ANALYSIS OF DC POWER SYSTEMS CONTAINING
INDUCTION MOTORDRIVE 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 MOTORDRIVE 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 MOTORDRIVE 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 MOTORDRIVE 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 motordrive 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 inverterfed induction motor, which is based on the
steadystate Ttype harmonic equivalent circuit model of the induction motor and the
inputoutput relationships of the inverter. In the simplified model, a VI load
characteristic curve is established that allows the inverter, motor, and load to be replaced
by a currentcontrolled voltage source. This simplified model can be utilized in the
vi
analysis of a multiplebus DC power system containing motordrive loads by
incorporating the VI load characteristic curve of each motordrive load into an iterative
procedure based on the NewtonRaphson method. The analytical method presented is
capable of analyzing DC power systems containing induction motordrive 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 motordrive 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 SixStep 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 TwoLevel PWM Inverter ..................................................................15
2.1.1.1 Harmonic Analysis of the TwoLevel Inverter Using the
Method of Pulse Pairs .....................................................................18
2.1.1.2 Simulation Results for the TwoLevel PWM Inverter....................25
2.1.2 The ThreeLevel PWM Inverter .................................................................30
2.1.2.1 Harmonic Analysis of the ThreeLevel Inverter Using the
Method of Pulse Pairs .....................................................................32
2.1.2.2 Simulation Results for the ThreeLevel 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 CarrierBased Approach .............................................................................43
2.2.2 Method of Multiple Pulses..........................................................................46
2.2.3 Simulation Results for the Space Vector PWM Inverter............................51
2.3 LineNeutral Voltage Fourier Series Development ...............................................55
2.3.1 The SixStep Inverter..................................................................................56
2.3.1.1 120? Conduction .............................................................................56
2.3.1.2 180? Conduction .............................................................................57
2.3.2 The TwoLevel Sinusoidal PWM Inverter .................................................58
2.3.3 The Space Vector PWM Inverter................................................................68
2.4 Summary................................................................................................................73
x
CHAPTER 3 THE INVERTERFED INDUCTION MOTOR ........................................75
3.1 Induction Motor Equivalent Circuit.......................................................................75
3.2 Verification of Induction Motor Harmonic Model ................................................83
3.3 MotorDrive System Model...................................................................................83
3.3.1 Simplified Model Simulation Results.........................................................88
3.3.2 SixStep Inverter Results ............................................................................89
3.3.3 TwoLevel 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 SixStep Simulation Results for a 10Bus System...............................................115
4.4 TwoLevel 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 LINETONEGATIVE DC BUS VOLTAGE
COMPONENTS FOR m
a
=1.4 and m
f
=15...................................................67
TABLE 2.16 LINETONEUTRAL VOLTAGE
COMPONENTS FOR m
a
=1.4 and m
f
=15...................................................68
TABLE 2.17 LINETONEGATIVE DC BUS VOLTAGE
COMPONENTS FOR M=0.7 and m
f
=15....................................................72
TABLE 2.18 LINETONEUTRAL VOLTAGE
COMPONENTS FOR M=0.7 and m
f
=15....................................................73
TABLE 3.1 50 HP, 3PHASE, INDUCTION MOTOR PARAMETERS .......................84
TABLE 3.2 MATLAB AND EMAP SIXSTEP INVERTER RESULTS.......................84
TABLE 3.3 DIFFERENCES AND PERCENT ERRORS ...............................................85
TABLE 4.1 4BUS SYSTEM LINE RESISTANCES AND LOAD TORQUES..........101
TABLE 4.2 POWER FLOW RESULTS FOR THE 4BUS SYSTEM..........................115
TABLE 4.3 SYSTEM LINE RESISTANCES AND LOAD TORQUES ......................118
TABLE 4.4 POWER FLOW RESULTS FOR THE SIXSTEP INVERTER ...............120
TABLE 4.5 SYSTEM LINE RESISTANCES AND LOAD TORQUES .......................121
TABLE 4.6 POWER FLOW RESULTS FOR THE TWOLEVEL
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 MotorDrive System Model ..............................................................................2
Figure 1.2 DC Power System Model ..................................................................................2
Figure 1.3 ThreePhase 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 TType Equivalent Circuit ...................................................12
Figure 2.1 Triangular Waveform and Control Signal.......................................................16
Figure 2.2 SinglePhase Inverter.......................................................................................17
Figure 2.3 TwoLevel 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 ThreeLevel PWM Output Waveform...........................................................31
Figure 2.11 ThreeLevel 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 ThreePhase Inverter Block Model ...............................................................55
Figure 2.23 SixStep Phase a Voltage Waveform with 120? Conduction........................57
Figure 2.24 SixStep Phase a Voltage Waveform with 180? Conduction........................57
Figure 2.25 ThreePhase Sinsusoidal PWM Control Signals and Carrier Waveform......59
Figure 2.26 LinetoNegative DC Bus Voltage Waveforms ............................................60
Figure 2.27 Waveform v
aN
(t) with Pulses Labeled ...........................................................61
Figure 2.28 Phase a LinetoNeutral Voltage Produced using MATLAB .......................62
Figure 2.29 Harmonic Spectrum of the Phase a LinetoNegative DC Bus Voltage .......62
xiv
Figure 2.30 Harmonic Spectrum of the Phase a LinetoNeutral Voltage .......................63
Figure 2.31 Space Vector PWM Control Signal and Carrier Waveform .........................69
Figure 2.32 LinetoNegative DC Bus Voltage Waveforms ............................................63
Figure 2.33 Phase a LinetoNeutral Voltage Waveform.................................................70
Figure 3.1 (a) Induction Motor TType 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 PositiveSequence Harmonic Equivalent Circuit ............................................82
Figure 3.4 NegativeSequence Harmonic Equivalent Circuit...........................................82
Figure 3.5 MotorDrive System Model ............................................................................85
Figure 3.6 VI 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 VI Load Curve Produced From MATLAB Code .........................................90
Figure 3.11 VI Load Curve Produced From PSPICE Simulations..................................91
Figure 3.12 VI Characteristic Curve For a Sinusoidal PWM Inverter
with T
L
=100 Nm...........................................................................................92
Figure 3.13 Quadratic Curve Fit for T
L
=100 Nm............................................................92
Figure 3.14 VI Curve For a Space Vector PWM Inverter with T
L
=80 Nm...................94
Figure 3.15 Quadratic Curve Fit for T
L
=80 Nm..............................................................94
Figure 4.1 DC Power System Model ................................................................................98
Figure 4.2 FourBus DC Power System .........................................................................101
Figure 4.3 VI Characteristic Curve for T
L
=75 Nm ......................................................105
Figure 4.4 Quadratic Curve Fit for T
L
=75 Nm..............................................................106
Figure 4.5 VI Characteristic Curve for T
L
=40 Nm ......................................................106
Figure 4.6 Quadratic Curve Fit for T
L
=40 Nm..............................................................107
Figure 4.7 PSPICE 4bus System Model........................................................................113
Figure 4.8 PSPICE SixStep MotorDrive Model ..........................................................114
Figure 4.9 PSPICE Induction Motor Part .......................................................................115
Figure 4.10 10Bus DC Power System Model................................................................116
Figure 4.11 PSPICE 10Bus Power System Model........................................................119
Figure 4.12 PSPICE Sinusoidal PWM MotorDrive Model...........................................122
Figure 4.13 SixStep Inverter System with a Low Line Resistance Value.....................124
Figure 4.14 LinetoLine Voltages with Low Line Resistance.......................................125
Figure 4.15 Inverter DC Input Current and Voltage with Low Line Resistance............125
Figure 4.16 SixStep Inverter System with a Higher Line Resistance Value.................127
Figure 4.17 LinetoLine 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 threephase inverter driving a threephase
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 motordrive loads such as the one shown in Figure 1.2.
The speed advantage of the analytical method is evident when multiple motordrive
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 threephase voltage source inverter is illustrated in Figure 1.3. Depending
2
DC Voltage
Source
Voltage
Source
Inverter
3Phase
Induction
Motor
i
I
i
V
+

N
a
b
c
Load
Figure 1.1: MotorDrive System Model.
Source
MotorDrive 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: ThreePhase Voltage Source Inverter.
upon the method for controlling the switches of the inverter in Figure 1.3, the inverter can
operate as a sixstep 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, lookup 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 steadystate Ttype 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 inputoutput relationships of the
voltage source inverter. In the simplified model a VI 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 currentcontrolled voltage source. The
simplified model developed for the system in Figure 1.1 was shown to be applicable to a
multiplebus DC power system such as that shown in Figure 1.2 by forming a VI load
characteristic curve for each motordrive 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 SixStep Inverter
The sixstep inverter is perhaps the simplest form of threephase inverter. A circuit
diagram of a threephase voltage source inverter is shown in Figure 1.3. The output of
a sixstep 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 sixstep 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 sixstep
voltage source inverter [17].
5
Murphy and Turnbull [8] discussed AC motor operation when supplied by a sixstep
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 sixstep inverter.
Abbas and Novotny [9] utilized a fundamental component approximation to develop
equivalent circuits that represent the transfer relations of the sixstep voltage source and
current source inverters during steadystate 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 rectifierinverter
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, steadystate operation.
Novotny [12] used time dependent functions called switching functions to represent
transfer properties of sixstep voltage source and current source inverters. The switching
functions were expanded as complex Fourier series and applied to steadystate 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. Steadystate analysis of the sixstep
voltage and current source inverterdriven 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 twolevel sinusoidal PWM inverter. This technique is widely used in
industrial applications such as variablespeed 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, 1618].
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 3D
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 sinetriangle 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
singlephase and threephase 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 JacobiAnger substitution. Analytical solutions were provided for
various PWM strategies including space vector modulation.
Tseng, et al. [26] used a 3D modulation model and the double Fourier series as first
proposed by Bennett to analyze the harmonic characteristics of a threephase twolevel
PWM inverter. Models of the threephase inverter system were constructed in PSPICE
and MATLAB for harmonic analysis purposes. Equations from the theoretical analysis
using the 3D 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 3D modulation
model and the double Fourier series.
Mohan et al. [27] conducted an analysis of twolevel PWM inverters in Chapter 8 of
their book. Design considerations for the twolevel 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 mid1980?s [3234] 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, 3537]. 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 linetoline 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 inverterfed AC motor
drives. A carrierbased 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 carrierbased approach to analyze
the space vector PWM inverter.
1.2.4 The Induction Motor
Fitzgerald, et al. [40] provided a detailed analysis of the steadystate Ttype 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 TType Equivalent Circuit.
Giesselmann [42] developed a PSPICE dq model of the induction motor for analysis and
simulation purposes. The PSPICE model was based on the Ttype equivalent circuit
model of the induction motor. Implementation of the dq 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 dq 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 inverterfed 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 inverterfed
induction motor are discussed in Chapter 3. Multiple motordrive 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 motordrive 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 sixstep
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 singlephase inverter topologies, while those analyzed in section
2.3 are typical waveforms produced by a threephase 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 JacobiAnger expansions to establish closed form solutions. The end result of
using JacobiAnger 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 twolevel and a threelevel 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, lookup 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 inverterdrive systems such as the
one presented in [44].
2.1.1 The TwoLevel PWM Inverter
In a twolevel PWM inverter with sinetriangle 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 (2.12.4) 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: SinglePhase Inverter.
18
Figure 2.3: TwoLevel PWM Output Waveform.
2.1.1.1 Harmonic Analysis of the TwoLevel 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
halfwave symmetry, the Fourier coefficient a
0
is zero. This is due to the fact that the
average value of a function with halfwave symmetry is always zero. The Fourier
coefficients a
n
and b
n
are also zero for n even due to halfwave 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 twolevel 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 P1P3 in the figure are positive pulse pairs and N1N3 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 (2.112.13) 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 twolevel modulation can be expressed in a
singlecosine 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 2L in
(2.17) stands for twolevel.
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 halfwave 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 TwoLevel 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 twolevel 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 2.12.4. 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 2.12.4. 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 twolevel PWM inverter output waveform.
TwoLevel 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 ThreeLevel PWM Inverter
In a threelevel PWM inverter with sinusoidal modulation, a control signal at a desired
output frequency is compared with a multilevel 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 twolevel 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 threelevel
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: ThreeLevel PWM Output Waveform.
32
of the sinusoidal control signal as described in [27]. This alternative method of
generating a threelevel PWM output signal is shown in Figure 2.11.
Figure 2.11: ThreeLevel PWM Alternative Method.
2.1.2.1 Harmonic Analysis of the ThreeLevel 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 halfwave
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 halfwave 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 (2.62.11). 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 threelevel 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 P1P3 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 threelevel 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 3L in (2.24) stands for
threelevel.
35
v
3L
(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 halfwave 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 ThreeLevel 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 threelevel PWM simulation model. A PSPICE ABM block was used to
compare the sinusoidal control signal and the multilevel 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 2.52.8. 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
ThreeLevel 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 threelevel 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 threelevel PWM inverter from [23] were coded in MATLAB. A
MATLAB simulation was conducted using the threelevel PWM inverter equations from
[23] with V
i
=270V, m
a
=1.4, and m
f
=18.
A threelevel PWM simulation model was constructed in PSPICE for comparison
purposes. A PSPICE ABM block was used to compare the sinusoidal control signal and
the multilevel 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 inverterdrive systems such as the one
presented in [44].
43
2.2.1 CarrierBased 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 sixswitch,
threephase, 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 ?carrierbased? approach can be used to implement the
space vector PWM as shown by different researchers in the literature [45, 46]. The
carrierbased 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 carrierbased 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
lookup 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 halfwave symmetry in the output waveform of the space vector PWM
inverter as described in [16, 35]. A function has halfwave symmetry if it satisfies
f(t)=  f(tT/2). The method of pulse pairs would fail if halfwave 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 P1P6 are positive pulses and N1N5 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 (2.392.42) 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 singlecosine 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 (2.442.47) 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 2.112.14. The
results shown in Tables 2.112.14 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 LinetoNeutral 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 singlephase inverters.
The focus of Chapter 3 and Chapter 4 will be on analyzing a threephase, 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 lineneutral voltage produced from the threephase inverter system shown
in Figure 2.22 that can be used in the harmonic analysis of an induction motor supplied
by a threephase 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: ThreePhase Inverter Block Model.
56
2.3.1 The SixStep Inverter
In this section, the Fourier series of the sixstep voltage source inverter linetoneutral
voltage for both 180? and 120? conduction will be presented. The Fourier series of the
linetoneutral voltage of the sixstep 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 sixstep 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 linetoneutral voltage of
the sixstep voltage source inverter.
2.3.1.1 120? Conduction
A plot of the phase a linetoneutral voltage of the sixstep inverter with 120?
conduction is shown in Figure 2.23. The Fourier series of the sixstep inverter phase a
linetoneutral 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: SixStep Phase a Voltage Waveform with 120? Conduction.
2.3.1.2 180? Conduction
A plot of the phase a voltage of the sixstep inverter with 180? conduction is shown in
Figure 2.24. The Fourier series of the basic sixstep 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: SixStep Phase a Voltage Waveform with 180? Conduction.
58
2.3.2 TwoLevel Sinusoidal PWM Inverter
When the motor in Figure 2.22 is supplied from a threephase twolevel sinusoidal
PWM inverter, the linetonegative 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 linetonegative 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 linetonegative DC bus voltage produced by twolevel 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 linetonegative DC bus voltage has now been
established in (2.50). However, a trigonometric Fourier series representation of the phase
59
Figure 2.25: ThreePhase Sinusoidal PWM Control Signals and Carrier Waveform.
60
Figure 2.26: LinetoNegative 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 linetoneutral voltage of the system shown in Figure 2.22 is needed. The phase a line
toneutral voltage waveform for this system while supplied by a twolevel sinusoidal
PWM inverter is shown in Figure 2.28. Before beginning to develop a Fourier series
representation of the linetoneutral voltage, it is appropriate to first look at the harmonic
spectrums of the phase a linetonegative DC bus voltage and the phase a linetoneutral
voltage by creating the waveforms in MATLAB and using the FFT command to produce
the harmonic spectrums. The harmonic spectrum of the phase a linetonegative DC bus
voltage is shown in Figure 2.29 and the harmonic spectrum of the phase a linetoneutral
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 linetoneutral voltage does not contain a DC component nor
any zerosequence harmonics (triplen harmonics). For maximum cancellation of
dominant harmonics in the line voltages of a threephase 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 LinetoNeutral Voltage Produced using MATLAB.
Figure 2.29: Harmonic Spectrum of the Phase a LinetoNegative DC Bus Voltage.
63
Figure 2.30: Harmonic Spectrum of the Phase a LinetoNeutral Voltage.
spectrums can be proven mathematically by first considering some basic threephase
relationships for the system shown in Figure 2.22. The inverter linetoneutral 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 (2.512.53) into (2.54):
64
[])()()(
3
1
)( tvtvtvtv
cNbNaNsN
++= . (2.55)
By substituting (2.54) into (2.50), the phase a linetoneutral 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 linetoneutral
voltage by considering the DC component of each linetonegative 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 linetoneutral voltage. It can
also be shown that the magnitudes of the harmonic components in the harmonic spectrum
of the linetoneutral voltages are the same as the magnitudes of the linetonegative DC
bus voltages. This can be accomplished by considering a balanced set of fundamental
linetonegative 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
linetoneutral and the linetonegative 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 linetoneutral 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 (2.652.67) 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 zerosequence (or triplen) harmonics exist in the line
toneutral voltages. No zerosequence 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 linetoneutral 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 linetonegative DC
bus voltage produced by twolevel sinusoidal modulation (2.50) and the equation used to
calculate the Fourier series of the phase a linetoneutral voltage produced by twolevel
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 twolevel sinusoidal PWM inverter simulation model using PSPICE ABM blocks. A
Fourier analysis was then performed in PSPICE on the phase a linetonegative DC bus
voltage and the phase a linetoneutral 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 twolevel sinusoidal PWM inverter.
TABLE 2.15
LINETONEGATIVE 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
LINETONEUTRAL 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 threephase space vector PWM
inverter, the linetonegative 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 linetonegative DC bus voltage waveforms are shown in Figure
2.32. The phase a linetoneutral 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: LinetoNegative DC Bus Voltage Waveforms.
70
Figure 2.33: Phase a LinetoNeutral 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 linetonegative 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 linetonegative 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 linetonegative DC bus voltage and
the phase a linetoneutral 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 linetonegative DC bus voltage
and the phase a linetoneutral 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
LINETONEGATIVE 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
LINETONEUTRAL 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 twolevel sinusoidal PWM
inverter and the threelevel sinusoidal PWM inverter. The method allowed direct
calculation of harmonic magnitudes and angles without having to use linear
approximations, iterative procedures, lookup 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 linetoneutral voltage Fourier series of the sixstep, twolevel sinusoidal
PWM, and space vector PWM inverters were presented. The method of multiple pulses
can be used to determine the harmonic content of the linetoneutral 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 INVERTERFED INDUCTION MOTOR
The focus of this chapter is on the inverterfed induction motor. A steadystate
harmonic model of the induction motor operating under balanced conditions is presented.
The harmonic model is based on the Ttype 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 inverterfed induction motor that is based on the
steadystate Ttype equivalent circuit of the motor and the inputoutput relationships of
the voltage source inverter is presented. A VI load characteristic curve that allows the
inverter, motor, and load to be replaced by a currentcontrolled 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 steadystate Ttype
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 positivesequence
equivalent circuit of the induction motor where balanced threephase 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 TType 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 positivesequence linetoneutral voltage, X
m
is the magnetizing
reactance, R
1
is the stator resistance, and X
1
is the stator leakage reactance. The
Theveninequivalent stator impedance is:
)(
)(
11
11
111
XXjR
jXRjX
jXRZ
m
m
eee
++
+
=+= . (3.2)
77
From the Theveninequivalent 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 (Nm), 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
steadystate 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 steadystate 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 (3.13.13) are valid for the steadystate 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 (3.13.13) 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 inverterfed 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 linetoneutral 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 positivesequence harmonics and the
negative torque in (3.19) is produced by negativesequence 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 positivesequence 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 positivesequence harmonic,
which may be calculated using (3.23):
82
p
p
k
k
sk
s
p
)1(
1
??
= . (3.23)
The negativesequence harmonic equivalent circuit is shown in Figure 3.4, where
n
k is
the negativesequence harmonic number and s
k
n
is the slip for the
th
n
k negativesequence
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. PositiveSequence 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. NegativeSequence 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 sixstep inverter output voltage
Fourier series. A harmonic analysis was performed on a 50 HP, 3phase 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 sixstep
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 MotorDrive System Model
The proposed motordrive system to be analyzed is shown in Figure 3.5. This figure
shows a DC source connected to an inverter driving a threephase 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, 3phase, 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 kgm
2
Machine Ratings:
V
LL
= 460V
Rated Speed = 1710 rpm
Rated Torque = 200 Nm
Note: All quantities in Table 3.1 have been reflected to the stator.
TABLE 3.2
MATLAB AND EMAP SIXSTEP 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
3Phase
Induction
Motor
i
I
i
V
+

N
a
b
c
Load
s
V
Figure 3.5. MotorDrive 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 linetoneutral 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 lineto
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 linetoneutral voltage and the load torque (or the linetoneutral
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 VI load characteristic curve can be generated at the input terminals of the
inverter in Figure 3.5. For a sixstep inverter, PSPICE and MATLAB simulations have
shown that the resulting VI 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 VI data. To illustrate why a quadratic was used to
curve fit the generated VI data, the equations of the induction motor and sixstep 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 478V577V with all other parameters remaining unchanged. The motor used in
the simulation was a 50 Hp, threephase induction motor having parameters as listed in
Table 3.1. A graph of the generated VI 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 VI 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 currentcontrolled voltage source having the characteristics of
(3.27). The simplified model of the inverter drive system is shown in Figure 3.9. The
currentcontrolled 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: VI 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 VI load characteristic curve that allows
the inverter, motor, and load to be replaced by a currentcontrolled voltage source.
89
+

s
V
i
I
)(
i
IV
Figure 3.9: Simplified System Model.
Simulation results are shown for the sixstep inverter (180? conduction), the twolevel
sinusoidal PWM inverter, and the space vector inverter.
3.3.2 SixStep Inverter Results
The equations of the induction motor and sixstep 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 240V480V 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 VI 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: VI Load Curve Produced From MATLAB Code.
A PSPICE model of the system shown in Figure 3.5 was simulated in order to produce
a VI characteristic curve. The load applied to the motor during simulations was a pulsed
torque load with the following characteristics: T
L
=30Nm, 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 240V480V 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 VI 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 VI
characteristic curve to represent a motordrive 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: VI Load Curve Produced From PSPICE Simulations.
3.3.3 TwoLevel Sinusoidal PWM Inverter Simulation Results
The equations of the induction motor and the twolevel 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 401V500V with all other
parameters remaining unchanged. The parameters of the 50 HP, threephase, 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 Nm. The VI characteristic curve that results from the
MATLAB simulation is shown in Figure 3.12. A quadratic curve fit of the VI
characteristic curve is shown in Figure 3.13. Using the polyfit command in MATLAB,
the following VI 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: VI Characteristic Curve for a Sinusoidal PWM Inverter with T
L
=100 Nm.
Figure 3.13: Quadratic Curve Fit for T
L
=100 Nm.
93
curve very well, which illustrates that the VI characteristic curve of a twolevel
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 401V500V with all other parameters
remaining unchanged. The parameters of the 50 HP, threephase, 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 Nm. The VI characteristic curve that results from the MATLAB
simulation is shown in Figure 3.14. A quadratic curve fit of the VI characteristic curve
is shown in Figure 3.15. Using the polyfit command in MATLAB, the VI 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 VI characteristic curve of a space vector PWM
inverter drive can be fit with a quadratic curve with good results.
94
Figure 3.14: VI Curve for a Space Vector PWM Inverter with T
L
=80 Nm.
Figure 3.15: Quadratic Curve Fit for T
L
=80 Nm.
95
3.4 Summary
A harmonic model of the induction motor operating under balanced, steadystate
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 motordrive system can be represented by a simplified model. In this
simplified model, a VI load characteristic curve was established that allowed the
inverter, motor, and load to be replaced by a currentcontrolled voltage source. It was
determined through model simulations that the currentcontrolled voltage source should
be a quadratic function of the inverter current. The model was shown to be applicable to
sixstep, sinusoidal PWM, and space vector PWM inverters.
96
CHAPTER 4
MULTIPLE MOTORDRIVE 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 4bus DC power system. The algorithm is then coded in MATLAB and
power flow analyses are conducted on a 10bus DC power system containing sixstep
inverterdrive loads and PWM inverterdrive 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 sixstep 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 motordrive system. A 10bus DC power
system containing sixstep 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 motordrive 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 VI 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
motordrive 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
MotorDrive 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 noncontributing buses
using the following formula [47]:
kjinji
G
GG
GG
kk
kjik
ij
new
ij
?=?= ,,,..,1,, . (4.5)
It should be noted that noncontributing buses are buses that have no external load or
source connected. The voltage is normally not of interest at a noncontributing bus, and
the bus can be eliminated. A Kronreduced system can now be formed as follows:
99
KronKronKron
VGI
~~
= (4.6)
where
Kron
I
~
is an (nm)x1 vector, G
Kron
is an (nm)x(nm) matrix,
Kron
V
~
is an (nm)x1
vector, and m is the number of noncontributing buses.
An iterative method based on the NewtonRaphson 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 (nm) nonlinear equations in (nm) 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 noncontributing 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, (4.84.11) 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: LinetoLine 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 linetoline voltage
waveform V
ab
that resulted from the simulation is shown in Figure 4.17. As can be seen
in Figure 4.17, the linetoline 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: SixStep 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: LinetoLine 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 multiplebus 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 (Nm) 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 [5456]. 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 [5456] 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 motordrive loads. It was shown
that a VI load characteristic curve can be developed for each motordrive 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 4bus DC power system using hand calculations. The algorithm was coded
in MATLAB and power flow results were presented for a 10bus DC power system
containing sixstep voltage source inverter drive loads and a 10bus 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 sixstep 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 inverterfed induction motor has been developed to be used
in the analysis of a DC power system containing motordrive loads. The model was
based on the steadystate Ttype equivalent circuit of an induction motor and the input
output relationships of a voltage source inverter. In the simplified model, a VI load
characteristic curve was established that allowed the inverter, motor, and load to be
replaced by a currentcontrolled 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 multiplebus DC power system by
incorporating the VI load curves of each motordrive load in a particular system into a
NewtonRaphson 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 sixstep
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 twolevel sinusoidal PWM inverter and the
134
threelevel 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
singlecosine 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 lookup 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 sixstep 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, steadystate
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 inverterfed induction motor that was based on the
steadystate Ttype equivalent circuit and the inputoutput relationships of the voltage
source inverter was developed. A VI load characteristic curve was established that
allowed the inverter, motor, and load to be replaced by a currentcontrolled voltage
source. The model was coded in MATLAB and compared with PSPICE simulations.
The model was shown to be applicable to sixstep, 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 motordrive loads. The simplified model
presented in Chapter 3 was shown to be applicable to the analysis of a multiplebus DC
power system containing motordrive loads by forming the VI characteristic curve of
each motordrive load in a given system. The VI load characteristic curve developed for
each motordrive 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 4bus DC power system
using hand calculations. The algorithm was then coded in MATLAB and power flow
analyses were conducted on a 10bus DC power system containing sixstep inverterdrive
loads and PWM inverterdrive 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 sixstep 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 multiplebus 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 10bus
DC power system containing sixstep 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 linetoline voltages.
Various researchers have developed methods for calculating the inverter input current
of a sixstep voltage source inverter [913 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.
In the methods that used instantaneous power balance [10, 11, and 59], electric
machine reference frame transformations and the Fourier series of the sixstep inverter
137
voltage waveforms were used to represent the inverterdrive system in the synchronously
rotating reference frame. An expression for the inverter input current was then developed
in terms of the synchronously rotating reference frame currents. However, the results
presented in [10, 11, and 59] are based on the assumption that the inverter input voltage is
a stiff DC voltage. As noted by [59], the determination of harmonics on both the input
and output sides of an inverter that has even harmonics present in the input voltage is a
complex problem and normally requires a detailed computer simulation using PSPICE or
other computer circuit simulation packages to produce accurate results.
138
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