-value modeling the 120-degree VSI-driven BLDC motor system due to the complicated commutation-conduction pattern of the stator current. • It shows that only for the motors with large stator resistance (small electrical time constant) the commutation interval may be neglected. • The paper also proposes a new explicit AVM that includes both commutation and ' conduction intervals, and is therefore more accurate in predicting both large-signal time-domain transients and small-signal frequency-domain characteristics. • Since it is not practical to analytically derive a closed-form solution for the commutation angle, this solution has been obtained numerically using the detailed simulation. This approach reduces laborious analytical derivations and has been shown to give accurate and practical results [19], [20], [21]. • The conducted studies are based on two typical industrial BLDC motors with large/small electrical time constants and include hardware measurements, detailed simulation, and comparison with a previously published average-value model [18]. 29 4.2 System Operation And Detailed Modeling 4.2.1 System Operation To set the stage for the discussion in this paper, the system operation is briefly described here. A cross-sectional view of an equivalent 2-pole PMSM is shown in Fig. 4.1, where the three Hall sensors Hl, H2, and H3 are used to detect the rotor position. The rotor electrical position and speed are denoted by 0r and cor. The advance in firing angle is denoted here by ^ [4], [6]. The machine is assumed to be driven by a typical inverter circuit shown in Fig. 4.2. The logical signals from sensors Hx, H2, H3 are used to control the inverter switches - transistors S\-S6. These signals are assumed to have duration of 180 degrees and are shifted relative to each other by 120 degrees as depicted in Fig. 4.3. The Hall sensor signals partition the rotation circle into six 60-degree switching intervals (SI), which are numbered as I—VI. The time duration of such prototypical switching interval is denoted here by Ts. Since the inverter switching is determined by the rotor position, the time duration of the switching interval is related to the rotor electrical speed as 7^  = n I Zcor. If the motor is driven according to the 180-degree logic, the switching signals for the upper transistors S{, S2, S3 coincide with the Hall sensor outputs Hx, H2, H3, and the signals for the lower transistors are complementary S4 , S5 , S6,as shown in Fig. 4.3 (a). With this logic, the'advance in firing angle is often set to zero, tj> = 0 [4]-[6], which corresponds to in-phase alignment of the applied voltage and the back emf (their fundamental components). Figure 4.1 PMSM with Hall sensors. 30 of the applied voltage by 30 electrical degrees. To maintain' the same alignment of the fundamental components of the applied voltage and the back emf, the Hall sensors may be shifted to make the switching happen 30 degrees earlier resulting in </> = 30°. This is in fact one of the most commonly used switching logic, which makes the Hall sensor signals aligned with the zero-crossings of the line-to-line back emf [5], [16]. However, operation with other advance angles is also possible [3], [4], [17]. 4.2.2 Detailed Modeling Since the detailed model is utilized in constructing the proposed AVM, it is also reviewed here for consistency. Although the BLDC machines are often considered to have a trapezoidal back emf, some machines are especially designed to have low cogging torque and close to sinusoidal back emf [22], [23], [24]. The assumptions of sinusoidal back emf and non-salient rotor are commonly used in the literature [4], [6], [17], [18] and were found to be very close approximations for the actual BLDC motors considered here (see Appendix B, Motors A and B). The detailed model is expressed in physical variables and coordinates [6]. In particular,^ he stator voltage equation is ^abcs ^s^abcs +' dk abcs dt (1) where fabcs = [fas fbs fcs f, and / may represent voltage, current or flux linkage. The stator resistance matrix is expressed by rs=diag[rs,rs,rs] \ (2) The flux linkages are expresses as ^•abcs = L/lahcs + k'm (3) where the inductance matrix and the rotor flux are Lis + Lm -0.5Lm -0.5Lm~ -Q.5Lm ks+Lm -0.5Lm (4) -0.5Lm -0.5Lm Lls + v sin 6r v s infe . -120°] (5) sin(t9r + 120°) 32 where LIs and Lm are the stator leakage and magnetizing inductances, and X'm is the amplitude of theflux linkages established by the permanent magnet referred to the stator/ Since the stator windings are assumed to be wye-connected, the three phase currents add up to zero. Thus (3) may be further simplified as ^•abcs - ^Aabcs + 09 3 where Ls=Lls+-Lm. The developed electromagnetic torque is given by P , e 2 I*as c o s 0 r + ^ - ~ i c s ) s i n d r (7) The mechanical subsystem is assumed here as a single rigid body for which the dynamics may be expressed as da, ~dt • = ( £ ) > . - r . ) « where wr is the rotor electrical angular speed; J is the combined moment of inertia of the load and the rotor; and P is the number of magnetic poles. The combined mechanical torque is approximated as Tm = K^n + K2 ; where n represents the mechanical speed in rpm . The terms Kxn and K2 represent the dynamometer torque and the torque due to mechanical losses in coupling and friction. The stator windings, together with the inverter circuit depicted in Fig. 4.2, represent a switched network. The detailed models of 120- and 180-degree voltage source inverters are developed and implemented in Simulink [10] using the approach described in [25] and the toolbox [9]. 33 4.3 Average-Value Modeling The approach of deriving AVM presented here is based on averaging the inverter voltages. To better understand the challenges of averaging the 120-degree inverter and for consistency of derivation, first the 180-degree inverter is briefly described. Without loss of generality, averaging of voltages is carried out with respect to the switching interval II (see Fig. 4.3) [17], [18], whereas the results are applicable for all switching intervals. 4.3.1 Averaging Stage In the AVM, it is advantageous to have the state variables constant in steady state. To achieve this, the stator variables are transformed into the qd -rotor reference frame using Park's Transformation [6] lqdOs I^slabc (9) where K: ( 2n cos 6r cos 6r ^ 3 sin sin| 0r 3 cos 2n "T • (a 2 n sin 6r H I 3 1 (10) As usual, applying transformation (9) to (1) yields diqs (11) (12) Since the stator windings are wyerconnected, the zero sequence quantities are not present. The variables in (11) and (12) are then averaged with respect to a prototypical switching interval Ts as (13) 34 where / may represent voltages or currents. The bar symbol is used to denote the average value. Replacing the variables in (11) and (12) with their fast averages and rearranging the terms yields digs _ -rsVqs-corL~irds-corXm v, + (14) dt Ls Ls dirds = -r/^-w.L/^ | yrs dt I , Ls The averaged developed electromagnetic torque is given by f -|'3JPW ' o - | — J ^ V (16) The averaged dc current that the inverter draws from the dc voltage source may be approximated as [6] t _ 3 VqJqS +VjJds ( l* = 2 Vdc M Equations (8), (14), (15) and (16) form the average-value state model of the PMSM with vrqs and vrds as inputs. These voltages are determined by the inverter switching logic. 4.3.2 Average-Value Modeling of 180-Degree System In this case, the transistors in the same phase leg are switched complementarity and each phase is always connected to either the positive or the negative rail. Therefore, the stator phase voltages are uniquely determined [6]. Typical phase voltage and current waveforms for this operation are shown in Fig. 4.4. As can be observed in Fig. 4.4, the voltage waveform has six steps of 60 electrical degrees. An important characteristic of the \ 80-degree operation is that the phase current is always continuous - does not go to zero. Since the switching pattern is determined, the transformed averaged voltages may be expressed as [6] 2 v ^ - v ^ c o s ^ (18) 2 vrds=-vdcsm</> (19) 71 The AVM is formed by (18), (19), together with the state model of PMSM - (8), (14)-(16). To 35 previously analyzed and is referred to as the negative-zero (NZ) mode [17]. Here, the commutation time is denoted by tcom , whereas the conduction time is tcond = Ts - tcom during which the other two phases as and cs conduct the full current. The commutation time tcom depends on the motor parameters (phase winding electrical time constant) and operating conditions, but in general cannot be zero because the current in inductor cannot be switched "off instantaneously. The presence of operating-point- and parameter-dependant time interval tcom makes it very difficult (if not impossible) to derive closed-form explicit analytical expressions for the average phase voltages and currents. The authors of [3] proposed an implicit transcendental equation for tcom considering the steady state operation only, which required iterative numerical solution. Figure 4.6 Typical voltage and current waveforms for the 120-degree BLDC motor-inverter system. A close observation of Fig. 4.6 suggests that commutation time tcom may be small relative to the conduction interval tcond . If, however, the commutation time tcom is neglected - assuming that the gated-off phase current goes to zero instantaneously, then it is possible to carry out the averaging of voltages over the entire switching interval form <j> to <j>. This is the bases 6 2 of the dynamic AVM proposed in [18]. In particular, if we assume that tcond - Ts and the speed is constant over the switching interval, the average voltages in the qd -rotor reference frame may be expressed as 37 which requires the instantaneous voltages vrqdscond . To obtain these voltages in the qd reference frame, the phase voltages in direct abc coordinates should be known first. It is shown in [3], [ 17] that during the conduction interval we have vabc,cond -Vdc ~-K<>>r c o s ^r - — Ammr cos 0r 2n 1 v 1 i (a 2 7 1 Z 2. \ J Transforming (22) into the rotor reference frame yields (22) r — — V ^qs,cond ~ « ' dc f 2n cos 6r - cosl 6r + — 2 /-. 2^ -+ Amar cos dr—— (23) Vds,cond ^ Vdc sin 6r - sinf 9r + • \ 3 + imoyr sin[V - -yj cos[V - y After substituting (23)-(24) into (20)-(21), we obtain (24) = y v r f c cos^ - + »rAOT 1 3V3 ft cos 2d> 2 4/r { 3 vdS — Vdc C0SI $ ~ I - °>rK C0SI 2<P + (25) (26) Thereafter, the AVM is formed by (25)-(26), together with (8), (\4)-(l6). To demonstrate the performance and accuracy of this AVM, a start-up transient of the Motor A (see Appendix B) was simulated again. The corresponding responses produced by the detailed and averaged models are superimposed in Fig. 4.7. As can be observed, the dynamic response produced by the AVM is not very accurate. 38 1200, § 800; cd 400 5, < 0 0 60 0 -60 -ww Detailed Model — — AVM, Commutation Neglected [A 0.05 0.1 Time (s) 0.15 0.2 Figure 4.7 Start-up transient of the Motor A driven by 120-degree inverter. It is important to recall that since this AVM neglects the commutation time, its accuracy depends on how large or small the commutation interval is, which in turn depends on the motor parameters. After all, this model was shown to give very good results [18]. To verify this point, we have implemented a start-up transient of the Motor B (see Appendix B), which has very similar ratings but larger stator resistance and somewhat smaller inductance. The corresponding transient responses produced by the detailed and averaged models are shown in Fig. 4.8. As can be observed here, the AVM with neglected commutation predicts the transient quite well. 300 I 200 0 s z 10 S o -10 -vw Detailed Model — — AVM, Commutation Neglected / •\MAMAAA 0.05 0.1 Time (s) 0.15 0.2 Figure 4.8 Start-up transient of the Motor B driven by 120-degree inverter. 39 4.4 Proposed Average-Value Model As was demonstrated in the previous Section, neglecting the commutation interval introduces errors especially noticeable when the commutation time is relatively large. To establish a more accurate AVM, it is important to include both commutation and conduction subintervals. In particular, the total averaged stator voltages should include the averages calculated over both subintervals as Vqs ~ vqs,com + vqs,cond (27) Vds ~ %s,com + Vds.cond (28) The applied voltages in the conduction subinterval vrqdscond have been already presented. Therefore, it remains to derive the voltages for the commutation interval. Since in this interval all three phases are carrying current (the current ibs is still flowing through the upper diode), the phase voltages in direct abc coordinates can be readily determined based on the inverter topology. Considering the beginning of SI II, after some derivations [3], the phase voltages can be found as Micron, =^[1 1 " 2 f (29) 3 Transforming (29) into the rotor reference frame yields ^qs,com - ' dc 2 „ (a 2 ^ — KJ„COS 6> + — 2 „ . (n 2n (30) Vdi.com =~^Vdc Sin|0, + -yj (3!) These voltages must be averaged over the commutation subinterval. Here, it is more convenient to carry out the averaging in terms of the corresponding commutation angle denoted here by ft . In particular, the voltages (30)—(31) must be averaged from <j> to <j> + (3 as 6 6 Vqs,com = ~ j T ^ ^qs,am fa K (32) : 7i ^--A 6 40 3 r—4+P \ 6 7t 7t Similarly, the voltages (23)-(24) must be averaged from $ + fi to ^ as 6 2 (33) Vqs,cond L- ^qs.cond {0r)d8r (34) Vds.cond L V ds,cond(f^r)fl®i It 4-<*+/? (35) The results of the averaging in (32)-(35) should then be combined according to (27)-(28). After some analytical manipulations, the total average voltages may be expressed as 2^3 . (n fi\ (, 71 fi •qs •corXm If, 3fi\ 3 (n _\ 71 ' — 1-— cos — + fi cos\2d> fi (36) 4 . (fi -v r f c s in — 71 12 5n__fi_ 6 2 j COS^I 2V3 . (it fi\ (, 2K fi" - — corXm cosf — + fi jcosf 2<b + — - fi lit \6 ) V 6 (37) 4 . (fi) . , 57t fi 71 \2 V 6 2 The averages of and in other switching intervals are the same. Also, (36) and (37) are more general and if we set fi = 0 , the result becomes (25) and (26), as required for consistency. The commutation angle fi plays an important role in (36) and (37) and the AVM. This angle depends on the machine-inverter operation condition (state) that is defined by the speed cor, the phase currents irqds, the applied dc voltage vdc, and of course the machine parameters. If one considers steady state, then the derivation of /?(•) may be carried out using the commutation time tcom , for which there is no closed-form analytical solution [3]. In general, obtaining a 41 closed-form explicit analytical expression for p\for,irds,vdc) that is valid for transient analysis may be even more challenging. Instead, the approach proposed in this paper is based on finding the function /?(•) numerically. In fact, the authors recognize that the presence of commutation-conduction current in the BLDC-motor-inverter-system is somewhat similar to the discontinuous conduction mode in the DC-DC converters [19]. In this regard, establishing tcom and/or p is similar to establishing the duty ratio constraint, which may be done very effectively numerically using the available detailed simulation of the original system [19]. This is also similar to numerically establishing the parametric functions for the average-value modeling of the synchronous machine-rectifier systems [20], which was shown to be a very effective and practical method for dealing with complex systems [21]. It is assumed that /?(•) can be expressed as an algebraic function of the state and input variables wr, iqds,ax\d vdc , respectively. To establish p{cor,vdc,iqs,idsY the detailed simulation can be run in a loop spanning a range of operating conditions while the currents are averaged according to (13) and cor, vdc, and p are recorded for the future use in a look-up table. To reduce the dimension of the look-up table, and therefore further reduce the model complexity, it is advantageous to define a dynamic impedance of the inverter switching cell as Thereafter, instead of a 4-dimentional look-up table p\cor, vdc,ir ids j , the commutation angle The detailed simulation of both Motor A and B has been run in a loop spanning a range of interested operating conditions. The resulting functions p[cor,z) are shown in Figs. 4.9 and 4.10, respectively. These functions are then used in the look-up tables, wherein interpolation/extrapolation may be used as necessary. Examining Figs. 4.9 and 4.10, it can be noticed that the commutation angle of the Motor A is significantly larger than that of the Motor B, which is consistent with the observation made earlier in Section III C. Once the algebraic function p(z, a>r) is available, the proposed AVM is implemented according to the block diagram shown in Fig. 4.11. It is important to point out that the resulting state model z = (38) lqds can be represented as a 2-dimentional look-up table p(a>r, z). 42 is proper and explicit, and therefore can be readily implemented using various simulation platforms. 4.5 Case Studies The previously described detailed model, the average-value model [18], and the proposed A V M have been implemented in Simulink [10]. It is important to keep in mind that since the proposed AVM has been extracted from the detailed model, it only approaches the detailed model in terms of accuracy. Therefore, the detailed model may be considered as a reference. Figure 4.9 Commutation angle p for the Motor A Figure 4.10 Commutation angle p for the Motor B. Permanent Magnet Synchronous Machine (8),(14-16) ^qds (35-36) U (37) k Jvdc Figure 4.11 Average-value model block diagram. 43 4.5.1 Time Domain Model Verification From the first inspection of the BLDC motors summarized in the Appendix B, it can be noticed that these machines have many similar features, except that their electrical time constants are very different. To verify the detailed models, the simulated waveforms of the phase current have been compared with the measurements obtained with both motors. In both cases motor inverter was supplied by the same voltage source, vdc = 40V . For the Motor A, an operating point defined by 200W mechanical load and 2350 rpm was considered. The corresponding measured and simulated phase currents are superimposed in Fig. 4.12. For the Motor B, a somewhat close operating point defined by 220W load and 2200 rpm was considered. The corresponding measured and simulated phase currents for Motor B are superimposed in Fig. 4.13. As can be observed in Figs. 4.12 and 4.13, the electrical frequency in two cases is different, which is consistent with the number of magnetic poles in each motor. However, both detailed models produce an excellent match of the waveforms with the corresponding hardware. This confirms that the accuracy of detailed models is very good and acceptable for further studies. A closer look at Figs. 4.12 and 4.13 suggests that although the two motors operate in close conditions, their current waveforms are qualitatively different. In particular, it is determined based on the measured and simulated data that the Motor A operates with a commutation angle of 10.5°, while the Motor B operates with a much smaller commutation angle of 1.7°. These angles correspond to 17.5% and 2.8% of the 60-degree switching interval, respectively. Comparing the machine parameters, it can be noted that the stator resistance of the Motor A is much smaller than that of the Motor B, and the inductance of the motor A is larger than that of the Motor B. Smaller resistance and larger inductance in the Motor A results in a larger electrical time constant and thus a larger commutation angle /?. At the same time, the commutation happens much faster in the Motor B due to its large resistance and small electrical time constant. The proposed AVM is further compared with the actual BLDC motors in the following transient study. Here the motors are initially supplied by a voltage source vdc = 25V , driving a mechanical load (dynamometer). Then, the input voltage is stepped up to 30K and the system continues to run. The transient responses of both motors are shown in Fig. 4.14. Here, the corresponding measured traces of DC current are carefully aligned in time and superimposed with the ones predicted by the proposed AVM. As can be seen in Fig. 4.14, the motors have different initial part of the transient, which is predominantly determined by the fast electrical dynamics, and is well predicted by the AVM for each motor. A good match with the measurements shows the validity of 44 60°4-'rAA - - - Detailed Model — Measured "t r i the proposed model. 10 5 < a 0 -5 y-10l Figure 4.12 Measured and simulated phase current of the Motor A. 10r 5-Com. Angle P= 10.5° 0.002s | Time < S 0 -10 60°4-• Detailed Model Measured Com. Angle P = 1.7° "1 n o.oi s VNKM Time Figure 4.13 Measured and simulated phase current of the Motor B. 15 < io| . i s ,0 -5 0.02s Time < • i 4 i 0-0.1s -wvi Measurement Proposed AVM (a) (b) Figure 4.14 Measured and simulated response in current idc: (a)-Motor A; (b)-Motor B. The AVM presented in this paper are then compared again. In the following study, the motor assumed to start up. Then, at t = 0.6s, a mechanical load 1 Nm is applied. The transient responses produced by Motor A and Motor B models are shown in Figs. 4.15 and 4.16, respectively. In each case, the response produced by the detailed model can be considered as reference. Analyzing the study with the Motor A, it can be observed in Fig. 4.15 (dashed line) that the AVM which neglects the commutation interval underestimates the developed torque (and therefore speed) during the transient and the steady state. This is consistent with the fact that this motor operates with a relatively large commutation interval (angle), and neglecting this interval results in incorrectly predicted averaged phase currents and torque. At the same time, the proposed AVM includes both commutation and conduction intervals and predicts the entire transient very well as can be seen in Fig. 4.15 (red solid line). The Motor Bis analyzed next in Fig. 4.16. As can be seen here, the developed torque predicted by both AVMs is very close and goes through the ripple of the waveform predicted by the detailed simulation (green line). This study shows that in the case of the Motor B, neglecting the commutation interval may be acceptable since this interval is very brief due to the small electrical time constant of this motor. However, the proposed AVM again demonstrates an excellent match throughout the transient study. -vfn Detailed Model Proposed AVM — — AVM, Commutation Neglected 0.2 0.4 0.6 Time, s 0.8 Figure 4.15 Speed, torque, ids response of Motor A 46 300 ' 0 0.2 0.4 0.6 0.8 1 Time, s Figure 4.16 Speed, torque, ids response of Motor B 4.5.2 Frequency Domain One of the main advantages of AVMs is their use for the small-signal frequency analysis using linearization (in a way that is not possible with detailed switching models or the hardware). In the following studies, the transfer function from the input voltage vdc to the developed output torque Te of the system is considered. Although it is more difficult to measure such transfer function in practice, it may be particularly useful for design of electromechanical systems with BLDC motors and it may be easily extracted from the corresponding model. To evaluate the accuracy of the developed AVMs, both motors were considered. The same previously defined operating points at the speed of 2350 rpm and 2200 rpm were considered for the Motor A and Motor B, respectively. Since the detailed model is switching and discontinuous, the small-signal injection and subsequent frequency sweep method has been implemented to extract the frequency domain information. The corresponding transfer functions obtained from the detailed and average-value models are shown in Figs. 4.17 and 4.18 for the Motor A and B, respectively. Since the AVMs approximate the detailed models only in the low-frequency range, the transfer functions were evaluated to about 1/3 of the switching frequency, which is different for each motor (940Hz for Motor A, and 22QHz for Motor B). The magnitude plot corresponding to the Motor B (see Fig. 4.18) also appears more flat since this motor has smaller electrical time constant of the stator winding and operates at lower switching 47 frequency. As it can be seen in Figs. 4.17 and 4.18, the AVM that neglects the commutation interval is not accurate in predicting the frequency domain response of the Motor A (which operates with large commutation angle), but it is acceptable for the Motor B (which operates with small commutation angle). However, the proposed AVM predicts the transfer functions of both motors very accurately, which is achieved by appropriately including commutation and conduction intervals. 4.6 Conclusion This paper examined average-value modeling of the BLDC motor 120-degree inverter systems. It is demonstrated that neglecting the commutation interval may lead to degradation of the model accuracy especially with the BLDC motors that have low stator resistance (high electrical time constant) and operate with large commutation angle. A new average-value model that appropriately includes both commutation and conduction subintervals has been presented here. Since it is not practical to analytically derive a closed-form solution for the commutation angle, a nonlinear algebraic function that represents the commutation angle has been obtained numerically using the detailed simulation. This approach avoids laborious analytical derivations and has been shown to give accurate and practical results. The proposed model is shown to be accurate in time-and frequency-domain and applicable for the motors with.large and small electrical time constant. -10 -o -a a -20 I 2 -30 10 50 T3 -50 -100 i — i — i — i — i i 11 1 ' : : : : : : / v / ^ ^ ^ -\ ' i 1—. • • • • ' . 10 - Detailed Model - Proposed AVM - AVM, Commutation Neglected: 10 f(Hz) Figure 4.17 Voltage-to-torque transfer function for the Motor A. 48 -10 3 'I ca s -20 -301 10 101 l(f C3 JS Ph ! -10 -20 Detailed Model • Proposed AVM > - AVM, Commutation Neglected 10 1 10 f(Hz) 10 Figure 4.18 Voltage-to-torque transfer function for the Motor B. 4.7 References [1] T. Nehl, F. Fouad, N. Demerdash, "Digital simulation of power conditioner-machine interaction for electronically commutated DC permanent magnet machines," IEEE Trans. Magnetics, vol. 17, pp. 3284-3286, Nov. 1981. [2] P. Pillay, R. Krishnan, "Modeling, simulation, and analysis of permanent-magnet motor drives. Part II. The brushless DC motor drive," IEEE Trans. Industry Applications, vol. 25, pp. 274-279, March-April 1989. [3] R. R. Nucera, S. D. Sudhoff, and P. C. Krause, "Computation of Steady-state Performance of an Electronically Commutated Motor", IEEE Trans: Industrial Applications, vol. 25, pp. 1110-1117, Nov.-Dec. 1989. [4] P. C. Krause, Analysis of Electric Machinery, McGraw-Hill, 1996. [5] R. Krishnan, Electric Motor Drives: Modeling, Analysis, and Control, Prentice-Hall, Upper Saddle River, NJ, 2001. [6] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, IEEE Press, Piscataway, NJ, 2002. [7] SimPowerSystems: Model and simulate electrical power systems User's Guide, The Math Works Inc., 2006 (www.mathworks.com'). [8] Piecewise Linear Electrical Circuit Simulation (PLECS) User Manual, Version. 1.4, Plexim GmbH (www.plexim.com). [9] Automated State Model Generator (ASMG) Reference Manual, Version 2, P C Krause & Associates, Inc. 2003 (www.pcka.com'). . ' 49 [10] Simulink: Dynamic System Simulation for Matlab, using Simulink, Version 6, The MathWorks Inc., 2006 (www.mathworks.com). [1 \~\Advanced Continuous Simulation Language (acslXtreme) Language Reference Guide, Version 2.3, The AEgis Technologies Group, Inc., Huntsville, AL, 2006. [12] K. A. Corzine, S. D. Sudhoff, H. J. Hegner, "Analysis of a current-regulated brushless DC drive," IEEE Trans. Energy Conversion, vol. 10, pp. 438^ 145, Sep. 1995. [13] K. A. Corzine, S. D. Sudhoff, "A hybrid observer for high performance brushless DC motor drives," IEEE Trans. Energy Conversion, vol. 11, pp. 318-323, June 1996. [14] P. L. Chapman, S. D. Sudhoff, and C. A. Whitcomb, "Multiple Reference Frame Analysis of Non-Sinusoidal Brushless DC Drives," IEEE Trans. Energy Conv., vol. 14, pp. 440-446, Sep. 1999. [15] Lei Hao, H. A.. Toliyat, "BLDC motor full-speed operation using hybrid sliding mode observer," In Proc. 18th IEEE Applied Power Electronics Conference and Exposition (APEC'03), 9-13 Feb. 2003, vol. 1, pp. 286-293. [16] P. Yedamale, "Brushless DC (BLDC) Motor Fundamentals," Application Notes AN885, Microchip Technology Inc., 2003 (available: http://www.microchip.com). [17] S. D. Sudhoff, P. C. Krause, "Operation Modes of the Brushless DC Motor with a 120° Inverter," IEEE Trans. Energy Conversion, vol. 5, pp. 558-564, 1990. [18] S. D. Sudhoff, P. C. Krause, "Average-value Model of the Brushless DC 120° Inverter System," IEEE Trans. Energy Conversion, vol. 5, pp 553-557, 1990. [19] A. Davoudi, J. Jatskevich, and T. DeRybel, "Numerical State-Space Average-Value Modeling ofPWM DC-DC Converters Operating in DCM and CCM," IEEE Trans. Power Electronics, vol. 21, pp. 1002-1012, Jul. 2006. [20] J. Jatskevich, S. D. Pekarek, and A. Davoudi, "Parametric average-value model of a synchronous machine-rectifier system," IEEE Trans. Energy Conversion., vol. 21, pp. 9-18, Mar. 2006. [21] J. Jatskevich, "Practical Average-Value Modeling of Power Electronic Systems," Professional Education Seminar, IEEE Applied Power Electronics Conference (APEC'07), Anaheim, California, February 25 - March 1, 2007, 102 pages. [22]T.W. Nehl, F.A. Fouad, N.A. Demerdash, and E.A. Maslowski, "Dynamic simulation of radially oriented permanent magnet-type electronically operated synchronous machines with parameters obtained from finite element field solution," IEEE Trans. Industrial Application, vol. IA-18, pp. 172-182, Mar./Apr. 1982. [23]T.W. Nehl, N.A. Demerdash, and F.A.Fouad, "Impact of winding inductances and other parameters on the design and performance of brushless dc motors," IEEE Trans. Power Application And System, vol PAS-104, pp. 2206-2213, Aug. 1985. [24] Maxon EC Motor: Technology - Short and to the Point (available at: www.maxonmotorusa.com). [25] O. Wasynczuk, S.D. Sudhoff, "Automated state model generation algorithm for power circuits and systems," IEEE Trans. Power Systems, vol. 11, p. 1951-1956, Nov. 1996. 50 5 Summary For analysis of electro-mechanical systems with machine-drives, it is very useful to represent the power electronic modules with their average-value models that can accurately approximate the system's behavior in both time- and frequency-domains in the range below the switching frequency. Average-value modeling of BLDC motor drives becomes particularly valuable today as these motors are finding wider application in various industries. Although several average-value models have been proposed in the literature, none of the previously established models are capable of accurately representing the dynamic behavior of BLDC machine with 120-degree VSI when the stator winding resistance is small and the electrical time constant is large. 5.1 Conclusions This thesis first analyzes the challenges in deriving the AVM due to the complicated commutation-conduction pattern of the stator currents, and shows that the commutation interval/angle should not be neglected especially for the motors with small stator resistance. Since it is not practical to analytically derive a closed-form explicit AVM that appropriately includes commutation and conduction intervals, this thesis extends a recently proposed parametric average-value modeling approach to the BLDC motor-inverter systems where the required parametric functions are calculated numerically. This approach has been shown to give accurate i results for practical systems including DC-DC converters and synchronous machine-rectifier sets. This thesis then presents two new AVMs. The first AVM represents the inverter as a composite switching cell which is then replaced by three algebraic functions - numerically constructed lookup tables. The second AVM is based on averaging the inverter voltages in commutation and conduction intervals and requires only one parametric function to represent the commutation angle, which is an advantage over the first model. We demonstrate that the proposed AVMs are more accurate in predicting large-signal time-domain transients and small-signal frequency-domain characteristics than the best previously published model known to us. The thesis also demonstrates that the proposed AVM is applicable to motors with both large and small electrical time constant. 51 5.2 Future Work In this thesis we have considered only one operating mode of BLDC motor, namely the so called negative-zero (NZ) mode in which the overall switching interval is divided into two subintervals. We have also assumed operation with advance firing angle fixed at 30 degrees. However, in general, 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. Analysis and average-value modeling of other operating modes represents a future research that is perhaps left to my successor - another graduate student who will join our group. I also believe that in the future the methodology presented in this thesis may be extended to other power-electronic-based systems. 52 APPENDICES Appendix A Arrow Precision Motor Co., L T D . , Model 86EMB3S98F-B1, 36 V D C , 210 W, 2000 rpm, 8 poles, rs =0.15fi , Ls = 0A5mH , im=2\.5mV • s , J = le - 3N • m • s2 , Tfriclion = 10~7a? . Appendix B Motor A : Arrow Precision Motor Co., L T D . , Model 86EMB3S98F, 36 V D C , 210 W, 2000 rpm, 8 poles, rs = 0A5£l,Ls =0A5mH ,Am = 2\.5mV • s , J = 12e - 4kgm 2 Kx = 4.0e - 4Nm/rpm K2 =0.UNm, Motor B : Maxon Precision Motors, Inc., Model EC 167131, 48 V D C , 400 W, 2680 rpm, 2 poles, ^ = 0 . 6 7 4 f i , Ls=0.4lmH, Am = 86.2mV • s , J = 12e - 4kgm 2 , Kx = 4.0e - 4Nm/rpm , K2 = 0.27Nm . / 53 

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