ANALYSIS OF DC POWER SYSTEMS CONTAINING INDUCTION MOTOR-DRIVE LOADS Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include proprietary or classified information. ______________________________ Aleck Wayne Leedy Certificate of Approval: ______________________________ ______________________________ S. Mark Halpin R. Mark Nelms, Chair Professor Professor Electrical and Computer Engineering Electrical and Computer Engineering ______________________________ ______________________________ Charles A. Gross John Y. Hung Professor Associate Professor Electrical and Computer Engineering Electrical and Computer Engineering ______________________________ Stephen L. McFarland Dean Graduate School ANALYSIS OF DC POWER SYSTEMS CONTAINING INDUCTION MOTOR-DRIVE LOADS Aleck Wayne Leedy A Dissertation Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Auburn, Alabama May 11, 2006 iii ANALYSIS OF DC POWER SYSTEMS CONTAINING INDUCTION MOTOR-DRIVE LOADS Aleck Wayne Leedy Permission is granted to Auburn University to make copies of this dissertation at its discretion, upon request of individuals or institutions and at their expense. The author reserves all publication rights. ______________________________ Signature of Author ______________________________ Date of Graduation iv VITA Aleck Wayne Leedy, son of Robert Aleck and Jane (Pigmon) Leedy, was born February 10, 1973, in Pennington Gap, Virginia. He graduated from Lee High School in Jonesville, Virginia in 1991. He entered the University of Kentucky in August, 1991, and graduated with a Bachelor of Science degree in Electrical Engineering with a Minor in Mathematics on May 5, 1996. After working for Mountain Empire Community College and The Trane Company, he entered Graduate School at the University of Kentucky in May, 1998. He graduated from the University of Kentucky with a Master of Science in Mining Engineering (Electrical Engineering emphasis) on May 6, 2001. Following his thesis defense, he entered Graduate School at Auburn University in March, 2001. He is a registered Professional Engineer in the Commonwealth of Kentucky. v DISSERTATION ABSTRACT ANALYSIS OF DC POWER SYSTEMS CONTAINING INDUCTION MOTOR-DRIVE LOADS Aleck W. Leedy Doctor of Philosophy, May 11, 2006 (M.S., University of Kentucky, 2001) (B.S., University of Kentucky, 1996) 157 Typed Pages Directed by R. Mark Nelms The development of an analytical method used for conducting a power flow analysis on a DC power system containing multiple motor-drive loads is presented. The method is fast, simplistic, easy to implement, and produces results that are comparable to software packages such as PSPICE and Simulink. The method presented utilizes a simplified model of a voltage source inverter-fed induction motor, which is based on the steady-state T-type harmonic equivalent circuit model of the induction motor and the input-output relationships of the inverter. In the simplified model, a V-I load characteristic curve is established that allows the inverter, motor, and load to be replaced by a current-controlled voltage source. This simplified model can be utilized in the vi analysis of a multiple-bus DC power system containing motor-drive loads by incorporating the V-I load characteristic curve of each motor-drive load into an iterative procedure based on the Newton-Raphson method. The analytical method presented is capable of analyzing DC power systems containing induction motor-drive loads fed from voltage source inverters with various types of switching schemes. The speed advantage of the analytical method presented versus simulation packages such as PSPICE is apparent when analyzing multiple motor-drive systems. vii ACKNOWLEDGMENTS I would like to thank my advisor, Dr. R. Mark Nelms, for his advice and guidance throughout my graduate studies at Auburn University. I am grateful to Dr. Charles A. Gross for providing an EMAP simulation that was used for comparison with my induction motor harmonic model that was used in this dissertation. I would also like to thank Dr. Gross for his helpful suggestions and his willingness to share some of his knowledge of electric machines with me. I want to thank the other members of my committee, Dr. S. Mark Halpin and Dr. John Y. Hung, for their time and suggestions during the proposal and review of my dissertation. I would like to thank my parents, Jane E. (Pigmon) Leedy and the late Robert A. Leedy, for always stressing to me the importance of a sound education. Most of all, I want to thank my wife, Stephanie J. Leedy, for her love, support, and encouragement during my graduate studies at Auburn University. viii Style manual or journal used Graduate School: Guide to Preparation and Submission of Theses and Dissertations. Computer software used: Microsoft Word 2003, Microsoft Excel 2003, Microsoft Visio 2000, MATLAB 6.5, and PSPICE 9.2. ix TABLE OF CONTENTS LIST OF TABLES............................................................................................................. xi LIST OF FIGURES ......................................................................................................... xiii CHAPTER 1 INTRODUCTION ........................................................................................1 1.1 Introduction..............................................................................................................1 1.2 Background..............................................................................................................4 1.2.1 The Six-Step Inverter....................................................................................4 1.2.2 The Sinusoidal PWM Inverter ......................................................................6 1.2.3 The Space Vector PWM Inverter..................................................................9 1.2.4 The Induction Motor...................................................................................11 1.3 Organization of Dissertation..................................................................................12 CHAPTER 2 HARMONIC ANALYSIS OF THE VOLTAGE SOURCE INVERTER ............................................................14 2.1 The Sinusoidal PWM Inverter ...............................................................................14 2.1.1 The Two-Level PWM Inverter ..................................................................15 2.1.1.1 Harmonic Analysis of the Two-Level Inverter Using the Method of Pulse Pairs .....................................................................18 2.1.1.2 Simulation Results for the Two-Level PWM Inverter....................25 2.1.2 The Three-Level PWM Inverter .................................................................30 2.1.2.1 Harmonic Analysis of the Three-Level Inverter Using the Method of Pulse Pairs .....................................................................32 2.1.2.2 Simulation Results for the Three-Level PWM Inverter...................36 2.1.2.3 Comparison of New and Old Methods ........................................... 40 2.2 The Space Vector PWM Inverter...........................................................................42 2.2.1 Carrier-Based Approach .............................................................................43 2.2.2 Method of Multiple Pulses..........................................................................46 2.2.3 Simulation Results for the Space Vector PWM Inverter............................51 2.3 Line-Neutral Voltage Fourier Series Development ...............................................55 2.3.1 The Six-Step Inverter..................................................................................56 2.3.1.1 120? Conduction .............................................................................56 2.3.1.2 180? Conduction .............................................................................57 2.3.2 The Two-Level Sinusoidal PWM Inverter .................................................58 2.3.3 The Space Vector PWM Inverter................................................................68 2.4 Summary................................................................................................................73 x CHAPTER 3 THE INVERTER-FED INDUCTION MOTOR ........................................75 3.1 Induction Motor Equivalent Circuit.......................................................................75 3.2 Verification of Induction Motor Harmonic Model ................................................83 3.3 Motor-Drive System Model...................................................................................83 3.3.1 Simplified Model Simulation Results.........................................................88 3.3.2 Six-Step Inverter Results ............................................................................89 3.3.3 Two-Level Sinusoidal PWM Inverter Simulation Results .........................91 3.3.4 Space Vector PWM Inverter Simulation Results........................................93 3.4 Summary................................................................................................................95 CHAPTER 4 MULTIPLE MOTOR DRIVE SYSTEMS.................................................96 4.1 DC Power Flow......................................................................................................97 4.2 Verification of the Power Flow Algorithm..........................................................100 4.3 Six-Step Simulation Results for a 10-Bus System...............................................115 4.4 Two-Level Sinusoidal PWM Simulation Results................................................120 4.5 Power Flow Results for Systems with Higher Line Resistance Values...............123 4.6 Summary..............................................................................................................132 CHAPTER 5 CONCLUSIONS ......................................................................................133 5.1 Summary..............................................................................................................133 5.2 Recommendations for Future Work.....................................................................136 REFERENCES ................................................................................................................138 xi LIST OF TABLES TABLE 2.1 MATLAB AND PSPICE RESULTS FOR m a =0.3 and m f =9 ......................27 TABLE 2.2 MATLAB AND PSPICE RESULTS FOR m a =0.6 and m f =15 ....................27 TABLE 2.3 MATLAB AND PSPICE RESULTS FOR m a =1.4 and m f =15 ....................28 TABLE 2.4 MATLAB AND PSPICE RESULTS FOR m a =2.2 and m f =25 ....................28 TABLE 2.5 MATLAB AND PSPICE RESULTS FOR m a =0.8 and m f =10 ....................38 TABLE 2.6 MATLAB AND PSPICE RESULTS FOR m a =1.4 and m f =16 ....................38 TABLE 2.7 MATLAB AND PSPICE RESULTS FOR m a =1.8 and m f =20 ....................39 TABLE 2.8 MATLAB AND PSPICE RESULTS FOR m a =2.2 and m f =20 ....................39 TABLE 2.9 BESSEL FUNCTION METHOD AND PSPICE RESULTS FOR m a =1.4 and m f =18..............................................................41 TABLE 2.10 METHOD OF PULSE PAIRS AND PSPICE RESULTS FOR m a =1.4 and m f =18............................................................42 TABLE 2.11 MATLAB AND PSPICE RESULTS FOR M=0.5 and m f =9.....................52 TABLE 2.12 MATLAB AND PSPICE RESULTS FOR M=0.866 and m f =9.................53 TABLE 2.13 MATLAB AND PSPICE RESULTS FOR M=0.7 and m f =15...................53 TABLE 2.14 MATLAB AND PSPICE RESULTS FOR M=0.65 and m f =15.................54 TABLE 2.15 LINE-TO-NEGATIVE DC BUS VOLTAGE COMPONENTS FOR m a =1.4 and m f =15...................................................67 TABLE 2.16 LINE-TO-NEUTRAL VOLTAGE COMPONENTS FOR m a =1.4 and m f =15...................................................68 TABLE 2.17 LINE-TO-NEGATIVE DC BUS VOLTAGE COMPONENTS FOR M=0.7 and m f =15....................................................72 TABLE 2.18 LINE-TO-NEUTRAL VOLTAGE COMPONENTS FOR M=0.7 and m f =15....................................................73 TABLE 3.1 50 HP, 3-PHASE, INDUCTION MOTOR PARAMETERS .......................84 TABLE 3.2 MATLAB AND EMAP SIX-STEP INVERTER RESULTS.......................84 TABLE 3.3 DIFFERENCES AND PERCENT ERRORS ...............................................85 TABLE 4.1 4-BUS SYSTEM LINE RESISTANCES AND LOAD TORQUES..........101 TABLE 4.2 POWER FLOW RESULTS FOR THE 4-BUS SYSTEM..........................115 TABLE 4.3 SYSTEM LINE RESISTANCES AND LOAD TORQUES ......................118 TABLE 4.4 POWER FLOW RESULTS FOR THE SIX-STEP INVERTER ...............120 TABLE 4.5 SYSTEM LINE RESISTANCES AND LOAD TORQUES .......................121 TABLE 4.6 POWER FLOW RESULTS FOR THE TWO-LEVEL SINE PWM INVERTER ............................................................................123 TABLE 4.7 HARMONIC CONTENT OF INVERTER CURRENT AND VOLTAGE ....................................................................130 xii TABLE 4.8 SYSTEM LINE RESISTANCES AND LOAD TORQUES ......................130 TABLE 4.9 POWER FLOW RESULTS WITH LARGER LINE RESISTANCES......131 xiii LIST OF FIGURES Figure 1.1 Motor-Drive System Model ..............................................................................2 Figure 1.2 DC Power System Model ..................................................................................2 Figure 1.3 Three-Phase Voltage Source Inverter................................................................3 Figure 1.4 Carrier Waveform and Control Signal for a Sinusoidal PWM Inverter............7 Figure 1.5 Carrier Waveform and Control Signal for a Space Vector PWM Inverter .....10 Figure 1.6 Induction Motor T-Type Equivalent Circuit ...................................................12 Figure 2.1 Triangular Waveform and Control Signal.......................................................16 Figure 2.2 Single-Phase Inverter.......................................................................................17 Figure 2.3 Two-Level PWM Output Waveform...............................................................18 Figure 2.4 Positive Pulse Pair ...........................................................................................19 Figure 2.5 Negative Pulse Pair..........................................................................................19 Figure 2.6 PWM Output Signal with Positive and Negative Pulse Pairs Labeled ...........23 Figure 2.7 Special Case Crossing Points ..........................................................................25 Figure 2.8 Harmonic Spectrum with m a =1.0 and m f =25..................................................29 Figure 2.9 Carrier Waveform and Control Signal ............................................................31 Figure 2.10 Three-Level PWM Output Waveform...........................................................31 Figure 2.11 Three-Level PWM Alternative Method ........................................................32 Figure 2.12 Positive Pulse Pair .........................................................................................33 Figure 2.13 PWM Output Signal with Pulse Pairs Labeled..............................................35 Figure 2.14 Special Case Crossing Points ........................................................................36 Figure 2.15 Harmonic Spectrum with m a =0.9 and m f =16................................................40 Figure 2.16 Triangular Waveform and Space Vector Control Signal ..............................44 Figure 2.17 Space Vector PWM Output Waveform.........................................................45 Figure 2.18 Positive Pulse.................................................................................................47 Figure 2.19 Negative Pulse...............................................................................................47 Figure 2.20 PWM Output Signal with Positive and Negative Pulses Labeled.................50 Figure 2.21 Harmonic Spectrum with M=1.1 and m f =27.................................................54 Figure 2.22 Three-Phase Inverter Block Model ...............................................................55 Figure 2.23 Six-Step Phase a Voltage Waveform with 120? Conduction........................57 Figure 2.24 Six-Step Phase a Voltage Waveform with 180? Conduction........................57 Figure 2.25 Three-Phase Sinsusoidal PWM Control Signals and Carrier Waveform......59 Figure 2.26 Line-to-Negative DC Bus Voltage Waveforms ............................................60 Figure 2.27 Waveform v aN (t) with Pulses Labeled ...........................................................61 Figure 2.28 Phase a Line-to-Neutral Voltage Produced using MATLAB .......................62 Figure 2.29 Harmonic Spectrum of the Phase a Line-to-Negative DC Bus Voltage .......62 xiv Figure 2.30 Harmonic Spectrum of the Phase a Line-to-Neutral Voltage .......................63 Figure 2.31 Space Vector PWM Control Signal and Carrier Waveform .........................69 Figure 2.32 Line-to-Negative DC Bus Voltage Waveforms ............................................63 Figure 2.33 Phase a Line-to-Neutral Voltage Waveform.................................................70 Figure 3.1 (a) Induction Motor T-Type Equivalent Circuit; (b) Thevenin Equivalent of (a)........................................................................76 Figure 3.2 (a) Induction Motor Harmonic Equivalent Circuit; (b) Thevenin Equivalent of (a)........................................................................80 Figure 3.3 Positive-Sequence Harmonic Equivalent Circuit ............................................82 Figure 3.4 Negative-Sequence Harmonic Equivalent Circuit...........................................82 Figure 3.5 Motor-Drive System Model ............................................................................85 Figure 3.6 V-I Data Points................................................................................................87 Figure 3.7 Linear Curve Fit ..............................................................................................88 Figure 3.8 Quadratic Curve Fit .........................................................................................88 Figure 3.9 Simplified System Model................................................................................89 Figure 3.10 V-I Load Curve Produced From MATLAB Code .........................................90 Figure 3.11 V-I Load Curve Produced From PSPICE Simulations..................................91 Figure 3.12 V-I Characteristic Curve For a Sinusoidal PWM Inverter with T L =100 N-m...........................................................................................92 Figure 3.13 Quadratic Curve Fit for T L =100 N-m............................................................92 Figure 3.14 V-I Curve For a Space Vector PWM Inverter with T L =80 N-m...................94 Figure 3.15 Quadratic Curve Fit for T L =80 N-m..............................................................94 Figure 4.1 DC Power System Model ................................................................................98 Figure 4.2 Four-Bus DC Power System .........................................................................101 Figure 4.3 V-I Characteristic Curve for T L =75 N-m ......................................................105 Figure 4.4 Quadratic Curve Fit for T L =75 N-m..............................................................106 Figure 4.5 V-I Characteristic Curve for T L =40 N-m ......................................................106 Figure 4.6 Quadratic Curve Fit for T L =40 N-m..............................................................107 Figure 4.7 PSPICE 4-bus System Model........................................................................113 Figure 4.8 PSPICE Six-Step Motor-Drive Model ..........................................................114 Figure 4.9 PSPICE Induction Motor Part .......................................................................115 Figure 4.10 10-Bus DC Power System Model................................................................116 Figure 4.11 PSPICE 10-Bus Power System Model........................................................119 Figure 4.12 PSPICE Sinusoidal PWM Motor-Drive Model...........................................122 Figure 4.13 Six-Step Inverter System with a Low Line Resistance Value.....................124 Figure 4.14 Line-to-Line Voltages with Low Line Resistance.......................................125 Figure 4.15 Inverter DC Input Current and Voltage with Low Line Resistance............125 Figure 4.16 Six-Step Inverter System with a Higher Line Resistance Value.................127 Figure 4.17 Line-to-Line Voltage (V ab ) with a Higher Line Resistance ........................128 Figure 4.18 Inverter Input Voltage with a Higher Line Resistance................................128 Figure 4.19 Inverter Input Current with a Higher Line Resistance ................................129 1 CHAPTER 1 1.1 Introduction This effort has been focused on the analysis of the system shown in Figure 1.1. In this figure, a DC voltage source is connected to a three-phase inverter driving a three-phase induction motor with a load attached. The goal was to develop an analytical method to analyze this system that is faster than simulation packages such as PSPICE and Simulink and produces comparable results. The method can be utilized in the analysis of a DC power system containing multiple motor-drive loads such as the one shown in Figure 1.2. The speed advantage of the analytical method is evident when multiple motor-drive systems are analyzed. Some possible applications for DC power systems such as the one shown in Figure 1.2 are: transit systems, U.S Navy ships and submarines, and some coal mining operations. The induction motor was utilized in Figure 1.2 because it is employed in some of the applications mentioned previously. Induction motors are used in a wide range of industrial settings as they are capable of operating in dusty and harsh environments such as in underground coal mines. The output voltage waveforms produced by the inverter shown in Figure 1.1 will contain harmonics. The harmonic content of the output waveforms will depend on the switching scheme utilized in the voltage source inverter of Figure 1.1. A more detailed drawing of a three-phase voltage source inverter is illustrated in Figure 1.3. Depending 2 DC Voltage Source Voltage Source Inverter 3-Phase Induction Motor i I i V + - N a b c Load Figure 1.1: Motor-Drive System Model. Source Motor-Drive Loads Distribution Network . . . Figure 1.2: DC Power System Model. 3 i I i V 1S 5S 4S 6S 2S 3S a b c ? + Figure 1.3: Three-Phase Voltage Source Inverter. upon the method for controlling the switches of the inverter in Figure 1.3, the inverter can operate as a six-step inverter, sinusoidal PWM inverter, or a space vector PWM inverter. Two methods for determining the harmonic components of the output waveforms of the voltage source inverter in Figure 1.3 were developed in this dissertation. Both methods can be used to determine the harmonic content of the inverter output waveforms for different switching schemes. The two harmonic analysis methods developed allow direct calculation of harmonic magnitudes and angles without having to use linear approximations, iterative procedures, look-up tables, or Bessel functions. These methods can also be extended to other types of multilevel inverters and PWM schemes. Because the voltages at the terminals of the induction motor shown in Figure 1.1 will contain harmonics produced by the inverter, a harmonic model of the induction motor 4 was developed that is based on the steady-state T-type equivalent circuit model of the induction motor. A simplified model of the system shown in Figure 1.1 was developed using the induction motor harmonic model and the input-output relationships of the voltage source inverter. In the simplified model a V-I load characteristic curve was established that allows all of the system components to the right of V i (inverter, motor, and load) in Figure 1.1 to be replaced by a current-controlled voltage source. The simplified model developed for the system in Figure 1.1 was shown to be applicable to a multiple-bus DC power system such as that shown in Figure 1.2 by forming a V-I load characteristic curve for each motor-drive load in the system and incorporating them into an iterative procedure used to conduct a power flow analysis. 1.2 Background 1.2.1 The Six-Step Inverter The six-step inverter is perhaps the simplest form of three-phase inverter. A circuit diagram of a three-phase voltage source inverter is shown in Figure 1.3. The output of a six-step inverter can be produced by using one of two types of gate firing sequences: three switches in conduction at the same time (180? conduction), or two switches in conduction at the same time (120? conduction). With either case, the gating signals are applied and removed every 60? of the output voltage waveform. The switches in Figure 1.3 are gated in the sequence S1, S2, S3, S4, S5, and S6 every cycle. The result of this type of gating produces six steps in each cycle. Even though the six-step inverter is simplistic compared to the various types of PWM inverters, many articles have been written covering different applications and various aspects of the operation of the six-step voltage source inverter [1-7]. 5 Murphy and Turnbull [8] discussed AC motor operation when supplied by a six-step voltage source inverter in Chapter 4 of their book. Voltage waveforms were provided along with the Fourier series representations. Current waveforms were also provided with detailed discussions of motor operation when supplied by a six-step inverter. Abbas and Novotny [9] utilized a fundamental component approximation to develop equivalent circuits that represent the transfer relations of the six-step voltage source and current source inverters during steady-state operation. Development of the equivalent circuits was based on the idealized switching constraints of the inverter circuits. Only the fundamental component of the voltage and current Fourier series was retained in the development of the equivalent circuits presented. This simplification was made due to the harmonics resulting in small amounts of average torque. Krause and Lipo [10] presented simplified representations of a rectifier-inverter induction motor drive system. The first simplified representation was developed by neglecting the harmonic components due to the switching in the rectifier. The second simplified representation resulted when the harmonic components due to the switching in the inverter were neglected. The final simplification was made by neglecting all harmonic components and representing the system in the synchronously rotating reference frame. In the analysis leading to the final simplified system representation, the operation of the inverter was expressed analytically in the synchronously rotating reference frame with the harmonic components due to the switching in the inverter included. Krause and Hake [11] used the method of multiple reference frames and the equations of transformation of the inverter to establish a method of calculating the inverter input 6 current. The method presented allows the current flowing into the inverter to be determined during constant speed, steady-state operation. Novotny [12] used time dependent functions called switching functions to represent transfer properties of six-step voltage source and current source inverters. The switching functions were expanded as complex Fourier series and applied to steady-state inverter operation. The concepts presented can be extended to PWM inverters. Novotny [13] used time domain complex variables to represent the inverter and the induction motor. Time domain complex variables result from applying the symmetrical component concept to instantaneous quantities. Steady-state analysis of the six-step voltage and current source inverter-driven induction motor is provided. Closed form solutions for the instantaneous voltages, currents, and torques were presented. 1.2.2 The Sinusoidal PWM Inverter Pulse width modulation is a popular technique used to control the magnitude and frequency of the AC output voltages of an inverter. In a sinusoidal PWM inverter, the gate signals used to control the switches of the inverter in Figure 1.3 are produced by comparing a sinusoidal control signal with a high frequency carrier waveform as shown in Figure 1.4 for a two-level sinusoidal PWM inverter. This technique is widely used in industrial applications such as variable-speed electric drives [14, 15] and has been the focus of research interests in power electronics applications for many years. Most of the research to date has been focused on determining the harmonic components produced as a result of the modulation process due to various schemes and techniques [14, 16-18]. 7 Figure 1.4: Carrier Waveform and Control Signal for a Sinusoidal PWM Inverter. Analysis of modulated pulses was first introduced by Bennett [19] in 1933. Bennett used the double Fourier series to analyze modulated pulses in his study of rectified waves. Bennett?s method was shown to be applicable to various types of waveforms and complex modulation processes. A detailed explanation of Bennett?s method as applied to communications systems was presented by Black [20]. Bowes [21,22] was the first to use Bennett?s method in power electronics applications. Bowes used a 3-D modulation model based on the double Fourier series to apply Bennett?s method to inverter systems. The method introduced by Bennett and applied by Black and Bowes is valid only for amplitude modulation ratios less than one. Using the waveforms of a two- level sinusoidal PWM inverter with sine-triangle modulation in Figure 1.4, the amplitude modulation ratio is defined as: 8 tri con a V V m = (1.1) where V con is the peak amplitude of the control signal in Figure 1.4 and V tri is the peak amplitude of the triangular carrier waveform in Figure 1.4. Extensions of Bennett?s method to calculate the harmonic content of the output voltage of a PWM inverter for amplitude modulation ratios greater than one were presented by Franzo et al. [15] and Mazzucchelli et al. [23]. Carrara et al. [24] used an extension of Bennett?s method to find analytical expressions of the output voltage of single-phase and three-phase inverters. Calculations of the harmonic components of the output voltage of the inverter were possible for any operating condition, including the over modulation region m a >1.0. The analysis presented was applied to various multilevel modulation techniques. Holmes [25] presented a generalized analytical approach for calculating the harmonic components of various fixed carrier frequency PWM schemes. The method was based on the double Fourier series of the switched waveform. Holmes produced closed form solutions using a Jacobi-Anger substitution. Analytical solutions were provided for various PWM strategies including space vector modulation. Tseng, et al. [26] used a 3-D modulation model and the double Fourier series as first proposed by Bennett to analyze the harmonic characteristics of a three-phase two-level PWM inverter. Models of the three-phase inverter system were constructed in PSPICE and MATLAB for harmonic analysis purposes. Equations from the theoretical analysis using the 3-D modulation model and the double Fourier series were coded in MATLAB for comparison with PSPICE and Simulink results. It was shown that the harmonic 9 content of waveforms produced from the PSPICE and Simulink models are in good agreement with the harmonic content of waveforms calculated using the 3-D modulation model and the double Fourier series. Mohan et al. [27] conducted an analysis of two-level PWM inverters in Chapter 8 of their book. Design considerations for the two-level PWM were discussed in Chapter 8 as well. Harmonic analysis of the induction motor was discussed in Chapter 14. Various schemes using pulse width modulation for the purpose of shaping the AC output voltages of an inverter to be as close to sinusoidal as possible have been studied and continue to be the focus of many power electronics research activities. For the interested researcher, a detailed literature review on pulse width modulation that includes various modulation techniques and schemes can be found in [16]. 1.2.3 The Space Vector PWM Inverter Space vector modulation is a PWM technique that has become extremely popular in recent years. In a space vector PWM inverter, the gate signals used to control the switches of the inverter in Figure 1.3 are produced by comparing the control signal shown in Figure 1.5 with a high frequency triangular waveform. The space vector PWM inverter is commonly used in vector control drive applications [28] where microprocessors are used to generate voltage waveforms [29]. Even though many articles are available in the literature [16], space vector pulse width modulation continues to be the focus of many power electronics researchers [30, 31]. Space vector modulation was first introduced in the mid-1980?s [32-34] and was greatly advanced by Van Der Broeck [33] in 1988. The method was initially developed as a vector approach to pulse 10 Figure 1.5: Carrier Waveform and Control Signal for a Space Vector PWM Inverter. width modulation. The approach used by Van Der Broeck was based on representing voltages using space vectors in the ?, ? plane. Harmonic analysis of the space vector PWM inverter has been investigated by various researchers [16, 29, 35-37]. Boys and Handley [29] decomposed a general regularly sampled asymmetric PWM waveform into symmetrical components that simplified the harmonic analysis of the PWM output waveform. The technique was extended by Boys and Handley to analyze waveforms generated by space vector modulation. Bresnahan et al. [35] conducted a harmonic analysis of space vector line-to-line voltages generated by a microcontroller. An FFT analyzer and MATLAB/Simulink routines were used to conduct the harmonic analysis. Moynihan et al. [36] used an extension of the geometric- wall model to conduct a harmonic analysis on space vector modulated waveforms. 11 Harmonic analysis of two different space vector PWM methods was presented by Halasz et al. [37]. Holmes and Lipo presented a technique used to analyze the harmonic content of space vector PWM waveforms using a double Fourier series method [16]. A detailed explanation of the technique was provided along with the mathematical derivation of the analytical results. Panaitescu and Mohan [38] presented an analysis and hardware implementation of space vector pulse width modulation used for voltage source inverter-fed AC motor drives. A carrier-based approach was used without the need for sector calculations or vector decomposition. Mohan [39] presents a detailed explanation of space vector PWM inverters in Chapter 7 of his book. A CD was provided with examples and Simulink? models that are helpful in understanding space vector concepts. Mohan used a carrier-based approach to analyze the space vector PWM inverter. 1.2.4 The Induction Motor Fitzgerald, et al. [40] provided a detailed analysis of the steady-state T-type equivalent circuit model of the induction motor in Chapter 7 of their book. The model presented in Chapter 7, and shown in Figure 1.6, can easily be modified in order to perform a harmonic analysis on the induction motor. Ozpineci and Tolbert [41] presented a modular Simulink implementation of an induction motor model. In the model presented, each block solved one of the model equations. This ?modular? system model allowed all of the machine parameters to be accessible for control and verification of results. 12 - R 1 jX 1 jX 2 jX m 1 V + 1 I 2 I 1 2 s R Figure 1.6: Induction Motor T-Type Equivalent Circuit. Giesselmann [42] developed a PSPICE d-q model of the induction motor for analysis and simulation purposes. The PSPICE model was based on the T-type equivalent circuit model of the induction motor. Implementation of the d-q model equations in PSPICE was accomplished using Analog Behavioral Modeling (ABM) devices. Expression based ABM devices allow the user to enter mathematical expressions that can be used in PSPICE circuit models. Krause [43] used reference frame theory for the analysis of electric machines in Chapter 3 of his book. In Chapter 4, a detailed d-q analysis of the induction motor is presented. Reference frame theory as applied to the analysis of electric drives is discussed in Chapter 13. 1.3 Organization of the Dissertation In this introductory chapter, a description of the problem to be investigated, the goals of the dissertation, and background information on previous work relating to voltage source inverter-fed induction motor drives have been presented. Harmonic analysis of the voltage source inverter and two methods for determining the harmonic components of the output of a voltage source inverter are discussed in Chapter 2. A harmonic model of 13 the induction motor and the development of a simplified model of an inverter-fed induction motor are discussed in Chapter 3. Multiple motor-drive systems are the focus of Chapter 4, with a presentation of an iterative procedure that can be used to conduct a power flow analysis on a DC power system containing multiple motor-drive loads. The dissertation concludes with a summary of the dissertation and recommendations for future work in Chapter 5. 14 CHAPTER 2 HARMONIC ANALYSIS OF THE VOLTAGE SOURCE INVERTER The focus of this chapter is on the harmonic analysis of different types of voltage source inverters. The types of inverters analyzed in this chapter include: the six-step inverter, the sinusoidal PWM inverter, and the space vector PWM inverter. Methods for determining the harmonic content of the output waveforms of the sinusoidal PWM and the space vector PWM voltage source inverters are presented and can be used to conduct a harmonic analysis on an induction motor while supplied by a voltage source inverter. The waveforms analyzed in sections 2.1and 2.2 are typical voltage source inverter output waveforms produced by single-phase inverter topologies, while those analyzed in section 2.3 are typical waveforms produced by a three-phase voltage source inverter. The equations used to determine the harmonic content of the voltage source inverter output waveforms were coded in MATLAB and compared with PSPICE simulation models. The chapter concludes with a summary of the harmonic analysis techniques presented in the chapter. 2.1 The Sinusoidal PWM Inverter A method to analyze the harmonic content of modulated pulses was first introduced by Bennett in 1933 [19]. Bennett?s method and other methods based on Bennett?s work used the double Fourier series to analyze the output PWM signal. Using a double 15 Fourier series to determine the harmonic components of the PWM output signal required the use of Jacobi-Anger expansions to establish closed form solutions. The end result of using Jacobi-Anger expansions was the appearance of Bessel functions in the final expression of the output PWM signal. Understanding and applying these methods can be cumbersome, leading to computer programming errors when attempting to implement a particular method. Methods that use the double Fourier series also result in final voltage expressions that typically contain three terms: one term to calculate the amplitude of the fundamental harmonic, one term to calculate the carrier frequency harmonic and harmonics of the carrier frequency, and another term to calculate the sideband frequency harmonics. The purpose of this section is to present a method to calculate the harmonic components of the output voltage of a two-level and a three-level sinusoidal PWM inverter that is capable of being applied to various types of multilevel inverters and PWM schemes. This method allows direct calculation of harmonic magnitudes and angles without the use of linear approximations, iterative procedures, look-up tables, Bessel functions, or the gathering of harmonic terms. The method is valid in the overmodulation region (m a >1.0) and has the potential to be extended to inverter-drive systems such as the one presented in [44]. 2.1.1 The Two-Level PWM Inverter In a two-level PWM inverter with sine-triangle modulation, a sinusoidal control signal at a desired output frequency is compared with a triangular waveform as shown in Figure 2.1. The control signal shown in Figure 2.1 can be expressed as: tVtv concontrol 1 sin)( ?= (2.1) 16 Figure 2.1: Triangular Waveform and Control Signal. where V con is the peak amplitude of the control signal and ? 1 is the angular frequency. The angular frequency is given as: 11 2 f?? = (2.2) where f 1 is the desired fundamental frequency of the inverter output. The triangular waveform v triangle in Figure 2.1 is normally kept at a constant frequency f s and a constant amplitude V tri . The frequency f s is also known as the switching frequency or carrier frequency of the inverter. The amplitude modulation ratio is defined as: tri con a V V m = . (2.3) The frequency modulation ratio is defined as: 17 1 f f m s f = . (2.4) If the variables listed in (2.1-2.4) are known, the output PWM signal can be produced by comparing the waveforms shown in Figure 2.1. Referring to Figure 2.2, when v control > v triangle , T A+ and T B- are closed and the value of the output PMW signal is +V i (where V i is the DC input voltage of the inverter). When v control < v triangle , T A- and T B+ are closed and the value of the output PWM signal becomes -V i . As noted in [23], the output voltage of the inverter can be considered to be a voltage switching from +V i to -V i . The output PWM signal produced from comparing the waveforms in Figure 2.1 is shown in Figure 2.3. i I i V +A T ?A T ?B T +B T ? + ? + )(tv o Figure 2.2: Single-Phase Inverter. 18 Figure 2.3: Two-Level PWM Output Waveform. 2.1.1.1 Harmonic Analysis of the Two-Level Inverter Using the Method of Pulse Pairs It is desirable to find a general technique to calculate the harmonic components of a PWM waveform such as the one shown in Figure 2.3. To accomplish this task, it can be observed that the waveform in Figure 2.3 is made up of multiple positive and negative pulse pairs. Also, another observation that will be helpful in the derivation of the analysis technique presented is the fact that the waveform in Figure 2.3 possesses half- wave symmetry. This means that for each positive pulse during the first half of the period of the PWM signal, there is a corresponding negative pulse in the second half of the PWM signal period. This is illustrated by the arbitrary positive pulse pair shown in Figure 2.4 where A is the amplitude of the pulse, a P is the initial time delay of the positive pulse, b P is the pulse width of the positive pulse, and T is the period of the 19 f(t) t T/2 Ta P b P a P b P A -A Figure 2.4: Positive Pulse Pair. PWM waveform. For each negative pulse in the first half of the PWM signal period, there is a corresponding positive pulse in the second half of the period. This is illustrated by the arbitrary negative pulse pair shown in Figure 2.5. In this figure, a N is the initial time delay of the negative pulse, and b N is the pulse width of the negative pulse. g(t) t T/2 Ta N b N a N b N A -A Figure 2.5: Negative Pulse Pair. 20 The first step in the analysis is to find the trigonometric Fourier series of the waveform shown in Figure 2.4. Since it is known that the waveform in Figure 2.3 has half-wave symmetry, the Fourier coefficient a 0 is zero. This is due to the fact that the average value of a function with half-wave symmetry is always zero. The Fourier coefficients a n and b n are also zero for n even due to half-wave symmetry. Using the above simplifications, the trigonometric Fourier series of the function f(t) shown in Figure 2.4 can be expressed as: ? ? = ? ? ? ? ? ? += oddn n nn t T n bt T n atf POSPOS 1 2 sin 2 cos)( ?? (2.5) where a n POS and b n POS are the Fourier coefficients of the positive pulse pair. The coefficient a n POS can be found from Figure 2.4 as follows: dtt T n tf T a T n POS ? = 0 2 cos)( 2 ? , (2.6) . 2 cos)( 22 cos)( 2 2 2 dtt T n A T dtt T n A T a PP P PP P POS ba T a T ba a n ?? ++ + + ?+= ?? (2.7) Integrating (2.7) and using the identity sin?-sin ? = 2cos 1/2(?+?) sin 1/2(?-?), (2.7) becomes: .sin 2 cos 2 sin 2 cos 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ++? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += PPP PPPn b T n b T n a T n n n A b T n b T n a T n n A a POS ??? ? ? ??? ? (2.8) 21 The coefficient b n POS can be found from Figure 2.4 as follows: , 2 sin)( 2 0 dtt T n tf T b T n POS ? = ? (2.9) . 2 sin)( 22 sin 2 2 2 dtt T n A T dtt T n A T b PP P PP P POS ba T a T ba a n ?? ++ + + ?+= ?? (2.10) Integrating (2.10), using the identity cos ?-cos ? = -2sin 1/2(?+?) sin 1/2(?-?), and using the fact that sin(-?) = -sin ?, (2.10) becomes: .sin 2 sin 2 sin 2 sin 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ++? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += PPP PPPn b T n b T n a T n n n A b T n b T n a T n n A b POS ??? ? ? ??? ? (2.11) Equations (2.8) and (2.11) can now be substituted into (2.5) and the trigonometric Fourier series of the waveform f(t) is established. The trigonometric Fourier series of the waveform g(t) shown in Figure 2.5 is the same as the waveform f(t) in Figure 2.4 except that the magnitudes are the negative of each other. The Fourier coefficients for g(t) are as follows: ,sin 2 cos 2 sin 2 cos 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +++ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +?= NNN NNNn b T n b T n a T n n n A b T n b T n a T n n A a NEG ??? ? ? ??? ? (2.12) 22 ,sin 2 sin 2 sin 2 sin 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +++ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +?= NNN NNNn b T n b T n a T n n n A b T n b T n a T n n A b NEG ??? ? ? ??? ? (2.13) where a n NEG and b n NEG are the Fourier coefficients of the negative pulse pair. The trigonometric Fourier series for g(t) can be expressed in the same form as f(t) in (2.5): . 2 sin 2 cos)( 1 ? ? = ? ? ? ? ? ? += oddn n nn t T n bt T n atg NEGNEG ?? (2.14) Because the Fourier series of arbitrary positive and negative pulse pairs has been established, the Fourier series of a given PWM signal produced by two-level modulation can be found by application of the principle of superposition. A PWM waveform like the one in Figure 2.3 is made up of the sum of positive and negative pulse pairs as shown in Figure 2.6 where P1-P3 in the figure are positive pulse pairs and N1-N3 are negative pulse pairs. All that is required to find the Fourier series of a signal like the one shown in Figure 2.6 is to find the Fourier coefficients of each individual positive and negative pulse pair contained in the PWM signal and add them to get the Fourier coefficients of the entire PWM signal. The total a n and b n coefficients of the entire PWM signal can be found using (2.8) and (2.11-2.13) as follows: () ?? ? == += oddn n K j nnn j POS j NEG aaa 11 , (2.15) 23 t T/2 T Vi -Vi P1 P1 P2 P3 P2 P3 N3 N1 N2 N1 N2 N3 )( 2 tv L? Figure 2.6: PWM Output Signal with Positive and Negative Pulse Pairs Labeled. () ?? ? == += oddn n K j nnn j POS j NEG bbb 11 , (2.16) where K is the number of positive or negative pulse pairs (Note: the number of positive pulse pairs will equal the number of negative pulse pairs due to symmetry.). The Fourier series of a given PWM signal produced by two-level modulation can be expressed in a single-cosine series as: ? ? ? ? ? ? += ? ? = ? n oddn n nL t T n Ctv ? ?2 cos)( 1 2 (2.17) where 22 nnn baC += and ? ? ? ? ? ? ? ? ?= ? n n n a b 1 tan? . It should be noted that the subscript 2-L in (2.17) stands for two-level. 24 The final step in implementing this method is to find the crossing points of the waveforms shown in Figure 2.2 that determine the edges of the PWM signal pulses. In order to determine the crossing points, an equation for the triangular wave in Figure 2.2 must be established. The signal can be thought of as being made up of straight lines having alternating positive and negative slopes with shifted intercepts on the time axis. To implement this idea in a computer software package, the triangular waveform can be expressed as: ())1(2)1( 4 )1(),( 21 ?+?+ ? ? ? ? ? ? ? ? ?= ++ nVVt T V tnV tritri n s trin triangle (2.18) where n is the index number used in a computer program and T s is the period of the triangular wave. Since the PWM signal has half-wave symmetry, only the crossing points that occur in the first half of the PWM signal period need to be considered when using the method of pulse pairs. To find the crossing points, set v control = v triangle and solve the transcendental equation for t. To easily solve the transcendental equation in MATLAB, declare t as a symbolic object using the syms command. The solve command can then be used to find the crossing points. However, the use of (2.18) results in some special cases where crossing points occur above the peak amplitude V tri of the triangular wave as shown in Figure 2.7. These special cases occur due to the fact that the straight lines used to represent the triangular signal extend beyond the value of V tri 25 Figure 2.7: Special Case Crossing Points. and will intersect the control signal at crossing points that are undesired. These undesired points can be eliminated using the find command in MATLAB, leaving the crossing points that determine the edges of the PWM signal pulses. At this point, the only requirement to implement the method of pulse pairs is to use the crossing points to determine the time delays and the pulse widths. 2.1.1.2 Simulation Results for the Two-Level PWM Inverter The equations of the control signal, the carrier waveform, and the equations used to implement the method of pulse pairs were coded in MATLAB for the purpose of computing the harmonic components of a PWM signal such as the one shown in Figure 2.3. MATLAB code was also written to find the crossing points, time delays, and pulse widths. Four MATLAB simulations were conducted using different values of m a and m f . 26 The following parameter values were used for all simulations: V i = 270 V, V tri = 10 V, and f 1 = 60 Hz. The other parameters used for the first simulation were as follows: V con = 3 V and f s = 540 Hz. The parameters used for the second MATLAB simulation were: V con = 6 V and f s = 900 Hz. The parameters used for the third simulation were: V con = 14 V and f s = 900 Hz. The fourth simulation was conducted using the following parameters: V con = 22 V and f s = 1.5 kHz. PSPICE was used to verify the results from the MATLAB calculations by constructing a two-level PWM simulation model. A PSPICE ABM block was used to compare the sinusoidal control signal and the triangular carrier wave. A Fourier analysis was then performed in PSPICE on the PWM output signal of the ABM block. The parameters used in the PSPICE simulations were the same as the ones used in the four MATLAB simulations. Results of the MATLAB and PSPICE simulations are shown in Tables 2.1-2.4. The results shown in Table 2.1 and Table 2.2 are for dominant carrier frequency and sideband harmonics. Because the results shown in Table 2.3 and Table 2.4 are for simulations conducted in the overmodulation region, all harmonics up to the 31 st harmonic were included. The harmonic number of individual sidebands can be found using the following formula [27]: qpmh f ?= (2.19) where p and q are integers. When p is odd, sideband harmonics exist only for even values of q. When p is even, sideband harmonics exist only for odd values of q. The use of (2.19) is not required when applying the method of pulse pairs and is provided here as 27 TABLE 2.1 MATLAB AND PSPICE RESULTS FOR m a =0.3 and m f =9 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 81.03 80.999 0.031 -0.030 0.0006 0.03049 7 9.395 9.3652 0.0298 90.190 90.0002 0.1898 9 324.9 324.9511 0.0511 89.910 90.0001 0.0901 11 9.362 9.3652 0.0032 89.980 90.0077 0.0277 25 24.18 24.1504 0.0296 -90.130 269.9971 0.1271 27 64.07 64.1064 0.0364 -90.260 -89.9995 0.2605 29 24.17 24.1504 0.0196 -90.370 269.9822 0.3522 35 49.96 49.9735 0.0135 179.600 180.0004 0.4004 37 49.98 49.9735 0.0065 -0.382 0.0057 0.388 41 4.187 4.1754 0.0116 83.030 83.7006 0.6706 43 29.14 29.1326 0.0074 89.620 89.9774 0.3574 45 1.728 1.7524 0.0244 90.210 89.9252 0.2848 53 22.92 22.9487 0.0287 -0.556 0.002 0.5578 55 22.93 22.9487 0.0187 179.600 180.1294 0.5294 57 15.95 15.942 0.008 -177.400 183.3271 0.7271 TABLE 2.2 MATLAB AND PSPICE RESULTS FOR m a =0.6 and m f =15 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 162 161.9981 0.0019 0.003 0.0013 0.001369 13 35.38 35.4205 0.0405 89.830 89.9971 0.1671 15 271.5 271.5686 0.0686 89.850 90.0002 0.1502 17 35.45 35.4205 0.0295 89.790 90.0056 0.2156 27 19.1 19.1058 0.0058 -0.394 -0.0128 0.3815 29 99.99 99.947 0.043 -0.306 -0.0033 0.303 31 99.93 99.947 0.017 179.700 180.0033 0.3033 33 19.17 19.1058 0.0642 179.7 180.0125 0.3125 41 12.63 12.606 0.024 -90.72 269.9957 0.7157 43 54.94 54.9466 0.0066 -90.45 269.9967 0.4467 45 22.52 22.4717 0.0483 89.62 89.9978 0.3778 47 54.9 54.9466 0.0466 -90.47 -89.9949 0.4751 49 12.57 12.6061 0.0361 -90.77 -89.9948 0.7752 28 TABLE 2.3 MATLAB AND PSPICE RESULTS FOR m a =1.4 and m f =15 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 311.8 311.8012 0.0012 -0.239 -0.2342 0.0047 3 39.2 39.2488 0.0488 3.519 3.5087 0.0103 5 8.765 8.7275 0.0375 176.100 176.1209 0.0209 7 7.559 7.5407 0.0183 -132.400 227.4985 0.1015 9 4.01 4.0148 0.0048 33.060 33.788 0.728 11 37.26 37.2808 0.0208 87.700 87.8266 0.1266 13 83.63 83.6026 0.0274 91.390 91.5182 0.1282 15 105.4 105.3281 0.0719 89.820 89.9667 0.1467 17 83.65 83.6208 0.0292 88.310 88.4674 0.1574 19 37.29 37.2907 0.0007 92.430 92.5775 0.1475 21 3.731 3.7163 0.0147 143.900 143.55 0.35 23 12.49 12.51 0.02 -26.680 -26.3705 0.3095 25 35.1 35.0952 0.0048 0.798 1.0319 0.2341 27 43.54 43.5175 0.0225 2.872 3.131 0.259 29 20.07 20.0505 0.0195 -4.290 -3.9921 0.2979 31 20.02 20.0147 0.0053 -176.800 183.4837 0.2837 TABLE 2.4 MATLAB AND PSPICE RESULTS FOR m a =2.2 and m f =25 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 331.5 331.5119 0.0119 -0.187 -0.1964 0.0091 3 80.94 80.8832 0.0568 0.163 0.1361 0.0273 5 21.78 21.7184 0.0616 4.276 4.2415 0.0345 7 2.064 2.0923 0.0283 130.800 131.7061 0.9061 9 7.324 7.3304 0.0064 -176.100 183.8417 0.0583 11 5.917 5.8938 0.0232 -146.200 213.9363 0.1363 13 4.024 4.0355 0.0115 -96.410 264.1293 0.5393 15 2.304 2.3063 0.0023 10.570 9.7675 0.8025 17 11.69 11.6474 0.0426 79.240 79.4863 0.2463 19 27.94 27.9149 0.0251 89.070 89.3683 0.2983 21 45.75 45.7661 0.0161 91.260 91.5431 0.2831 23 59.53 59.573 0.043 90.930 91.2003 0.2703 25 64.71 64.7653 0.0553 89.700 89.9511 0.2511 27 59.55 59.5887 0.0387 88.470 88.7123 0.2423 29 45.77 45.7829 0.0129 88.230 88.4476 0.2176 29 an aid in determining sideband harmonic numbers for the example simulations shown in Tables 2.1-2.4. Most techniques that use the double Fourier series approach must include a term in the final PWM output voltage expression dedicated to calculating sideband harmonics that requires (2.19). The harmonic spectrum of a PWM inverter output voltage waveform with m a =1.0 and m f =25 is shown in Figure 2.8 for the first 80 harmonics. The white bars on the graph in Figure 2.8 are PSPICE results and the gray bars on the graph are results from the derived equations that were coded in MATLAB. The harmonic components found using the equations coded in MATLAB are similar to the ones found using the PSPICE model as illustrated by the results in the tables and Figure 2.8. These results show that the method of pulse pairs is an accurate method used to find the harmonic components of a two-level PWM inverter output waveform. Two-Level PWM Output Voltage Harmonic Spectrum 0 50 100 150 200 250 300 1 2123252729454749505153556769717375777980 Har m onic Num be r M a gni t ude PSPICE Matlab Figure 2.8: Harmonic Spectrum with m a =1.0 and m f =25. 30 2.1.2 The Three-Level PWM Inverter In a three-level PWM inverter with sinusoidal modulation, a control signal at a desired output frequency is compared with a multi-level triangular waveform as shown in Figure 2.9. The control signal shown in Figure 2.9 can be expressed the same as (2.1). It should be noted that the carrier signal in Figure 2.9 is a different carrier signal than the one used for the two-level case in Figure 2.1. Therefore, a new notation for the carrier waveform is needed. The triangular waveform in Figure 2.9 will be referred to as v carrier and the amplitude of the carrier waveform will be denoted as V car . The amplitude modulation ratio is defined as: car con a V V m = . (2.20) The frequency modulation ratio is defined the same as in (2.4). If the variables listed in (2.1, 2.2, 2.4, and 2.20) are known, the output PWM signal can be produced by comparing the waveforms shown in Figure 2.9. The switches in Figure 2.2 are controlled based on the following conditions: v control v tri : T A+ is closed, and when v control >-v tri : T B- is closed. It should be noted that V tri is the upper half of the carrier waveform and -V tri is the lower half of the carrier waveform in Figure 2.9. Referring to Figure 2.2, when T A+ and T B- are closed, the value of the output PMW signal is +V i . When T A- and T B+ are closed in Figure 2.2, the value of the output PWM signal is -V i . When T A+ and T B+ are closed or when T A- and T B- are closed, the value of the output PWM signal is zero. A three-level PWM output waveform such as the one shown in Figure 2.10 can also be generated by comparing a triangular carrier waveform with a sinusoidal control signal and the negative 31 Figure 2.9: Carrier Waveform and Control Signal. Figure 2.10: Three-Level PWM Output Waveform. 32 of the sinusoidal control signal as described in [27]. This alternative method of generating a three-level PWM output signal is shown in Figure 2.11. Figure 2.11: Three-Level PWM Alternative Method. 2.1.2.1 Harmonic Analysis of the Three-Level Inverter Using the Method of Pulse Pairs A technique can be found to calculate the harmonic components of the PWM waveform shown in Figure 2.10 that is simple and easy to implement in a computer software package such as MATLAB. It can be observed that the waveform in Figure 2.10 is made up of multiple positive pulse pairs. This waveform also possesses half-wave symmetry. This means that for each positive pulse during the first half of the period of 33 the PWM signal, there is a corresponding negative pulse in the second half of the PWM signal period. This is illustrated by the arbitrary positive pulse pair shown in Figure 2.12 where A in the figure is the amplitude of the pulse, a P is the initial time delay of the positive pulse, b P is the pulse width of the positive pulse, and T is the period of the PWM waveform. h(t) t T/2 Ta P b P a P b P A -A Figure 2.12: Positive Pulse Pair. The first step in the analysis is to find the trigonometric Fourier series of the waveform shown in Figure 2.12. Because it is known that the waveform in Figure 2.10 has half-wave symmetry, the Fourier coefficient a 0 is zero. The trigonometric Fourier series of the function h(t) shown in Figure 2.12 can be expressed as: ? ? = ? ? ? ? ? ? += oddn n nn t T n bt T n ath POSPOS 1 2 sin 2 cos)( ?? (2.21) where a n POS and b n POS are the Fourier coefficients of the positive pulse pair. The 34 coefficients a n POS and b n POS can be found using (2.6-2.11). The Fourier coefficients can then be substituted into (2.21) and the trigonometric Fourier series of the waveform h(t) is established. Because the Fourier series of an arbitrary positive pulse pair has been established in (2.21), the Fourier series of a given PWM signal produced by three-level modulation can be found by application of the principle of superposition. A PWM waveform like the one in Figure 2.10 is made up of the sum of positive pulse pairs as shown in Figure 2.13 where P1-P3 in the figure are positive pulse pairs. All that is required to find the Fourier series of the signal in Figure 2.13 is to find the Fourier coefficients of each individual positive pulse pair contained in the PWM signal and add them to get the Fourier coefficients of the entire PWM signal. The total a n and b n coefficients of the entire PWM signal can be found using (2.8) and (2.11) as follows: () ?? ? == = oddn n K j nn P j POS aa 11 , (2.22) () ?? ? == = oddn n K j nn P j POS bb 11 , (2.23) where K P is the number of positive pulse pairs. The Fourier series of a given PWM signal produced by three-level modulation can be expressed in a single cosine series as: ? ? ? ? ? ? += ? ? = ? n oddn n nL t T n Dtv ? ?2 cos)( 1 3 (2.24) where 22 nnn baD += and ? ? ? ? ? ? ? ? ?= ? n n n a b 1 tan? . The subscript 3-L in (2.24) stands for three-level. 35 v 3-L (t) t T/2 T Vi -Vi P2 P3P1 P1 P2 P3 Figure 2.13: PWM Output Signal with Pulse Pairs Labeled. The final step in implementing this method is to find the crossing points of the waveforms shown in Figure 2.9 that determine the edges of the PWM signal pulses. In order to determine the crossing points, an equation for the carrier wave in Figure 2.9 must be established. The signal can be thought of as being made up of straight lines having alternating positive and negative slopes with shifted intercepts on the time axis in the first half cycle of the control signal. To implement this idea in a computer software package, the carrier waveform can be expressed as: t T V tV s car carrier ? ? ? ? ? ? ? ? = 2 ),1( , (2.25) even,; 2 ),( mmVt T V tmV car s car carrier + ? ? ? ? ? ? ? ?? = (2.26) 36 odd;)1( 2 ),( nVnt T V tnV car s car carrier ?? ? ? ? ? ? ? ? ? = , (2.27) where m and n are index numbers used in a computer program, and T s is the period of the triangular wave. Because the PWM signal has half-wave symmetry, only the crossing points that occur in the first half of the PWM signal period need consideration when using the method of pulse pairs. To find the crossing points, set v control = v carrier and solve the transcendental equation for t using MATLAB. Special cases exist as shown in Figure 2.14. Figure 2.14: Special Case Crossing Points. 2.1.2.2 Simulation Results for the Three-Level PWM Inverter The equations of the control signal, the carrier waveform, and the equations used to implement the method of pulse pairs were coded in MATLAB for the purpose of computing the harmonic components of a PWM signal such as the one shown in Figure 37 2.10. MATLAB code was also written to find the crossing points, time delays, and pulse widths. Four MATLAB simulations were conducted using different values of m a and m f . The following parameter values were used for all simulations: V i =270 V, V car =10 V, and f 1 =60 Hz. The other parameters used for the first simulation were as follows: V con = 8V and f s =600 Hz. The parameters used for the second MATLAB simulation were: V con =14V and f s =960 Hz. The parameters used for the third simulation were: V con =18 V and f s =1.2 kHz. The fourth simulation was conducted using the following parameters: V con =22 V and f s =1.2 kHz. PSPICE was used to verify the results from the MATLAB calculations by constructing a three-level PWM simulation model. A PSPICE ABM block was used to compare the sinusoidal control signal and the multi-level triangular carrier wave. A Fourier analysis was then performed in PSPICE on the PWM output signal of the ABM block. The parameters used in the PSPICE simulations were the same as the ones used in the four MATLAB simulations. Results of the MATLAB and PSPICE simulations are shown in Tables 2.5-2.8. The results shown in these tables include all harmonics up to the 31 st harmonic. The harmonic spectrum of a PWM inverter output voltage waveform with m a =0.9 and m f =16 is shown in Figure 2.15 for the first 61 harmonics. The white bars on the graph in Figure 2.15 are PSPICE results and the gray bars on the graph are results from the derived equations that were coded in MATLAB. 38 TABLE 2.5 MATLAB AND PSPICE RESULTS FOR m a =0.8 and m f =10 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 215.8 215.9948 0.1948 -0.012 0.0012 0.0132 7 37.59 37.6563 0.0663 179.900 179.9988 0.0988 9 84.96 84.9067 0.0533 179.900 180.0002 0.1002 11 84.46 84.382 0.078 -0.148 -0.0001 0.1477 13 32.93 32.9386 0.0086 -0.239 0.0035 0.2422 15 19.32 19.3161 0.0039 179.600 179.9957 0.3957 17 31.02 30.9192 0.1008 179.900 179.9998 0.0998 19 27.46 27.4984 0.0384 -0.304 -0.0085 0.2954 21 33.53 33.6488 0.1188 179.700 180.0046 0.3046 23 14.63 14.5081 0.1219 -0.134 0.017 0.1512 27 18.4 18.4752 0.0752 -0.200 -0.01 0.1899 29 13.18 13.2756 0.0956 179.800 180.0031 0.2031 31 4.428 4.376 0.052 179.500 179.9848 0.4848 TABLE 2.6 MATLAB AND PSPICE RESULTS FOR m a =1.4 and m f =16 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 310 310.1109 0.1109 -0.018 0.0014 0.0193 3 37.36 37.4979 0.1379 0.005 0.0012 0.0040 5 6.103 6.0378 0.0652 179.400 180.0546 0.6546 7 2.777 2.8288 0.0518 179.500 180.1123 0.6123 9 8.475 8.5533 0.0783 -179.700 180.0023 0.2977 11 32.62 32.6266 0.0066 -180.000 179.9948 0.0052 13 45.23 45.1691 0.0609 179.900 179.9989 0.0989 15 21.71 21.6852 0.0248 179.700 180.008 0.3080 17 21.85 21.7923 0.0577 -0.077 -0.0083 0.0690 19 44.53 44.445 0.085 -0.220 0.0015 0.2211 21 27.06 27.0322 0.0278 -0.288 0.0112 0.2996 23 7.486 7.4276 0.0584 179.700 179.9614 0.2614 25 22.21 22.1277 0.0823 179.600 179.9976 0.3976 27 8.243 8.2003 0.0427 179.600 180.0215 0.4215 29 9.591 9.5887 0.0023 -0.527 -0.0266 0.5003 31 6.533 6.5106 0.0224 -0.748 -0.0297 0.7187 39 TABLE 2.7 MATLAB AND PSPICE RESULTS FOR m a =1.8 and m f =20 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 323.9 324.0214 0.1214 -0.001 0.0016 0.0028 3 63.97 64.1346 0.1646 0.008 0.0031 0.0044 5 7.55 7.6989 0.1489 0.131 -0.0262 0.1575 7 5.1 5.0645 0.0355 180.000 180.0812 0.0812 9 4.785 4.8727 0.0877 -179.700 180.0647 0.235 11 8.907 9.0402 0.1332 -179.700 180.0037 0.296 13 21.08 21.1577 0.0777 -180.000 179.9912 0.009 15 32.33 32.3006 0.0294 179.900 179.9946 0.0946 17 30.68 30.5728 0.1072 179.900 180.0012 0.1012 19 12.55 12.4589 0.0911 179.800 180.0195 0.2195 21 12.98 12.9781 0.0019 -0.113 -0.0196 0.0933 23 30.45 30.3457 0.1043 -0.174 -0.0013 0.1731 25 29.61 29.4787 0.1313 -0.177 0.0086 0.1854 27 13.05 12.9782 0.0718 -0.062 0.0317 0.09339 29 6.237 6.1935 0.0435 179.300 179.9469 0.6469 31 15.05 14.9172 0.1328 179.600 179.9953 0.3953 TABLE 2.8 MATLAB AND PSPICE RESULTS FOR m a =2.2 and m f =20 Voltage Voltage Voltage Voltage Harmonic Magnitude Magnitude ?VAngle Angle ?? Number (PSPICE) (MATLAB) (PSPICE) (MATLAB) 1 334.3 334.3343 0.0343 -0.009 0.0019 0.0114 3 87.19 87.3399 0.1499 -0.034 0.0025 0.0362 5 26.6 26.7078 0.1078 -0.081 -0.0104 0.07055 7 3.268 3.2849 0.0169 -179.800 180.1529 -0.0471 9 19.23 19.3552 0.1252 179.900 180.0165 0.1165 11 26.1 26.2223 0.1223 179.900 179.9973 0.0973 13 26.27 26.2904 0.0204 179.800 179.9883 0.1883 15 21.58 21.4839 0.0961 179.800 179.9916 0.1916 17 13.83 13.6935 0.1365 179.800 180.0146 0.2146 19 4.882 4.8219 0.0601 179.900 180.1015 0.2015 21 3.428 3.3572 0.0708 -0.465 -0.1208 0.3441 23 9.581 9.4353 0.1457 -0.284 0.0037 0.2876 25 12.7 12.5928 0.1072 -0.247 0.041 0.2877 27 12.68 12.7009 0.0209 -0.251 0.053 0.3042 29 10.12 10.2469 0.1269 -0.302 0.0378 0.3398 31 5.985 6.1173 0.1323 -0.427 -0.0196 0.407 40 Three-Level PWM Output Voltage Harmonic Spectrum 0 50 100 150 200 250 1 3 5 7 9 1113151719212325272931333537394143454749515355575961 Harmonic Number M agn it u d e (PSPICE) (Matlab) Figure 2.15. Harmonic Spectrum with m a =0.9 and m f =16. 2.1.2.3 Comparison of New and Old Methods A paper written in 1981 by Mazzucchelli, et al. [23] claims to have a Fourier series representation for the output voltage waveform of a three-level PWM inverter based on an extension of Bennett?s method [19] that is valid for amplitude modulation ratios greater than one. The equations used to calculate the harmonic components of the output voltage waveform of a three-level PWM inverter from [23] were coded in MATLAB. A MATLAB simulation was conducted using the three-level PWM inverter equations from [23] with V i =270V, m a =1.4, and m f =18. A three-level PWM simulation model was constructed in PSPICE for comparison purposes. A PSPICE ABM block was used to compare the sinusoidal control signal and the multi-level triangular carrier waveform. A Fourier analysis was then performed in PSPICE on the PWM output signal of the ABM block. The parameters used in the PSPICE simulation were the same as the ones used in the MATLAB simulation. 41 The results from the MATLAB coded equations of the Bessel function method presented in [23] were compared with the PSPICE simulation. The results from the comparison are shown in Table 2.9 for a few harmonics. Table 2.9 shows that the method presented in [23] is not very accurate when used to calculate the 3 rd , 11 th , and 39 th harmonic components. A MATLAB simulation using the method of pulse pairs was conducted using the same parameter values that were used in the previous two simulations. Table 2.10 shows the results from the method of pulse pairs compared with the PSPICE simulation. This table shows that the method of pulse pairs is a more accurate method than the one presented in [23]. It should be noted that the PSPICE values in Tables 2.9 and 2.10 were assumed to be the base (or benchmark) values and the percent error was calculated as: %100% x valuePSPICE valueMATLABvaluePSPICE error ? = . (2.28) Unless otherwise noted, all percent error calculations shown in the tables in this dissertation will be calculated as in (2.28). TABLE 2.9 BESSEL FUNCTION METHOD AND PSPICE RESULTS FOR m a =1.4 and m f =18 Voltage Voltage Voltage Harmonic (Bessel Function Method) (PSPICE) ?V% Ero Number (V) (V) (V) (% of PSPICE values) 1 311.7518 311.6 0.1518 0.05 3 34.3703 38.3 3.9297 11.43 11 11.4749 9.509 1.9659 17.13 13 34.8965 35.6 0.7035 2.02 17 20.0398 20.19 0.1502 0.75 21 43.3644 44.36 0.9956 2.30 39 7.7607 13.35 5.5893 72.02 42 TABLE 2.10 METHOD OF PULSE PAIRS AND PSPICE RESULTS FOR m a =1.4 and m f =18 Voltage Voltage Voltage Harmonic (Method of Pulse Pairs) (PSPICE) ?V% Er Number (V) (V) (V) (% of PSPICE values) 1 311.7425 311.6 0.1425 0.05 3 38.5205 38.3 0.2205 0.57 11 9.6195 9.509 0.1105 1.15 13 35.6043 35.6 0.0043 0.01 17 20.1429 20.19 0.0471 0.23 21 44.2791 44.36 0.0809 0.18 39 13.5071 13.35 0.1571 1.16 2.2 The Space Vector PWM Inverter The analytical methods for determining the harmonic components of the output waveforms of a space vector PWM inverter presented in [16, 29, 36, 37] resulted in the appearance of Bessel functions in the final expression of the output PWM signal. Methods such as those presented in [16, 36] use the double Fourier series in the analysis. The purpose of this section is to present a method used to calculate the harmonic components of the output voltage waveforms of a space vector PWM inverter that is general and capable of being applied to various types of multilevel inverters and PWM schemes. This method allows direct calculation of harmonic magnitudes and angles without using the double Fourier series in the analysis. The final expression of the output voltage is compact, and does not contain Bessel functions. The method presented in this section also has the potential to be extended to inverter-drive systems such as the one presented in [44]. 43 2.2.1 Carrier-Based Approach Space vector modulation involves the vector decomposition of a desired voltage space vector into voltage vector components that can be generated using a typical six-switch, three-phase, voltage source inverter. The instantaneous output voltages are determined by the state of the inverter switches. Eight states are possible that correspond to the six possible instantaneous voltage vectors [29]. However, implementing this ?classical? space vector PWM approach can be a complex task to perform. The implementation requires the use of Park?s transformation, sector calculations, hexagon of states, and vector decomposition. A newer ?carrier-based? approach can be used to implement the space vector PWM as shown by different researchers in the literature [45, 46]. The carrier-based method is less complex, more intuitive, and easier to implement than the classical method and will be used to generate the space vector PWM output voltages. Space vector pulse width modulation can be realized by comparing a control signal with a triangular carrier signal as shown in Figure 2.16. The control signal shown in Figure 2.16 is the same control signal used in Mohan?s carrier-based approach [38, 39]. The control signal shown in Figure 2.16 can be expressed as [29]: 44 Figure 2.16: Triangular Waveform and Space Vector Control Signal. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ??? ? ? ? ? ? ? ??? ? ? ? ? ? + ?? ?? ??? ? ? ? ? ? ? ??? ? ? ? ? ? + ?? = ?? ? ? ? ? ?? ? ? ? ?? ? ? ??? ?? ? ? ? ? ?? ? ? ? ?? ? ? ?? 2 6 11 ,sin 2 3 6 11 2 3 , 6 sin 2 3 2 3 6 7 , 6 sin 2 3 6 7 ,sin 2 3 6 5 ,sin 2 3 6 5 2 , 6 sin 2 3 26 , 6 sin 2 3 6 0,sin 2 3 )( ttM ttM ttM ttM ttM ttM ttM ttM tv control (2.29) 45 where M is the modulation index, and ? is the angular frequency. The range of values of (2.29) is limited to M ?1.15. Once the level of M =1.15 is reached, different regions of overmodulation are defined as described in [16]. Each region of overmodulation requires a different space vector modulation strategy. Extension of space vector modulation into the overmodulation region above M =1.15 requires extensive computations and the use of look-up tables as noted in [16]. The output PWM signal can be produced by comparing the waveforms shown in Figure 2.16. Referring to Figure 2.2, when v control > v triangle , T A+ and T B- are closed and the value of the output PMW signal is +V i (where V i is the DC input voltage of the inverter). When v control < v triangle , T A- and T B+ are closed and the value of the output PWM signal becomes - V i . The output PWM signal produced from comparing the waveforms in Figure 2.16 is shown in Figure 2.17. Figure 2.17: Space Vector PWM Output Waveform. 46 2.2.2 Method of Multiple Pulses The method of multiple pulses was developed due to the fact that there is a possibility of a loss of half-wave symmetry in the output waveform of the space vector PWM inverter as described in [16, 35]. A function has half-wave symmetry if it satisfies f(t)= - f(t-T/2). The method of pulse pairs would fail if half-wave symmetry is lost, because there would not be corresponding positive and negative pulse pairs in the output waveform. There is no limitation due to a loss of symmetry when the method of multiple pulses is used. This method is a general method that is valid regardless of the scheme utilized to produce a PWM waveform. The method of multiple pulses is less complex and easier to implement than other methods found in the literature. To begin the analysis, it can be observed that the waveform in Figure 2.17 is made up of multiple positive and negative pulses. Harmonic analysis of the PWM waveform shown in Figure 2.17 can be conducted by breaking up the waveform into multiple positive and negative pulses analyzed individually. An arbitrary positive pulse is shown in Figure 2.18 where A in the figure is the amplitude of the pulse, a P is the initial time delay of the positive pulse, b P is the pulse width of the positive pulse, and T is the period of the PWM waveform. An arbitrary negative pulse is shown in Figure 2.19 where a N is the initial time delay of the negative pulse and b N is the pulse width of the negative pulse. The first step in the analysis is to find the trigonometric Fourier series of the waveform shown in Figure 2.18. The trigonometric Fourier series of the function x(t) can be expressed as: ? ? = ? ? ? ? ? ? ++= 1 0 2 sin 2 cos)( n nn t T n bt T n aatx POSPOSPOS ?? (2.30) 47 x(t) t T/2 Ta P b P A -A Figure 2.18: Positive Pulse. y(t) t T/2 Ta N b N A -A Figure 2.19: Negative Pulse. where a 0 POS , a n POS , and b n POS are the Fourier coefficients of the positive pulse. The coefficient a 0 POS can be found from Figure 2.18 as follows: ? = T dttx T a POS 0 0 )( 1 , (2.31) 48 ? + = PP P POS ba a dtA T a )( 1 0 , (2.32) [] P Ab T a POS 1 0 = . (2.33) The coefficient a n POS can be found from Figure 2.18 as follows: dtt T n tx T a T n POS ? = 0 2 cos)( 2 ? , (2.34) . 2 cos)( 2 dtt T n A T a PP P POS ba a n ? + = ? (2.35) Integrating (2.35) and using the identity )( 2 1 sin)( 2 1 cos2sinsin ?????? ?+=? , (2.35) becomes: .sin 2 cos 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += PPPn b T n b T n a T n n A a POS ??? ? (2.36) The coefficient b n POS can be found from Figure 2.18 as follows: , 2 sin)( 2 0 dtt T n tx T b T n POS ? = ? (2.37) . 2 sin 2 dtt T n A T b PP P POS ba a n ? + = ? (2.38) Integrating (2.38), using the identity )( 2 1 sin)( 2 1 sin2coscos ?????? ?+?=? , and the fact that sin(-?)= -sin(? ), (2.38) becomes: .sin 2 sin 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += PPPn b T n b T n a T n n A b POS ??? ? (2.39) 49 Equations (2.33), (2.36), and (2.39) can now be substituted into (2.30) and the trigonometric Fourier series of the waveform x(t) can be established. The trigonometric Fourier series of the waveform y(t) shown in Figure 2.19 is the same as the waveform x(t) in Figure 2.18 except that the magnitudes are the negative of each other. The Fourier coefficients for y(t) are as follows: [] P Ab T a NEG 1 0 ?= , (2.40) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +?= NNNn b T n b T n a T n n A a NEG ??? ? sin 2 cos 2 , (2.41) ,sin 2 sin 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +?= NNNn b T n b T n a T n n A b NEG ??? ? (2.42) where a 0 NEG , a n NEG , and b n NEG are the Fourier coefficients of the negative pulse. The trigonometric Fourier series for y(t) can be expressed in the same form as x(t) in (2.30): . 2 sin 2 cos)( 1 0 ? ? = ? ? ? ? ? ? ++= n nn t T n bt T n aaty NEGNEGNEG ?? (2.43) Because the Fourier series of arbitrary positive and negative pulses has been established, the Fourier series of a given PWM signal produced by space vector modulation can be found by application of the principle of superposition. A PWM waveform like the one in Figure 2.17 is made up of the sum of positive and negative pulses as shown in Figure 2.20 where P1-P6 are positive pulses and N1-N5 are negative pulses. All that is required to find the Fourier series of a signal like the one shown in Figure 2.20 is to find the Fourier coefficients of each individual positive pulse and negative pulse contained in the PWM signal and add them to get the Fourier coefficients 50 t v SV (t) T/2 T Vi -Vi P1 P2 P3 N4 N5N3 N1 N2 P4 P5 P6 Figure 2.20: PWM Output Signal with Positive and Negative Pulses Labeled. of the entire PWM signal. The total Fourier coefficients of the entire PWM signal can be found using (2.33), (2.36), and (2.39-2.42) as follows: () ? = += N j POS j NEG K j aaa 1 000 , (2.44) () () ???? ? == ? == += 1111 n K j n n K j nn P j POS N j NEG aaa , (2.45) () () ???? ? == ? == += 1111 n K j n n K j nn P j POS N j NEG bbb , (2.46) where K N is the number of negative pulses, and K P is the number of positive pulses. The Fourier series of a given PWM signal produced by space vector modulation can be expressed in a single-cosine series as: ? ? ? ? ? ? ++= ? ? = n oddn n nSV t T n CCtv ? ?2 cos)( 1 0 (2.47) 51 where 00 aC = , 22 nnn baC += , and ? ? ? ? ? ? ? ? ?= ? n n n a b 1 tan? . The subscript SV in (2.47) stands for space vector. The final step in implementing this method is to find the crossing points of the waveforms shown in Figure 2.16 that determine the edges of the PWM signal pulses. In order to determine the crossing points, an equation for the triangular wave in Figure 2.16 is needed. An expression used to represent this waveform is given in (2.18). 2.2.3 Simulation Results for the Space Vector PWM Inverter The equations of the control signal (2.29), the carrier waveform (2.18), and the equations used to implement the method of multiple pulses (2.33), (2.36), (2.39), (2.40- 2.42), and (2.44-2.47) were coded in MATLAB for the purpose of computing the harmonic components of a PWM signal such as the one shown in Figure 2.17. MATLAB code was also written to find the crossing points, time delays, and pulse widths. Four MATLAB simulations were conducted using different values of M and m f . The following parameter values were used for all simulations: V i = 270 V, V tri = 10 V, and f 1 = 60 Hz. The other parameters used for the first simulation were as follows: M=0.5 and f s = 540 Hz. The parameters used for the second MATLAB simulation were: M=0.866 and f s = 540 Hz. The parameters used for the third simulation were: M=0.7 and f s = 900 Hz. The fourth simulation was conducted using the following parameters: M=0.65 and f s = 900 Hz. PSPICE was used to verify the results from the MATLAB calculations by constructing a space vector PWM simulation model. A PSPICE ABM block was used to 52 compare the control signal and the triangular carrier wave. A Fourier analysis was then performed in PSPICE on the PWM output signal of the ABM block. The parameters used in the PSPICE simulations were the same as the ones used in the four MATLAB simulations. Results of the MATLAB and PSPICE simulations are shown in Tables 2.11-2.14. The results shown in Tables 2.11-2.14 include all harmonics up to the 31 st harmonic. The harmonic spectrum of a space vector PWM inverter output voltage waveform with M=1.1 and m f =27 is shown in Figure 2.20 for the first 61 harmonics. The light colored bars on the graph in Figure 2.21 are PSPICE results and the darker colored bars on the graph are results from the derived equations that were coded in MATLAB. TABLE 2.11 MATLAB AND PSPICE RESULTS FOR M=0.5 and m f =9 Voltage Voltage Voltage Voltage Magnitude Magnitude Angle Angle Harmonic PSPICE MATLAB ?VPSPICEMATLAB ?? Number (V) (V) (V) (degrees) (degrees) (degrees) 1 135 135.023 0.023 0.06184 0.0636 0.00176 3 28.2 28.156 0.044 2.501 2.4728 0.0282 5 10.26 10.2734 0.0134 90.5 90.7438 0.2438 7 15.01 14.997 0.013 90.94 91.0142 0.0742 9 290.2 290.2518 0.0518 90.59 90.6879 0.0979 11 14.69 14.7426 0.0526 84.83 84.975 0.145 13 12.47 12.4597 0.0103 62.8 62.9017 0.1017 15 24.97 24.9186 0.0514 4.298 4.4682 0.1702 17 101.3 101.2876 0.0124 1.046 1.2386 0.1926 19 101.5 101.4705 0.0295 182.2 182.3483 0.1483 21 24.68 24.6842 0.0042 194.5 194.576 0.076 23 23.31 23.2777 0.0323 256.2 256.5535 0.3535 25 30.9 30.9417 0.0417 267.66 267.9628 0.3028 27 7.847 7.832 0.015 105.3 105.5191 0.2191 29 31.75 31.7658 0.0158 261.47 261.8123 0.3423 31 27.25 27.2818 0.0318 237 237.3195 0.3195 53 TABLE 2.12 MATLAB AND PSPICE RESULTS FOR M=0.866 and m f =9 Voltage Voltage Voltage Voltage Magnitude Magnitude Angle Angle Harmonic PSPICE MATLAB ?VPSPICEMATLAB ?? Number (V) (V) (V) (degrees) (degrees) (degrees) 1 233.9 233.8479 0.0521 0.209 0.2088 0.0002 3 47.01 47.0461 0.0361 4.994 4.9453 0.0487 5 27.8 27.8464 0.0464 90.67 90.811 0.141 7 41.82 41.7877 0.0323 90.64 90.7053 0.0653 9 194.1 194.1243 0.0243 90.18 90.2867 0.1067 11 42.54 42.5242 0.0158 81.34 81.446 0.106 13 36.57 36.6615 0.0915 52.58 52.6732 0.0932 15 32.97 32.9496 0.0204 7.404 7.6133 0.2093 17 86.3 86.3222 0.0222 -0.635 -0.4072 0.2278 19 86.32 86.3061 0.0139 187.5 187.7082 0.2082 21 37.8 37.7688 0.0312 213.3 213.6108 0.3108 23 36.79 36.719 0.071 228.9 229.1608 0.2608 25 36.01 35.999 0.011 251.1 251.4222 0.3222 27 74.91 74.9134 0.0034 101.3 101.5967 0.2967 29 37.6 37.6243 0.0243 244.2 244.5293 0.3293 31 30.27 30.3025 0.0325 232.2 232.5469 0.3469 TABLE 2.13 MATLAB AND PSPICE RESULTS FOR M=0.7 and m f =15 Voltage Voltage Voltage Voltage Magnitude Magnitude Angle Angle Harmonic PSPICE MATLAB ?VPSPICEMATLAB ?? Number (V) (V) (V) (degrees) (degrees) (degrees) 1 189 189.001 0.001 -0.02108 -0.0173 0.00378 3 38.87 38.8784 0.0084 -0.5511 -0.4735 0.0776 5 1.459 1.4856 0.0266 268.14 268.6838 0.5438 7 1.856 1.8592 0.0032 90.16 91.2928 1.1328 9 4.675 4.6945 0.0195 141.7 141.6784 0.0216 11 19.65 19.6529 0.0029 89.55 89.7153 0.1653 13 28.19 28.1499 0.0401 89.47 89.7296 0.2596 15 242.3 242.2973 0.0027 89.57 89.7477 0.1777 17 28.07 28.1285 0.0585 90.49 90.6927 0.2027 19 19.88 19.8898 0.0098 91.16 91.5545 0.3945 21 3.032 3.0604 0.0284 85.27 85.1494 0.1206 23 4.105 4.0833 0.0217 31.64 31.7633 0.1233 25 13.85 13.8198 0.0302 -9.223 -8.7861 0.4369 27 29.9 29.8562 0.0438 -1.691 -1.3571 0.3339 29 103.7 103.8003 0.1003 -0.7829 -0.4568 0.3261 31 103.8 103.768 0.032 178.8 179.1586 0.3586 54 TABLE 2.14 MATLAB AND PSPICE RESULTS FOR M=0.65 and m f =15 Voltage Voltage Voltage Voltage Magnitude Magnitude Angle Angle Harmonic PSPICE MATLAB ?V PSPICE MATLAB ?? Number (V) (V) (V) (degrees) (degrees) (degrees) 1 175.5 175.501 0.001 -0.008324 -0.0141 0.005776 3 36.16 36.1635 0.0035 -0.5121 -0.4455 0.0666 5 1.29 1.321 0.031 268.19 268.697 0.507 7 1.576 1.6077 0.0317 91.21 91.3396 0.1296 9 4.255 4.2918 0.0368 144.9 144.7438 0.1562 11 17.18 17.1535 0.0265 89.68 89.7168 0.0368 13 24.52 24.536 0.016 89.49 89.7247 0.2347 15 255.4 255.4661 0.0661 89.55 89.7422 0.1922 17 24.54 24.4988 0.0412 90.56 90.6821 0.1221 19 17.36 17.3766 0.0166 91.36 91.5882 0.2282 21 2.663 2.6327 0.0303 96.75 96.0965 0.6535 23 3.399 3.3937 0.0053 34.62 34.5675 0.0525 25 11.56 11.5085 0.0515 -9.882 -9.618 0.264 27 28.77 28.8084 0.0384 -1.752 -1.3321 0.4199 29 105.7 105.7049 0.0049 -0.8627 -0.4768 0.3859 31 105.7 105.6959 0.0041 178.8 179.2077 0.4077 Space Vector PWM Output Voltage Harmonic Spectrum 0 50 100 150 200 250 300 350 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 Harmonic Number M agn i t u d e PSPICE MATLAB Figure 2.21: Harmonic Spectrum with M=1.1 and m f =27. 55 The harmonic components found using the equations coded in MATLAB are similar to the ones found using the PSPICE model as illustrated by the results in the tables and Figure 2.21. The method of multiple pulses is an accurate method used to find the harmonic components of a space vector PWM inverter output waveform as illustrated by the results. 2.3 Line-to-Neutral Voltage Fourier Series Development The focus of the previous sections of this chapter has been on determining the harmonic content of the output voltages of the sinusoidal PWM inverter and the space vector PWM inverter. The methods developed were shown to be effective methods for determining the harmonic content of the inverter output waveforms. However, the inverter output waveforms are typical waveforms produced from single-phase inverters. The focus of Chapter 3 and Chapter 4 will be on analyzing a three-phase, voltage source inverter supplying an induction motor. A general diagram of the system is shown in Figure 2.22. The purpose of this section is to develop a general Fourier series expression of the phase a line-neutral voltage produced from the three-phase inverter system shown in Figure 2.22 that can be used in the harmonic analysis of an induction motor supplied by a three-phase inverter. DC Voltage Source i I i V + - N a b c Voltage Source Inverter a i b i c i s Induction Motor Figure 2.22: Three-Phase Inverter Block Model. 56 2.3.1 The Six-Step Inverter In this section, the Fourier series of the six-step voltage source inverter line-to-neutral voltage for both 180? and 120? conduction will be presented. The Fourier series of the line-to-neutral voltage of the six-step voltage source inverter can be easily found in the literature [8, 47]. However, the Fourier series will be presented in this section due to the fact that the Fourier series of the six-step inverter will be used in analyses presented in Chapter 3 and 4. It should be noted that the method of pulse pairs or the method of multiple pulses can be used to produce the Fourier series of the line-to-neutral voltage of the six-step voltage source inverter. 2.3.1.1 120? Conduction A plot of the phase a line-to-neutral voltage of the six-step inverter with 120? conduction is shown in Figure 2.23. The Fourier series of the six-step inverter phase a line-to-neutral voltage waveform with 120? conduction can be expressed as [8, 47]: ? ? ? +?++?+? ? ? ? ?+??+= ...)3011cos( 11 1 )307cos( 7 1 )305cos( 5 1 )30cos( 3 )( tt ttVtv i ?? ?? ? ? (2.48) As can be seen from (2.48), harmonics exist at 16 ?= kh for ...,3,2,1=k . The other phase voltages can be found by substituting ?? 3 2 ?t and ?? 3 2 +t into (2.48) in place of ?. 57 t? )(tv ? 2 i V ?300 ?360?120 ?180 ?720 Figure 2.23: Six-Step Phase a Voltage Waveform with 120? Conduction. 2.3.1.2 180? Conduction A plot of the phase a voltage of the six-step inverter with 180? conduction is shown in Figure 2.24. The Fourier series of the basic six-step inverter representing the phase a voltage during normal, balanced operation with 180? conduction can be expressed as [8, 47]: ? ? ? ? ? ? +??+= ...11cos 11 1 7cos 7 1 5cos 5 1 cos 2 )( ttttVtv i ???? ? ? . (2.49) It is easy to recognize from (2.49) that harmonics exist at 16 ?= kh for ...,3,2,1=k The other phase voltages can be found by substituting ?? 3 2 ?t and ?? 3 2 +t into (2.49) in place of ?. )(tv ? t? i V 3 1 i V 3 2 ?360?180 ?720 Figure 2.24: Six-Step Phase a Voltage Waveform with 180? Conduction. 58 2.3.2 Two-Level Sinusoidal PWM Inverter When the motor in Figure 2.22 is supplied from a three-phase two-level sinusoidal PWM inverter, the line-to-negative DC bus voltage waveforms (the negative DC bus is denoted with an N in Figure 2.22) produced under balanced operating conditions for m a =1.4 and m f =15 can be produced by comparing the three sinusoidal control signals shifted 120? from each other with a triangular carrier waveform as illustrated in Figure 2.25. The resulting line-to-negative DC bus voltage waveforms are shown in Figure 2.26. A harmonic analysis can be conducted on these waveforms by using the method of multiple pulses that was discussed in Section 2.2.2. As can be observed from Figure 2.26, the waveforms shown can be broken up into multiple positive pulses as shown in Figure 2.27 and analyzed individually as in Section 2.2.2. The equation of the triangular carrier waveform used to find the crossing points is the same as in (2.18). It should be noted that the waveform in Figure 2.26 will contain a DC component. The Fourier series of the phase a line-to-negative DC bus voltage produced by two-level sinusoidal modulation can be expressed as: ? ? ? ? ? ? ++= ? ? = n n naN t T n CCtv ? ?2 cos)( 1 0 (2.50) where 00 aC = , 22 nnn baC += , and ? ? ? ? ? ? ? ? ?= ? n n n a b 1 tan? . The trigonometric Fourier series of the line-to-negative DC bus voltage has now been established in (2.50). However, a trigonometric Fourier series representation of the phase 59 Figure 2.25: Three-Phase Sinusoidal PWM Control Signals and Carrier Waveform. 60 Figure 2.26: Line-to-Negative DC Bus Voltage Waveforms. 61 v aN (t) T/2 T Vi 0 P1 P2 P3 P4 P5 P6 -10 Figure 2.27: Waveform v aN (t) with Pulses Labeled. a line-to-neutral voltage of the system shown in Figure 2.22 is needed. The phase a line- to-neutral voltage waveform for this system while supplied by a two-level sinusoidal PWM inverter is shown in Figure 2.28. Before beginning to develop a Fourier series representation of the line-to-neutral voltage, it is appropriate to first look at the harmonic spectrums of the phase a line-to-negative DC bus voltage and the phase a line-to-neutral voltage by creating the waveforms in MATLAB and using the FFT command to produce the harmonic spectrums. The harmonic spectrum of the phase a line-to-negative DC bus voltage is shown in Figure 2.29 and the harmonic spectrum of the phase a line-to-neutral voltage is shown in Figure 2.30. It can be observed from Figure 2.29 and Figure 2.30 that the magnitudes of the harmonics and the harmonic content of each voltage waveform is the same except that the line-to-neutral voltage does not contain a DC component nor any zero-sequence harmonics (triplen harmonics). For maximum cancellation of dominant harmonics in the line voltages of a three-phase inverter, m f should always be odd and a multiple of three [27]. The results from the comparison of the harmonic 62 Figure 2.28: Phase a Line-to-Neutral Voltage Produced using MATLAB. Figure 2.29: Harmonic Spectrum of the Phase a Line-to-Negative DC Bus Voltage. 63 Figure 2.30: Harmonic Spectrum of the Phase a Line-to-Neutral Voltage. spectrums can be proven mathematically by first considering some basic three-phase relationships for the system shown in Figure 2.22. The inverter line-to-neutral voltages can be expressed as [27]: )()()( tvtvtv sNaNas ?= , (2.51) )()()( tvtvtv sNbNbs ?= , (2.52) )()()( tvtvtv sNcNcs ?= . (2.53) The following condition for the inverter voltages must hold under balanced conditions [27]: 0)()()( =++ tvtvtv csbsas . (2.46) The following relationship can be obtained by substituting (2.51-2.53) into (2.54): 64 [])()()( 3 1 )( tvtvtvtv cNbNaNsN ++= . (2.55) By substituting (2.54) into (2.50), the phase a line-to-neutral voltage can be expressed as: )( 3 1 )( 3 1 )( 3 2 )( tvtvtvtv cNbNaNas ??= . (2.56) Using (2.56) it is easy to prove that no DC component exists in the phase a line-to-neutral voltage by considering the DC component of each line-to-negative DC bus voltage: 0 0 AV aN = , 0 0 AV bN = , and 0 0 AV cN = . These components can be substituted into (2.56) as follows: 0000 3 1 3 1 3 2 cNbNaNas VVVV ??= , (2.57) 0 3 1 3 1 3 2 000 0 =??= AAAV as . (2.58) As can be seen from (2.58), no DC component exists in the line-to-neutral voltage. It can also be shown that the magnitudes of the harmonic components in the harmonic spectrum of the line-to-neutral voltages are the same as the magnitudes of the line-to-negative DC bus voltages. This can be accomplished by considering a balanced set of fundamental line-to-negative DC bus voltages: tAtv aN ?cos)( 1 1 = , (2.59) ( )??= 120cos)( 1 1 tAtv bN ? , (2.60) ()?+= 120cos)( 1 1 tAtv cN ? . (2.61) These voltages can be substituted into (2.56) as follows: ()()?+????= 120cos 3 1 120cos 3 1 cos 3 2 )( 111 1 tAtAtAtv as ??? . (2.62) 65 Using the trigonometric identity ( ) ?????? sinsincoscoscos m=? , (2.62) can be written as: tAtAtAtv as ??? cos 6 1 cos 6 1 cos 3 2 )( 111 1 ++= , (2.63) tAtv as ?cos)( 1 1 = . (2.64) The result in (2.64) matches (2.59), verifying that the harmonic spectrums of the phase a line-to-neutral and the line-to-negative DC bus voltages are the same excluding the triplen harmonics and the DC component. Perhaps the most important result is to show that the triplen harmonics are not present in the line-to-neutral voltages. To prove this, consider the following balanced set of 3 rd harmonic voltages: tAtv aN ?3cos)( 3 3 = , (2.65) ()??= 3603cos)( 3 3 tAtv bN ? , (2.66) ()?+= 3603cos)( 3 3 tAtv cN ? . (2.67) The voltages in (2.65-2.67) can be substituted into (2.56) as follows: ()()?+????= 3603cos 3 1 3603cos 3 1 3cos 3 2 )( 333 3 tAtAtAtv as ??? . (2.68) The trigonometric identity ( ) ?????? sinsincoscoscos m=? can be used to express (2.68) as: tAtAtAtv as ??? 3cos 3 1 3cos 3 1 3cos 3 2 )( 333 3 ??= , (2.69) tAtAtv as ?? 3cos 3 2 3cos 3 2 )( 33 3 ?= , (2.70) 0)( 3 =tv as . (2.71) 66 The result in (2.71) proves that no zero-sequence (or triplen) harmonics exist in the line- to-neutral voltages. No zero-sequence current can flow in an ungrounded wye circuit under balanced or unbalanced conditions. At this point, the expression in (2.50) can be modified to produce a Fourier series representation for the phase a line-to-neutral voltage waveform shown in Figure 2.28 as: ? ? ? ? ? ? += ? ? = ? = n k kn n nas t T n Ctv ? ?2 cos)( ,...3,2,1 3 1 (2.72) where 22 nnn baC += , and ? ? ? ? ? ? ? ? ?= ? n n n a b 1 tan? . The equation used to calculate the Fourier series of the phase a line-to-negative DC bus voltage produced by two-level sinusoidal modulation (2.50) and the equation used to calculate the Fourier series of the phase a line-to-neutral voltage produced by two-level sinusoidal modulation (2.72) were both coded in MATLAB for the purpose of computing the harmonic content of each waveform for a given set of parameter values. The equations of the control signal, the carrier waveform, and the equations used to implement the method of multiple pulses were also coded in MATLAB. The following parameter values were used for the simulation: V i = 270 V, V tri = 10 V, V con = 14 V, f 1 = 60 Hz, and f s = 900 Hz. PSPICE was used to verify the results from the MATLAB calculations by constructing a two-level sinusoidal PWM inverter simulation model using PSPICE ABM blocks. A Fourier analysis was then performed in PSPICE on the phase a line-to-negative DC bus voltage and the phase a line-to-neutral voltage. The parameters used in the PSPICE simulations were the same as the ones used in the MATLAB simulation. 67 Results of the MATLAB and PSPICE simulations are shown in Tables 2.15 and 2.16. The harmonic components found using the equations coded in MATLAB are similar to the ones found using the PSPICE model as illustrated by the results in Tables 2.15 and 2.16. Based on these results, (2.50) and (2.72) are correct and the method of multiple pulses is an accurate method used to find the harmonic components of the voltage waveforms produced by a two-level sinusoidal PWM inverter. TABLE 2.15 LINE-TO-NEGATIVE DC BUS VOLTAGE COMPONENTS FOR m a =1.4 and m f =15 V aN V aN V aN V aN Magnitude Magnitude ?V Angle Angle ?? Harmonic PSPICE MATLAB PSPICE MATLAB Number (Volts) (Volts) (Volts) (degrees) (degrees) (degrees) DC 134.8817 134.9002 0.0185 1 155.9 155.9006 0.0006 -0.239 -0.2342 0.0047 3 19.6 19.6244 0.0244 3.519 3.5087 0.0103 5 4.3825 4.36375 0.01875 176.100 176.1209 0.0209 7 3.7795 3.77035 0.00915 -132.400 -132.5015 0.1015 9 2.005 2.0074 0.0024 33.060 33.788 0.728 11 18.63 18.6404 0.0104 87.700 87.8266 0.1266 13 41.815 41.8013 0.0137 91.390 91.5182 0.1282 15 52.7 52.66405 0.03595 89.820 89.9667 0.1467 17 41.825 41.8104 0.0146 88.310 88.4674 0.1574 19 18.645 18.64535 0.00035 92.430 92.5775 0.1475 21 1.8655 1.85815 0.00735 143.900 143.55 0.35 23 6.245 6.255 0.01 -26.680 -26.3705 0.3095 25 17.55 17.5476 0.0024 0.798 1.0319 0.2341 27 21.77 21.75875 0.01125 2.872 3.131 0.259 29 10.035 10.02525 0.00975 -4.290 -3.9921 0.2979 31 10.01 10.00735 0.00265 -176.800 -176.5163 0.2837 68 TABLE 2.16 LINE-TO-NEUTRAL VOLTAGE COMPONENTS FOR m a =1.4 and m f =15 V as V as V as V as Magnitude Magnitude ?VAngleAngle ?? Harmonic PSPICE MATLAB PSPICE MATLAB Number (V) (V) (V) (degrees) (degrees) (degrees) DC 1 155.9 155.9006 0.0006 -0.239 -0.2342 0.0047 3 5 4.3825 4.36375 0.01875 176.100 176.1209 0.0209 7 3.7795 3.77035 0.00915 -132.400 -132.5015 0.1015 9 11 18.63 18.6404 0.0104 87.700 87.8266 0.1266 13 41.815 41.8013 0.0137 91.390 91.5182 0.1282 15 17 41.825 41.8104 0.0146 88.310 88.4674 0.1574 19 18.645 18.64535 0.00035 92.430 92.5775 0.1475 21 23 6.245 6.255 0.01 -26.680 -26.3705 0.3095 25 17.55 17.5476 0.0024 0.798 1.0319 0.2341 27 29 10.035 10.02525 0.00975 -4.290 -3.9921 0.2979 31 10.01 10.00735 0.00265 -176.800 -176.5163 0.2837 2.3.3 The Space Vector PWM Inverter When the motor in Figure 2.22 is supplied from a three-phase space vector PWM inverter, the line-to-negative DC bus voltage waveforms produced under balanced operating conditions can be produced by comparing the three space vector control signals shifted 120? from each other with a triangular carrier waveform as illustrated in Figure 2.31. The resulting line-to-negative DC bus voltage waveforms are shown in Figure 2.32. The phase a line-to-neutral voltage waveform produced by the system in Figure 2.22 while supplied by a space vector PWM inverter is shown in Figure 2.33. It can be observed by comparing Figure 2.26 and Figure 2.28 with Figure 2.32 and Figure 2.33 that the waveforms are similar and the method of multiple pulses presented in Section 2.3.1 69 Figure 2.31: Space Vector PWM Control Signals and Carrier Waveform. Figure 2.32: Line-to-Negative DC Bus Voltage Waveforms. 70 Figure 2.33: Phase a Line-to-Neutral Voltage Waveform. can be used. The equation of the triangular waveform used to find the crossing points is the same as the one in (2.18) and the equation of the space vector control signal is given in (2.29). Using the method of multiple pulses from Section 2.3.1, the Fourier series of the phase a line-to-negative DC bus voltage produced by space vector modulation can be expressed as: ? ? ? ? ? ? ++= ? ? = n n naN t T n CCtv ? ?2 cos)( 1 0 (2.73) where 00 aC = , 22 nnn baC += , and ? ? ? ? ? ? ? ? ?= ? n n n a b 1 tan? . 71 Because mf should be selected to be odd and a multiple of three, in order to cancel dominant harmonics [27], no triplen harmonics will appear in the line voltages. Therefore, the Fourier series of the phase a line-to-negative DC bus voltage of the space vector PWM inverter can be expressed as: ? ? ? ? ? ? += ? ? = ? = n k kn n nas t T n Ctv ? ?2 cos)( ,...3,2,1 3 1 (2.74) where 22 nnn baC += , and ? ? ? ? ? ? ? ? ?= ? n n n a b 1 tan? . The Fourier series in (2.73) and (2.74) were both coded in MATLAB for the purpose of computing the harmonic content of the phase a line-to-negative DC bus voltage and the phase a line-to-neutral voltage produced by space vector modulation for a given set of parameter values. The equations of the space vector control signal, the carrier waveform, and the equations used to implement the method of multiple pulses were also coded in MATLAB. The following parameter values were used for the simulation: V i = 270 V, M=0.7, f 1 = 60 Hz, and f s = 900 Hz. PSPICE was used to verify the results from the MATLAB calculations by constructing a space vector PWM inverter simulation model using PSPICE ABM blocks. A Fourier analysis was then performed in PSPICE on the phase a line-to-negative DC bus voltage and the phase a line-to-neutral voltage. The parameters used in the PSPICE simulations were the same as the ones used in the MATLAB simulation. Results of the MATLAB and PSPICE simulations are shown in Tables 2.17 and 2.18. The harmonic components found using the equations coded in MATLAB are similar to the ones found using the PSPICE model as illustrated by the results in Tables 2.17 and 72 2.18. These results show that (2.73) and (2.74) are valid and that the method of multiple pulses is an accurate method used to find the harmonic components of the voltage waveforms produced by a space vector PWM inverter. TABLE 2.17 LINE-TO-NEGATIVE DC BUS VOLTAGE COMPONENTS FOR M=0.7 and m f =15 V aN V aN V aN V aN Magnitude Magnitude ?VAngleAngle ?? Harmonic PSPICE Matlab (PSPICE) (Matlab) Number (V) (V) (V) (degrees) (degrees) (degrees) DC 134.7298 134.7314 0.0016 1 94.49 94.5005 0.0008 -0.021 -0.0173 0.00378 3 19.44 19.4392 0.0132 -0.551 -0.4735 0.0776 5 0.7296 0.7428 0.0018 -91.860 -91.3162 0.5438 7 0.9278 0.9296 0.00925 90.160 91.2928 1.1328 9 2.338 2.34725 0.00055 141.700 141.6784 0.0216 11 9.827 9.82645 0.02505 89.550 89.7153 0.1653 13 14.1 14.07495 0.05135 89.470 89.7296 0.2596 15 121.2 121.14865 0.02425 89.570 89.7477 0.1777 17 14.04 14.06425 0.0039 90.490 90.6927 0.2027 19 9.941 9.9449 0.0142 91.160 91.5545 0.3945 21 1.516 1.5302 0.01035 85.270 85.1494 0.1206 23 2.052 2.04165 0.0131 31.640 31.7633 0.1233 25 6.923 6.9099 0.0219 -9.223 -8.7861 0.4369 27 14.95 14.9281 0.03015 -1.691 -1.3571 0.3339 29 51.87 51.90015 0.026 -0.783 -0.4568 0.3261 31 51.91 51.884 0.026 178.800 179.1586 0.3586 73 TABLE 2.18 LINE-TO-NEUTRAL VOLTAGE COMPONENTS FOR M=0.7 and m f =15 V as V as V as V as Magnitude Magnitude ?VAngleAngle ?? Harmonic PSPICE Matlab (PSPICE) (Matlab) Number (V) (V) (V) (degrees) (degrees) (degrees) DC 1 94.49 94.5005 0.0105 -0.021 -0.0173 0.00378 3 5 0.7296 0.7428 0.0132 -91.860 -91.3162 0.5438 7 0.9278 0.9296 0.0018 90.160 91.2928 1.1328 9 11 9.827 9.82645 0.00055 89.550 89.7153 0.1653 13 14.1 14.07495 0.02505 89.470 89.7296 0.2596 15 17 14.04 14.06425 0.02425 90.490 90.6927 0.2027 19 9.941 9.9449 0.0039 91.160 91.5545 0.3945 21 23 2.052 2.04165 0.01035 31.640 31.7633 0.1233 25 6.923 6.9099 0.0131 -9.223 -8.7861 0.4369 27 29 51.87 51.90015 0.03015 -0.783 -0.4568 0.3261 31 51.91 51.884 0.026 178.800 179.1586 0.3586 2.4 Summary Two methods for finding the harmonic components of the output voltage of sinusoidal PWM inverters and space vector PWM inverters were presented in this chapter. The method of pulse pairs was the first method discussed. This method was shown to be applicable to different multilevel inverter types such as the two-level sinusoidal PWM inverter and the three-level sinusoidal PWM inverter. The method allowed direct calculation of harmonic magnitudes and angles without having to use linear approximations, iterative procedures, look-up tables, or Bessel functions. The main limitation of the method of pulse pairs is the possibility of a loss of symmetry in the 74 output voltage waveform of the inverter. To rectify this problem, the method of multiple pulses was developed. This method is entirely general and has the potential to be used to analyze the harmonic content of inverter output waveforms produced by various types of multilevel inverters and PWM schemes. There is no limitation of the method of multiple pulses due to loss of symmetry or the harmonic content of the inverter output voltage waveform. The line-to-neutral voltage Fourier series of the six-step, two-level sinusoidal PWM, and space vector PWM inverters were presented. The method of multiple pulses can be used to determine the harmonic content of the line-to-neutral voltages of all of the voltage source inverter types studied, including the space vector PWM inverter. This method will be utilized during MATLAB simulations conducted in Chapters 3 and 4. 75 CHAPTER 3 THE INVERTER-FED INDUCTION MOTOR The focus of this chapter is on the inverter-fed induction motor. A steady-state harmonic model of the induction motor operating under balanced conditions is presented. The harmonic model is based on the T-type equivalent circuit of the induction motor, and is capable of being used to analyze induction motors supplied from nonsinusoidal sources. A simplified model of an inverter-fed induction motor that is based on the steady-state T-type equivalent circuit of the motor and the input-output relationships of the voltage source inverter is presented. A V-I load characteristic curve that allows the inverter, motor, and load to be replaced by a current-controlled voltage source is established. MATLAB and PSPICE simulation results are presented in order to validate the use of the simplified model. 3.1 Induction Motor Equivalent Circuit All analysis and simulation in this dissertation are based on the steady-state T-type equivalent circuit model of the induction motor [40], shown in Figure 3.1a (note: all quantities have been reflected to the stator). This model is the positive-sequence equivalent circuit of the induction motor where balanced three-phase operation is assumed. 76 - (b) R 1 j X 1 jX 2 1 2 s R 1 V + a b 1 I 2 I (a) R e1 jX e1 a V 1 + jX 2 1 2 s R a b 2 I m jX - Figure 3.1: (a) Induction Motor T-Type Equivalent Circuit; (b) Thevenin Equivalent of (a). Thevenin?s theorem can be used to transform the network to the left of points a and b in Fig. 3.1a into an equivalent voltage source a V 1 in series with an equivalent impedance R e1 +jX e1 as shown in Figure 4.1b. The equivalent source voltage can be expressed as [40]: )( 11 11 m m a XXjR jX VV ++ = (3.1) where 1 V is the stator positive-sequence line-to-neutral voltage, X m is the magnetizing reactance, R 1 is the stator resistance, and X 1 is the stator leakage reactance. The Thevenin-equivalent stator impedance is: )( )( 11 11 111 XXjR jXRjX jXRZ m m eee ++ + =+= . (3.2) 77 From the Thevenin-equivalent circuit of Figure 3.1 (b), the magnitude of the rotor current referred to the stator is: 2 21 2 1 2 1 1 2 )( XX s R R V I ee a ++ ? ? ? ? ? ? ? ? + = (3.3) where R 2 is the rotor resistance, X 2 is the rotor leakage reactance, and s 1 is the fundamental slip. The internal mechanical power developed by the motor can be expressed as [40]: 1 1 2 2 2 1 s s RmIP d ? = (3.4) where m is the number of stator phases. The internal power (3.4) can also be written as: sed sTP ?)1( 1 ?= (3.5) where T e is the internal electromagnetic torque (N-m), and ? s is the synchronous angular velocity (rad/s). The synchronous angular velocity is given as: P f s ? ? 4 = (3.6) where f is the excitation frequency and P is the number of poles. Substituting (3.5) into (3.4) and solving for T e yields an expression for the electromagnetic torque as follows: 1 22 2 s R I m T s e ? = . (3.7) Substituting (3.3) into (3.7) yields: 2 21 2 1 2 1 1 22 1 )( XX s R R s R V m T ee a s e ++ ? ? ? ? ? ? ? ? + = ? (3.8) 78 Equation (3.8) can be rearranged and solved in terms of the slip as follows: A ACBB s 2 4 2 1 ??? = (3.9) where 2 21 2 1 )( XXTRTA eseese ++= ?? , 2 1212 2 aese VmRRRTB ?= ? , and 2 2 RTC se ?= . The torque and rotor speed are related by [48]: Lrm r e T P B dt d P JT ++= ? ? 22 (3.10) where J is the inertia of the rotor and the connected load, ? r is the angular velocity of the rotor, B m is the damping coefficient associated with the rotational system of the machine and mechanical load, and T L is the load torque. The coefficient B m is typically small and often neglected. Some simplifications of (3.10) can be made when considering the steady-state operation of the induction motor [48]. The speed is constant during steady- state operation and the acceleration is zero. Using these simplifications and the fact that B m can be neglected, (3.10) becomes: Le TT = (3.11) during steady-state operation. Substituting (3.11) into (3.9) produces an equation for the slip in terms of variables that are generally known. The total impedance looking into the circuit of Figure 3.1 (a) is: ??= ++ ? ? ? ? ? ? ? ? + ++= 1 2 1 2 2 1 2 111 )( Z XXj s R jX s R jX jXRZ m m (3.12) The magnitude of the stator current can now be found using the following formula: 79 1 1 1 Z V I = . (3.13) The power factor can be found by taking the cosine of the angle from (3.12). The equations developed in (3.1-3.13) are valid for the steady-state analysis of the induction motor under balanced operating conditions when the motor is supplied from a pure sinusoidal source. These equations can easily be modified to perform a harmonic analysis on an induction motor when supplied from a nonsinusoidal source. It is necessary to account for the k th harmonic number in (3.1-3.13) and define the slip for both positive and negative sequence harmonics. It should be noted that the frequency dependence of the motor resistances will be ignored in all analyses in this dissertation. Ignoring the frequency dependence of the resistances is a typical practice [8, 27, 40, and 43] that produces reasonable results for the practicing electrical engineer. For the interested researcher, a paper that investigates the frequency dependence of the rotor resistance of an inverter-fed induction motor can be found in [49]. The equivalent source voltage for the k th harmonic can be determined by examining Figure 3.2 and using Thevenin?s theorem: )( 11 1 m m ka kXkXjR kXj VV k ++ = (3.14) where k V is the k th harmonic stator line-to-neutral voltage. The k th harmonic Thevenin- equivalent stator impedance is: )( )( 11 11 111 kXkXjR jkXRjkX jXRZ m m eee kkk ++ + =+= (3.15) The magnitude of the rotor current referred to the stator for the k th harmonic is: 80 - - R 1 k s R 2 k V 1 + a b k I 1 k I 2 (a) k a V 1 + k s R 2 a b (b) m jkX 1 jkX 2 jkX k e R 1 k e jX 1 2 jkX k I 2 Figure 3.2: (a) Induction Motor Harmonic Equivalent Circuit; (b) Thevenin Equivalent of (a). 2 21 2 2 1 1 2 )( kXkX s R R V I kk k k e k e a ++ ? ? ? ? ? ? ? ? + = (3.16) where s k is the k th harmonic slip. The internal mechanical power developed by the motor can be expressed as: k k d s s RmIP kk ? = 1 2 2 2 . (3.17) The internal power (3.17) can also be written as: sked sTP kk ?)1( ?= . (3.18) 81 Substituting (3.18) into (3.17) and solving for T ek yields an expression for the k th harmonic electromagnetic torque as follows: ks e s R I m T kk 22 2 ? ?= . (3.19) The positive torque in (3.19) is produced by positive-sequence harmonics and the negative torque in (3.19) is produced by negative-sequence harmonics [8]. Substituting (3.16) into (3.19) yields: 2 2 2 2 1 22 )( kXkX s R R s R V m T ek k e k ka s e k ++ ? ? ? ? ? ? ? ? + ?= ? . (3.20) The total impedance looking into the circuit of Figure 3.2a is: k m k k m kk Z kXkXj s R jkX s R jkX jkXRZ ??= ++ ? ? ? ? ? ? ? ? + ++= 1 2 2 2 2 111 )( . (3.21) The magnitude of the stator current for the k th harmonic can now be found using the following formula: k k Z V I k 1 1 = . (3.22) The positive-sequence harmonic equivalent circuit of the induction motor used for analysis and simulation purposes is shown in Figure 3.3, where k p is the positive- sequence harmonic number and s k P is the slip for the th p k positive-sequence harmonic, which may be calculated using (3.23): 82 p p k k sk s p )1( 1 ?? = . (3.23) The negative-sequence harmonic equivalent circuit is shown in Figure 3.4, where n k is the negative-sequence harmonic number and s k n is the slip for the th n k negative-sequence harmonic, which may be calculated using (3.24): n n k k sk s n )1( 1 ?+ = . (3.24) - p k V 1 + pk I 1 p k s R 2 1 Xkj p 1 R 2 Xkj p p k I 2 mp Xkj Figure 3.3. Positive-Sequence Harmonic Equivalent Circuit. - nk V 1 + n k I 1 n k s R 2 1 Xkj n 1 R 2 Xkj n n k I 2 mn Xkj Figure 3.4. Negative-Sequence Harmonic Equivalent Circuit. 83 3.2 Verification of Induction Motor Harmonic Model The equations of the induction motor based on the circuits shown in Figure 3.1 and Figure 3.2 were coded in MATLAB along with the six-step inverter output voltage Fourier series. A harmonic analysis was performed on a 50 HP, 3-phase induction motor with parameters listed in Table 3.1 using MATLAB. Results from a harmonic analysis of the induction motor operating at a speed of 1748.9 rpm while supplied by a six-step voltage source inverter with 180? conduction and a DC input voltage to the inverter of V i =461V are shown in Table 3.2. This table also shows results from an EMAP simulation [50] for the same motor and operating conditions. Table 3.3 compares the results of the two simulations by showing the differences and percent errors between MATLAB analysis and EMAP. The EMAP values in Table 3.2 were assumed to be the base (or benchmark) values and the percent error listed in Table 3.3 was calculated as: %100% x valueEMAP valueMATLABvalueEMAP error ? = . (3.25) From Table 3.2 and Table 3.3, it can be observed that the MATLAB code produces results that are comparable to EMAP. The MATLAB code can be used in the analysis of an induction motor supplied by nonsinusoidal voltages. 3.3 Motor-Drive System Model The proposed motor-drive system to be analyzed is shown in Figure 3.5. This figure shows a DC source connected to an inverter driving a three-phase induction motor with a load attached. In Figure 3.5, V i is the inverter DC input voltage and I i is the inverter DC input current. 84 TABLE 3.1 50 HP, 3-phase, Induction Motor Parameters f = 60 Hz number of poles = 4 R 1 = 0.087? R 2 = 0.228? X 1 = 0.302 ? X 2 = 0.302 ? X m = 13.08 ? J = 1.662 kg-m 2 Machine Ratings: V L-L = 460V Rated Speed = 1710 rpm Rated Torque = 200 N-m Note: All quantities in Table 3.1 have been reflected to the stator. TABLE 3.2 MATLAB AND EMAP SIX-STEP INVERTER RESULTS V as I a V as I a (V) (A) (V) (A) Harmonic Slip (RMS) (RMS) Slip (RMS) (RMS) Number (EMAP) (EMAP) (EMAP) (Matlab Code) (Matlab Code) (Matlab Code) 1 0.0284 207.53 29.75 0.0284 207.52 29.75 5 1.1943 41.52 13.85 1.1943 41.51 13.83 7 0.8612 29.66 7.07 0.8612 29.65 7.07 11 1.0883 18.89 2.87 1.0883 18.87 2.87 13 0.9253 16 2.06 0.9253 15.96 2.06 17 1.0572 12.25 1.21 1.0572 12.21 1.2 19 0.9489 10.97 0.97 0.9489 10.92 0.96 23 1.0422 9.08 0.66 1.0422 9.02 0.66 25 0.9611 8.37 0.56 0.9611 8.3 0.56 29 1.0335 7.23 0.42 1.0335 7.16 0.413 31 0.9687 6.78 0.37 0.9687 6.69 0.362 85 TABLE 3.3 DIFFERENCES AND PERCENT ERRORS Slip V as I a Harmonic ?Vas ?I a % error % error % error Number ?Slip (V) (A) (% of EMAP) (% of EMAP) (% of EMAP) 1 0 0.01 0 0 0.00 0.00 5 0 0.01 0.02 0 0.02 0.14 7 0 0.01 0 0 0.03 0.00 11 0 0.02 0 0 0.11 0.00 13 0 0.04 0 0 0.25 0.00 17 0 0.04 0.01 0 0.33 0.83 19 0 0.05 0.01 0 0.46 1.03 23 0 0.06 0 0 0.66 0.00 25 0 0.07 0 0 0.84 0.00 29 0 0.07 0.007 0 0.97 1.67 31 0 0.09 0.008 0 1.33 2.16 Voltage Source Inverter 3-Phase Induction Motor i I i V + - N a b c Load s V Figure 3.5. Motor-Drive System Model. It is possible to develop a simplified model of the system shown in Figure 3.5 using the induction motor equivalent circuits and a power balance at the input and output terminals of the voltage source inverter. If a value of V i is assumed at the input terminals of the inverter in Figure 3.5, a corresponding voltage value on the output side of the inverter can be found using a power balance as follows: kkk k ii IVIV ?cos 2 3 1 ? ? = = (3.26) 86 where V k is the k th harmonic stator line-to-neutral voltage and I k is the k th harmonic stator current. Power inverters used in practical applications are not 100% efficient and inverter losses would need to be included in a power balance. However, it should be noted that all inverters analyzed in this dissertation are assumed to be ideal inverters that are 100% efficient and (3.26) applies. Assuming a value of V i at the input terminals of the inverter will allow the line-to- neutral voltage at the input terminals of the induction motor to be found regardless of the PWM scheme employed in the inverter. The induction motor can be analyzed from knowledge of the line-to-neutral voltage and the load torque (or the line-to-neutral voltage and the motor speed) using the standard equations of the induction motor (3.1- 3.24). Once the harmonic analysis of the induction motor has been completed for an assumed value of V i , the corresponding value of the DC input current I i can be found from (3.26). For any value of V i in Figure 3.5, a corresponding value of I i can be found from (3.26) using the process described in the previous paragraph. If this process is continually repeated, a V-I load characteristic curve can be generated at the input terminals of the inverter in Figure 3.5. For a six-step inverter, PSPICE and MATLAB simulations have shown that the resulting V-I load characteristic curve has the following form: cbIaIIV iii ++= 2 )( (3.27) where a, b, and c are constants determined using the polyfit command in MATLAB which fits a curve to the generated V-I data. To illustrate why a quadratic was used to curve fit the generated V-I data, the equations of the induction motor and six-step inverter (180? conduction) relationships were coded in MATLAB for the purpose of simulating 87 the system shown in Figure 3.5. The source voltage of Figure 3.5 was varied over a range of 478V-577V with all other parameters remaining unchanged. The motor used in the simulation was a 50 Hp, three-phase induction motor having parameters as listed in Table 3.1. A graph of the generated V-I data is shown in Figure 3.6. The data was initially fit with a linear curve in Excel as shown in Figure 3.7. Excel calculates an R 2 value when a curve fit is performed. The R 2 value is the square of the correlation coefficient. The correlation coefficient provides a measure of the reliability of the curve fit. The closer the R 2 value is to 1, the better the curve fit. The R 2 value for the linear curve fit was R 2 =0.9973. The V-I data was then fit with a quadratic curve as shown in Figure 3.8. The R 2 value for the quadratic curve fit was R 2 =1. The system in Figure 3.5 can now be replaced by a current-controlled voltage source having the characteristics of (3.27). The simplified model of the inverter drive system is shown in Figure 3.9. The current-controlled voltage source shown in this figure represents all system components to the right of V i (inverter, motor, and load) in Figure 3.5. Inverter Voltage vs. Inverter Current 460 480 500 520 540 560 580 600 34 35 36 37 38 39 40 41 42 Ii (A) Vi ( V ) Figure 3.6: V-I Data Points. 88 Inverter Voltage vs. Inverter Current y = -14.15x + 1060.8 R 2 = 0.9973 460 480 500 520 540 560 580 600 34 35 36 37 38 39 40 41 42 Ii (A) Vi ( V ) Figure 3.7: Linear Curve Fit. Inverter Voltage vs. Inverter Current y = 0.4006x 2 - 44.457x + 1632.4 R 2 = 1 460 480 500 520 540 560 580 600 34 35 36 37 38 39 40 41 42 Ii (A) Vi (V ) Figure 3.8: Quadratic Curve Fit. 3.3.1 Simplified Model Simulation Results The purpose of this section is to demonstrate using PSPICE and MATLAB that the system shown in Figure 3.5 can be replaced by a V-I load characteristic curve that allows the inverter, motor, and load to be replaced by a current-controlled voltage source. 89 + - s V i I )( i IV Figure 3.9: Simplified System Model. Simulation results are shown for the six-step inverter (180? conduction), the two-level sinusoidal PWM inverter, and the space vector inverter. 3.3.2 Six-Step Inverter Results The equations of the induction motor and six-step inverter (180? conduction) relationships were coded in MATLAB for the purpose of simulating the system shown in Figure 3.5. The source voltage of Figure 3.5 was varied over a range of 240V-480V with all other parameters remaining unchanged. The parameters of the motor studied were: R 1 =0.25?, R 2 =0.28?, X 1 =0.754?, X 2 =0.85?, X m =18?, J=0.1kg m 2 , P=4, and HP=5. Using MATLAB, the V-I characteristic found for this motor and inverter is: 26.904526.447197.0)( 2 +?= iii IIIV . (3.28) A plot of (3.28) is shown in Figure 3.10. Equation (3.28) represents everything to the right of the inverter input voltage (V i in Figure 3.5). 90 Inverter Voltage vs. Inverter Current (Matlab) y = 0.7197x 2 - 44.526x + 904.26 180 230 280 330 380 430 480 12 14 16 18 20 22 24 26 28 30 32 Ii (Amps) V i (V o l tag e ) Figure 3.10: V-I Load Curve Produced From MATLAB Code. A PSPICE model of the system shown in Figure 3.5 was simulated in order to produce a V-I characteristic curve. The load applied to the motor during simulations was a pulsed torque load with the following characteristics: T L =30N-m, T=6s, and D=2/3. Where T is the pulse period and D is the duty cycle. During PSPICE simulation tests, the source voltage of Figure 3.5 was varied over a range of 240V-480V with all other parameters remaining unchanged. The motor parameters were the same as the ones used in the MATLAB analysis. After conducting each simulation, the DC components of the inverter input voltage and inverter input current were recorded. These components were used to produce a plot of inverter input voltage vs. inverter input current as shown in Figure 3.11. The V-I load characteristic curve that resulted is as follows: 34.899106.447089.0)( 2 +?= iii IIIV . (3.29) It can be seen from Figures 3.10 and 3.11 that the MATLAB code produces results that are similar to PSPICE. Based on these results, there is a potential to use a V-I characteristic curve to represent a motor-drive load in a DC power flow analysis. 91 Inverter Voltage vs. Inverter Current (PSPICE) y = 0.7089x 2 - 44.106x + 899.34 180 230 280 330 380 430 480 12 14 16 18 20 22 24 26 28 30 32 Ii (Amps) Vi ( V o l t s ) Figure 3.11: V-I Load Curve Produced From PSPICE Simulations. 3.3.3 Two-Level Sinusoidal PWM Inverter Simulation Results The equations of the induction motor and the two-level sinusoidal PWM inverter were coded in MATLAB for the purpose of simulating the system shown in Figure 3.5. The source voltage of Figure 3.5 was varied over a range of 401V-500V with all other parameters remaining unchanged. The parameters of the 50 HP, three-phase, induction motor used to conduct the simulation study presented in this section are listed in Table 3.1. Other parameters used for the simulation were: f 1 = 60 Hz, m a =1.4, m f =15, and a constant load torque of T L =100 N-m. The V-I characteristic curve that results from the MATLAB simulation is shown in Figure 3.12. A quadratic curve fit of the V-I characteristic curve is shown in Figure 3.13. Using the polyfit command in MATLAB, the following V-I characteristic can be developed for this motor and inverter: 13003124.0)( 2 +?= iii IIIV . (3.30) Equation (3.30) represents everything to the right of the inverter input voltage (V i in Figure 3.5). As can be observed from Figure 3.13, the quadratic fit matches the original 92 Figure 3.12: V-I Characteristic Curve for a Sinusoidal PWM Inverter with T L =100 N-m. Figure 3.13: Quadratic Curve Fit for T L =100 N-m. 93 curve very well, which illustrates that the V-I characteristic curve of a two-level sinusoidal PWM inverter drive can be fit with a quadratic curve with good results. 3.3.4 Space Vector PWM Inverter Simulation Results The equations of the induction motor and the space vector PWM inverter were coded in MATLAB for the purpose of simulating the system shown in Figure 3.5. The source voltage of Figure 3.5 was varied over a range of 401V-500V with all other parameters remaining unchanged. The parameters of the 50 HP, three-phase, induction motor used to conduct the simulation study presented in this section are listed in Table 3.1. Other parameters used for the simulation were: f 1 = 60 Hz, M=0.7, m f =15, and a constant load torque of T L =80 N-m. The V-I characteristic curve that results from the MATLAB simulation is shown in Figure 3.14. A quadratic curve fit of the V-I characteristic curve is shown in Figure 3.15. Using the polyfit command in MATLAB, the V-I characteristic for this motor and inverter is as follows: 13003937.0)( 2 +?= iii IIIV . (3.31) Equation (3.31) represents everything to the right of the inverter input voltage (V i in Figure 3.5). As can be observed from Figure 3.15, the quadratic fit matches the original curve very well. This shows that the V-I characteristic curve of a space vector PWM inverter drive can be fit with a quadratic curve with good results. 94 Figure 3.14: V-I Curve for a Space Vector PWM Inverter with T L =80 N-m. Figure 3.15: Quadratic Curve Fit for T L =80 N-m. 95 3.4 Summary A harmonic model of the induction motor operating under balanced, steady-state conditions was presented in this chapter. The model that was presented was shown to be applicable to induction motors supplied from nonsinusoidal sources. It was shown in this chapter that a motor-drive system can be represented by a simplified model. In this simplified model, a V-I load characteristic curve was established that allowed the inverter, motor, and load to be replaced by a current-controlled voltage source. It was determined through model simulations that the current-controlled voltage source should be a quadratic function of the inverter current. The model was shown to be applicable to six-step, sinusoidal PWM, and space vector PWM inverters. 96 CHAPTER 4 MULTIPLE MOTOR-DRIVE SYSTEMS This chapter focuses on the analysis of a DC power system containing multiple motor- drive loads. An iterative procedure is presented that incorporates the simplified model from Chapter 3 into an algorithm used to perform a power flow analysis on a DC power system. The power flow algorithm presented is verified by conducting a power flow analysis on a 4-bus DC power system. The algorithm is then coded in MATLAB and power flow analyses are conducted on a 10-bus DC power system containing six-step inverter-drive loads and PWM inverter-drive loads. PSPICE simulation results are compared to the MATLAB power flow results for verification purposes. This chapter also includes a study conducted on an individual six-step inverter drive system that examines the effects on a system caused by larger line resistance values. A system with higher line resistances is simulated in PSPICE and the results are used to examine the effects of higher line resistances on a multiple motor-drive system. A 10-bus DC power system containing six-step inverter drive loads and higher line resistance values is also investigated. The chapter concludes with a summary of simulation results and findings from the study conducted on a system containing higher line resistance values. 97 4.1 DC Power Flow A DC power system containing motor-drive loads is shown in Figure 4.1. The simplified model discussed in the previous chapter can be extended to a system containing more than one motor drive. MATLAB can be used to produce a V-I load characteristic curve for each motor drive load in a DC power system that can be incorporated into an iterative procedure to conduct a power flow analysis. The network shown in Figure 4.1 can be represented as [51]: VGI ~~ = (4.1) where I ~ is the current vector (nx1), G is the network conductance matrix (nxn), V ~ is the bus voltage vector (nx1), and n is the number of buses. The system studied contains motor-drive loads only and each bus voltage element of V ~ (except for the swing bus) will be of the same form as (3.26): ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ++ ++ ++ = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = nnnnn n cIbIa cIbIa cIbIa V V V V V V 2 333 2 33 222 2 22 1 3 2 1 ~ MM (4.2) where bus 1 was chosen as the swing bus. Note that the currents in (4.2) are the DC inverter input currents of each individual motor drive load at the specified bus. The conductance matrix can be formed using the following rules [52]: 98 Source Motor-Drive Loads Distribution Network . . . 1 2 3 n n I 2 I 3 I + - 1 V Figure 4.1: DC Power System Model. )(, 1 ji R G ij ij ??= , (4.3) ? ? = = n ij j ij ii R G 1 1 , (4.4) where R ij is the line resistance between bus number i and bus number j. When the conductance matrix has been formed and the DC network equations placed in the form of (4.1), Kron reduction can be used to eliminate all non-contributing buses using the following formula [47]: kjinji G GG GG kk kjik ij new ij ?=?= ,,,..,1,, . (4.5) It should be noted that non-contributing buses are buses that have no external load or source connected. The voltage is normally not of interest at a non-contributing bus, and the bus can be eliminated. A Kron-reduced system can now be formed as follows: 99 KronKronKron VGI ~~ = (4.6) where Kron I ~ is an (n-m)x1 vector, G Kron is an (n-m)x(n-m) matrix, Kron V ~ is an (n-m)x1 vector, and m is the number of non-contributing buses. An iterative method based on the Newton-Raphson method [51] is well suited to solve for the load currents, because (4.6) represents a system of simultaneous nonlinear algebraic equations [53]. Moving all of the variables in (4.6) to one side and setting them equal to zero will produce a system of (n-m) nonlinear equations in (n-m) unknowns as: 0),...,,( ,0),...,,( ,0),...,,( 43 433 431 = = = ?? ? ? mnmn mn mn IIIf IIIf IIIf M (4.7) where the notation in (4.7) is based on the assumption that bus 1 is the swing bus and bus 2 is a non-contributing bus. In vector form, (4.7) becomes: 0) ~ ( ~ )()( = ?? k mn k mn If (4.8) where k is the k th iteration value. The system Jacobian matrix (based on (4.7)) is: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ??? ? ? mn k mn k mn k mn mn kkk mn kkk k I f I f I f I f I f I f I f I f I f J )( 4 )( 3 )( )( 3 4 )( 3 3 )( 3 )( 1 4 )( 1 3 )( 1 )( L MMM L L . (4.9) The load current correction for the k th iteration is: 100 [] [] [] )( )( ~ )( 1 )( ~ 1 )( )( ) ~ , ~ ( ~~ k mn k mn I k mn k mn I k k mn IVfJI ? ? ?? ? ? ?=? . (4.10) The values of the new updated load currents are: )()()1( ~~~ k mn k mn k mn III ?? + ? ?+= . (4.11) Once the initial estimates for the load currents are made, (4.8-4.11) can be used to iteratively compute the load currents of a DC power system containing motor drive loads. Convergence of the power flow iterations is based on the following criteria: ?> Figure 4.14: Line-to-Line Voltages with Low Line Resistance. Figure 4.15: Inverter DC Input Current and Voltage with Low Line Resistance. Time 4.000s 4.002s 4.004s 4.006s 4.008s 4.010s 4.012s 4.014s 4.016s V(Rs:2,Vs:-) 450V 460V 470V 480V SEL>> I(Rs) -50A 0A 50A 126 The line resistance of the system in Figure 4.13 was changed to 0.3 ? as shown in Figure 4.16 and the system was simulated again in PSPICE to investigate the effects of increasing the line resistance on the behavior of the system. The line-to-line voltage waveform V ab that resulted from the simulation is shown in Figure 4.17. As can be seen in Figure 4.17, the line-to-line voltage is beginning to deviate from the shape shown in Figure 4.14. The inverter input voltage waveform that resulted from the simulation is shown in Figure 4.18. It can be seen from this figure that the inverter input voltage is no longer a stiff DC voltage. The inverter input current waveform is shown in Figure 4.19. 127 Figure 4.16: Six-Step Inverter System with a Higher Line Resistance Value. 128 Time 4.000s 4.002s 4.004s 4.006s 4.008s 4.010s 4.012s 4.014s 4.016s V(S4:3,S6:3) -500V 0V 500V Figure 4.17: Line-to-Line Voltage (V ab ) with a Higher Line Resistance. Time 4.000s 4.002s 4.004s 4.006s 4.008s 4.010s 4.012s 4.014s 4.016s V(Rs:2,Vs:-) 440V 450V 460V Figure 4.18: Inverter Input Voltage with a Higher Line Resistance. 129 Time 4.000s 4.002s 4.004s 4.006s 4.008s 4.010s 4.012s 4.014s 4.016s I(Rs) 0A 20A 40A 60A 74A Figure 4.19: Inverter Input Current with a Higher Line Resistance. A Fourier analysis was conducted as part of the PSPICE simulation on the inverter input current and inverter input voltage. The results of this Fourier analysis are shown in Table 4.7. It can be seen that the inverter input current and voltage are both rich in even harmonic content. Harmonics with multiples of six are present in both waveforms. It is obvious from these results that the distortion in the inverter input voltage will effect the output voltage waveforms of the inverter as shown in Figure 4.17. In order to examine the effects of the presence of even harmonics on the input side of the inverters in a multiple-bus DC power system, the system in Figure 4.10 was modeled using the line resistance and load torque values listed in Table 4.8. The PSPICE model was constructed the same as in Figure 4.11 except for the line resistance values. The new system was also coded in MATLAB using the new line resistance values shown in Table 4.8. The results of the PSPICE simulation and the MATLAB power flow are shown in Table 4.9. As can be seen in this table, the presence of even harmonics on the input side of the inverters produces some larger differences between the MATLAB and PSPICE 130 TABLE 4.7 HARMONIC CONTENT OF INVERTER CURRENT AND VOLTAGE Inverter Input Inverter Input Inverter Input Inverter Input Harmonic Voltage Voltage Harmonic Current Current Number Magnitude Angle Number Magnitude Angle (V) (degrees) (A) (degrees) DC 446.8914 DC 43.69524 6 5.598 -94 1 1.557 -100.3 12 3.254 153.5 2 1.35 99.05 18 2.23 45.13 6 18.66 86 24 1.692 -62.17 8 1.107 -20.55 30 1.362 -169.2 12 10.85 -26.54 14 1.013 -127.1 18 7.434 -134.9 22 1.002 32.53 24 5.638 117.8 28 1.057 -74.12 30 4.54 10.85 TABLE 4.8 SYSTEM LINE RESISTANCES AND LOAD TORQUES Load Line Bus Torque Line Resistance Number (N-m) Section (?) 1 2 1 - 2 0.1 3 70 2 - 3 0.0009 4 65 2 - 4 0.1 5 35 2 - 5 0.15 6 60 2 - 6 0.2 7 50 2 - 7 0.25 8 40 2 - 8 0.3 9 30 2 - 9 0.35 10 25 2 - 10 0.4 131 TABLE 4.9 POWER FLOW RESULTS WITH LARGER LINE RESISTANCES Current Converged Current Voltage Converged Voltage from Current Percent from Voltage Percent Bus PSPICE (MATLAB) ?I Error PSPICE (MATLAB) ?VEo Number (A) (A) (A) (% of PSPICE) (V) (V) (V) (% of PSPICE) 3 26.0388 26.0361 0.0026 0.0102 535.5256 535.7882 0.26260 0.049036 4 24.4279 24.3197 0.1082 0.4429 533.1062 533.3797 0.27350 0.051303 5 13.7282 13.3794 0.3488 2.5409 533.4898 533.8048 0.31500 0.059045 6 22.7914 22.5762 0.2152 0.9441 530.9907 531.2964 0.30570 0.057572 7 19.2409 18.9213 0.3196 1.6609 530.7388 531.0813 0.34250 0.064533 8 15.7097 15.2617 0.4480 2.8518 530.8361 531.2332 0.39710 0.074807 9 12.1906 11.6049 0.5857 4.8047 531.2823 531.7499 0.46760 0.088013 10 10.3826 9.7839 0.5987 5.7665 531.3959 531.8981 0.50220 0.094506 results. The code written in MATLAB does not model the effects of the even harmonics, but PSPICE does account for the impact of even harmonics on the system. However, it can be observed from Table 4.9 that the higher line resistance values and the presence of even harmonics on the input side of the inverter did not significantly impact the accuracy of the MATLAB results. In practical applications, the line resistances in a system such as the one shown in Figure 4.10 are small due to the fact that the cable length between each drive and motor is typically less than 50 feet [54-56]. With cable lengths greater than 50 feet, it is possible to experience a voltage wave reflection at the motor terminals up to two times the applied voltage [57, 58]. This effect can be shown by using transmission line theory [54]. The line resistances that would result from the cable requirements outlined in [54-56] would be in a range similar to the ones listed in Table 4.5. In this line resistance range, the MATLAB code produced excellent results as can be seen in Table 4.6. 132 4.6 Summary In this chapter, an iterative procedure was presented that can be used to conduct a power flow analysis on a DC power system containing motor-drive loads. It was shown that a V-I load characteristic curve can be developed for each motor-drive load and can then be incorporated into an iterative procedure to conduct a power flow analysis on a given system. The power flow algorithm was verified by conducting a power flow analysis on a 4-bus DC power system using hand calculations. The algorithm was coded in MATLAB and power flow results were presented for a 10-bus DC power system containing six-step voltage source inverter drive loads and a 10-bus DC power system containing sinusoidal PWM inverter drive loads. PSPICE models of each system were built and the results were compared to the MATLAB power flow results. A study was conducted on an individual six-step inverter drive system that had a larger line resistance value to examine the effects of higher line resistances on a multiple- bus system. Even harmonics were present in the inverter input voltage and current waveforms of the system with a higher line resistance. However, the higher line resistance and the presence of even harmonics on the input side of the inverter did not significantly impact the accuracy of the MATLAB results. 133 CHAPTER 5 CONCLUSIONS 5.1 Summary A simplified model of an inverter-fed induction motor has been developed to be used in the analysis of a DC power system containing motor-drive loads. The model was based on the steady-state T-type equivalent circuit of an induction motor and the input- output relationships of a voltage source inverter. In the simplified model, a V-I load characteristic curve was established that allowed the inverter, motor, and load to be replaced by a current-controlled voltage source. Power flow analyses were conducted in MATLAB using the simplified model and the results were comparable to PSPICE. The simplified model used in the analysis of a multiple-bus DC power system by incorporating the V-I load curves of each motor-drive load in a particular system into a Newton-Raphson type iterative procedure. The focus of Chapter 2 was on the harmonic analysis of different types of voltage source inverters. The types of inverters analyzed in Chapter 2 included: (1) the six-step inverter, (2) the sinusoidal PWM inverter, and (3) the space vector PWM inverter. Two methods for finding the harmonic components of the output voltage of sinusoidal PWM inverters and space vector PWM inverters were presented in Chapter 2. The method of pulse pairs was the first method discussed. This method was shown to be applicable to different multilevel inverter types such as the two-level sinusoidal PWM inverter and the 134 three-level sinusoidal PWM inverter. The main limitation of the method of pulse pairs was the possibility of the loss of symmetry in the output voltage of the inverter. In this scenario, there would no longer be corresponding pulse pairs. The method of multiple pulses was developed to overcome this limitation. This method was used to calculate the Fourier coefficients of individual positive and negative pulses of the output PWM waveform. The coefficients of the individual pulses were added together using the principle of superposition to calculate the Fourier coefficients of the entire PWM output signal. The final expression for the PWM output voltage can be expressed compactly in a single-cosine Fourier series that allows direct calculation of harmonic components and can easily be implemented in a computer software package such as MATLAB. This method allows direct calculation of harmonic magnitudes and angles without having to use look-up tables, linear approximations, iterative procedures, Bessel functions, or the gathering of harmonic terms required by other methods. The method of multiple pulses, presented in Chapter 2, is entirely general and has the potential to be used to analyze the harmonic content of inverter output waveforms produced by various types of multilevel inverters and PWM schemes. There is no limitation to the method of multiple pulses due to loss of symmetry or the harmonic content of the inverter output voltage waveform. The method of multiple pulses can also be used to calculate the harmonic content of inverter waveforms produced by the six-step inverter. This method can be extended to analyze other types of multilevel inverters and PWM schemes not studied in this dissertation. 135 A harmonic model of the induction motor operating under balanced, steady-state conditions was presented in Chapter 3. The model produced simulation results for an induction motor supplied from a nonsinusoidal source that was comparable to EMAP [49]. A simplified model of an inverter-fed induction motor that was based on the steady-state T-type equivalent circuit and the input-output relationships of the voltage source inverter was developed. A V-I load characteristic curve was established that allowed the inverter, motor, and load to be replaced by a current-controlled voltage source. The model was coded in MATLAB and compared with PSPICE simulations. The model was shown to be applicable to six-step, sinusoidal PWM, and space vector PWM inverters. An iterative procedure was presented in Chapter 4 that can be used to perform a power flow analysis on a DC power system containing motor-drive loads. The simplified model presented in Chapter 3 was shown to be applicable to the analysis of a multiple-bus DC power system containing motor-drive loads by forming the V-I characteristic curve of each motor-drive load in a given system. The V-I load characteristic curve developed for each motor-drive load in a DC power system can then be incorporated into an iterative procedure to perform a power flow analysis on a particular system. The power flow algorithm was verified by conducting a power flow analysis on a 4-bus DC power system using hand calculations. The algorithm was then coded in MATLAB and power flow analyses were conducted on a 10-bus DC power system containing six-step inverter-drive loads and PWM inverter-drive loads. PSPICE models of each system were constructed and simulated. The MATLAB power flow results were found to be comparable to PSPICE. 136 Chapter 4 also included a section on the impact of larger line resistance values for an individual six-step inverter drive system. The system was constructed in PSPICE for simulation purposes. The results of the PSPICE simulations were used to examine the effects of higher line resistances on a multiple-bus system. The larger line resistance was shown via PSPICE simulations to produce even harmonics in the inverter input voltage and inverter input current waveforms. Power flow results from simulation of a 10-bus DC power system containing six-step inverter drives demonstrated that the higher line resistance values and the presence of even harmonics in the inverter input current and voltage did not have a significant impact on the accuracy of results. 5.2 Recommendations for Future Work An area for future consideration is the study of the effects caused by higher line resistance values. Even harmonics appear in the inverter input voltage waveform when the line resistances are higher. The appearance of even harmonics in the inverter input voltage will affect other machine variables such as the line-to-line voltages. Various researchers have developed methods for calculating the inverter input current of a six-step voltage source inverter [9-13 and 59]. Most of these methods use a power balance between the inverter input and the inverter output to establish an expression for the inverter current. An instantaneous power balance between the inverter input and inverter output was used by some of the researchers [10, 11, and 59] to develop an expression for the inverter current in terms of the synchronously rotating reference frame currents. 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