
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00024524/00001
Material Information
 Title:
 Highorder linearizing pulsewidth modulator for threephase power converters
 Creator:
 Chen, Jun, 1966
 Publication Date:
 2000
 Language:
 English
 Physical Description:
 vii, 162 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Coordinate systems ( jstor )
Electric potential ( jstor ) Electronics ( jstor ) Equivalent circuits ( jstor ) Inductors ( jstor ) Inverters ( jstor ) Modulators ( jstor ) Signals ( jstor ) Simulations ( jstor ) Waveforms ( jstor ) Modulators (Electronics) ( fast ) Pulse techniques (Electronics) ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 2000.
 Bibliography:
 Includes bibliographical references (leaves 155161).
 General Note:
 Printout.
 General Note:
 Vita.
 Statement of Responsibility:
 by Jun Chen.
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 Source Institution:
 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
 Resource Identifier:
 45259781 ( OCLC )
ocm45259781

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Full Text 
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR
FOR THREEPHASE POWER CONVERTERS
BY
JUN CHEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to my research committee
chairman, Dr. Khai D. T. Ngo, for welcoming me to the power electronics group
at the University of Florida. He provided constant support and encourage
ment for my study and research. I learned not only the knowledge of power
electronics, but also a work attitude that has greatly reshaped my career. I
also wish to thank Dr. Dennis P. Carroll, Dr. Alexander Domijan, Dr. Vladimir
A. Rakov, and Dr. Loc VuQuoc for their participation on my research commit
tee.
I am very grateful to American Research Corporation for its financial
support and projects, and also Texas Instruments for a TI fellowship.
My special thanks go to my colleagues, Jun Xu and Paiboon Nakmah
achal, for their helpful discussions and suggestions in my project and disserta
tion.
There is no word for me to describe my gratitude to my wife, Yin Xie,
who spent her time taking care of our family when I was working at the lab
day and night. Without her help and patience, I would not know how to finish
my research and dissertation.
ii
TABLE OF CONTENTS
pages
ACKNOWLEDGMENTS ................................................................................. ii
ABSTRACT .................................................. ................................ v
CHAPTERS
1 INTRODUCTION ..................................... .................... 1
2 MODELING AND ANALYSIS OF THREEPHASE CONVERTERS 10
2.1 Derivation of StateSpace Equations of PWM Converters .......... 11
2.2 Equivalent Circuit in the ABC Coordinates .............................. 20
2.3 ABCOFB Transformation ........................................ ........... .. 26
2.4 Equivalent Circuit in the OFB Coordinates ................................ 28
2.5 Graphical SteadyState Analysis ............................................. 36
2.6 Graphical SmallSignal Analysis ............................................. 38
3 REVIEW OF PULSEWIDTH MODULATION .................................. 44
3.1 Pulsewidth Modulation for DC Converters ............................... 44
3.2 Pulsewidth Modulation for ThreePhase Converters ................ 48
3.3 Synthesis of Continuous Sinusoidal Pulsewidth Modulation ..... 55
3.4 Synthesis of SpaceVector Modulation ....................................... 57
4 HIGHORDER LINEARIZING PULSEWIDTH MODULATOR ........ 62
4.1 FirstOrder Linearizing Pulsewidth Modulator ........................ 63
4.2 Nonlinear Problem in ThreePhase Converters .......................... 66
4.3 LargeSignal Linearization of PWM Converters ......................... 69
4.4 Linearization by FirstOrder LPWM ........................................... 72
4.5 Linearization by HighOrder LPWM ........................................... 80
5 ANALYSIS OF HIGHORDER LINEARIZING
PULSEWIDTH MODULATOR ............................................ ......... 97
5.1 Analysis of HighOrder Linearizing PWM .................................. 98
5.2 Sampling Effects in HighOrder LPWM .................................... 107
iii
6 IMPLEMENTATION AND EXPERIMENTAL RESULTS OF
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR ...... 116
6.1 Analog Implementation of HighOrder LPWM .......................... 118
6.2 Experimental Results ...................................... 135
6.3 Practical Issues in Experiment .................................................. 145
7 SUMMARY AND CONCLUSION ..................................................... 151
R E FE R E N CE S ............................................................................................... 155
BIOGRAPHICAL SKETCH ........................................................................... 162
iv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR
FOR THREEPHASE POWER CONVERTERS
By
Jun Chen
August 2000
Chairman: Dr. Khai D. T. Ngo
Major Department: Electrical and Computer Engineering
The feasibility of using an analog pulsewidth modulator (PWM) to lin
earize balanced threephase converters is investigated in this dissertation.
Prototype circuits, models, and analysis techniques are developed.
Most balanced threephase PWM converters that are controlled by the
conventional analog PWMs have nonlinear relationships between the control
and output voltages/currents. This study shows that these nonlinear relation
ships can be linearized by an analog highorder linearizing pulsewidth modu
lator (LPWM) that makes the output voltage track the control voltage linearly.
Instead of multipliers/dividers, the highorder LPWM uses only integrators
with the reset, and sample/holds to compute the switching instants for the
switches in the converter. The inputs to the integrators of the LPWM are just
linear functions of the control and state variables, but are nonlinear functions
v
when other analog PWMs, such as feedforward PWMs and onecycle control
lers, are used.
The analog highorder LPWM is synthesized from switchingfunction
averaging (SFA) equations of the threephase PWM converter. Thanks to the
SFA model of the PWM switch, the derivation of SFA statespace equations of
the converter is simply done by inspection and application of definition of cir
cuit elements, Kirchhoffs law, and other electrical principles without probing
into topological details of the converter. The set of SFA equations can be trans
formed into an equivalent circuit in the stationary coordinates to make simu
lation more efficient.
In order to analyze threephase converters that are controlled by the
LPWM or other pulsewidth modulation techniques, all threephase component
models in the rotating coordinates, including PWM switches, sources, and pas
sive components, are developed. After threephase components are replaced
by their models in the rotating coordinates, the timevariant threephase cir
cuit is transformed into a timeinvariant equivalent circuit that makes analy
sis and design much easier. The model of the highorder PWM is also
developed. It is useful to analyze the LPWMcontrolled converter and evaluate
time delay caused by sampling effects.
The synthesis and analysis theories of the highorder LPWM are veri
fied by a 1 KW prototype of a threephase boost inverter. Both simulation and
experimental results agree with the analysis. The experimental results show
that the control circuit is simple, and the output voltages of the inverter can
vi
track the control voltages linearly, and they have lowdistortion sinusoidal
waveforms.
In summary, the synthesis and analysis techniques are developed for
linearization of a threephase boost inverter in the dissertation. As general
methods, they can be applied to other threephase topologies, multiphase or
multilevel converters.
vii
CHAPTER 1
INTRODUCTION
With the development of highspeed, highpower semiconductors, the
switching power converter has gradually replaced linear power amplifiers to
become the main power conversion product on the market. The switching
power converter not only provides more efficient power conversion than the
linear power converter, but also has more flexible control capability that
allows the converter to meet various power demands and requirements.
Therefore, research on switching power converters has received much atten
tion. A major research issue is the linearization of switching power converters
that makes the controlled variable track the control signal and improve the
performance of the converter.
A basic switching power converter consists of two sections, as shown in
Figure 11. The first section is called the power stage that usually consists of
semiconductor switches and energy storage components. The power stage
receives the unregulated energy from the utility power line or power convert
ers and provides the regulated energy to customer loads. The second section is
the modulator that provides control signals to the power stage.
1
2
UTILITY LINES POWER STAGE CUSTOMER
OR
CONVERTERS (switches, inductors, LOADS
and capacitors)
HIGHFREQUENCY MODULATOR CONTROL
CARRIER o (analog, or DSP  SIGNALS
circuits)
Figure 11 A basic switching power converter.
The modulator can be implemented by analog and digital means,
depending on the requirements, complexity, and costs in converter design. The
digital modulator is used mostly in threephase converters since it has more
computation capability. However, when the switching frequency is increased
by size and weight requirements, the digital modulator will be limited by its
clock speed. Meanwhile, when the reference voltage does not change smoothly,
the sample/hold circuit with the digital modulator would be restrained by res
olution. In contrast, the analog modulator is much faster, and it can handle
any frequency, limited only by the capability of power stage [1].
A conventional pulsewidth modulator (PWM), as shown in Figure 12,
consists of a comparator, a ramp carrier signal vrmp, and a control signal vc.
The carrier signal provides highswitching frequency to the control signal and
to the switches in the converter. The control signal is followed by the con
trolled variables, such as output voltages. In the conventional PWM, the car
rier signal has a constant slope. The control signal is compared with the
3
Vrmp V V
Vrm rmp
p VP dT____
T
vc.^
Figure 12 A conventional PWM.
carrier signal through the comparator. The output pulse vp (also called switch
ing function), generated by the comparator, is used to drive the switch in the
converter. It has the duty ratio of
d v (1.1)
Vm
where Vm is the amplitude of the ramp vrmp.
The output signal vp determines the switching patterns of the con
verter. The controlled variable of the converter, such as the output voltage, is
the function of the converter input and duty ratio of vp that is determined by
Equation (1.1). Therefore, the controlled variable of the converter can be regu
lated by adjusting the control signal.
The power stage in Figure 11 could be the dcdc converter in the dc
power conversion or the threephase converter in threephase ac power con
version. The most popular dc converters are shown in Figure 13. The conver
sion ratios between the output and input voltages are listed in Table 1.1. The
singlephase converters and dc converters with the transformer isolation are
4
not listed here, because they can be derived from these basic topologies and
have one independent control variable like basic dc converters.
Buck + l
V T T gT C RT 
+ SEPI +
V Boos V Vg BVo
T T  
+v+
vg T
BuckBoost Dual SEPIC
Figure 13 Basic dcdc converter topologies.
Table 1.1 Voltage conversion ratios.
Converter Topology Voltage Conversion Ratio, Vo/Vg
Buck D
Boost 1/(1D)
BuckBoost D/(1D)
Cuk D/(1D)
SEPIC D(1D)
Dual SEPIC D/(1D)
5
The dc converters are used mostly in delicate and lowpower applica
tions, such as computers and microprocessors. The threephase PWM convert
ers are usually used in rugged, highpower applications, such as active
filtering [2], UPS [3], VAR compensation [4], power generation [5], motor
drives [6, 7], and multilevel converters [8]. The most popular threephase
PWM converters [9] are shown in Figure 14. The voltage conversion ratios
\LJ JI ( ( ( T T T
Buck inverter Boost inverter
Buck rectifier Boost rectifier
IT T
Flyback rectifier Flyback inverter
Figure 14 The threephase inverters and rectifiers.
6
Vm/Vg are listed in Table 1.2, where Vm is the amplitude of the output volt
ages; Vg is the amplitude of the input voltages. The conversion ratios in Table
1.2 are derived from the balanced threephase converters, and the input volt
age and current are assumed in phase in the rectifiers. In the table, Dm is the
amplitude of the sinusoidal control signal. D is the duty ratio of the dc switch
in the flyback topology. It should be noted that these conversion ratios are
derived by assuming that the impedance of input/output reactive components
are small at input/output frequency and can be neglected.
Table 1.2 Voltage conversion ratios VmVg.
Converter Topology Inverter Rectifier
Buck Dm/2 Dm
Boost 1/Dm 2/Dm
Flyback D/Dm D/Dm
Although PWM converters are the most popular in various power con
versions, they have an inherent problem: nonlinearity. It keeps the output
voltage from tracking the control signal, gives rise to waveform distortion, and
degrades the performance of the converter. The reason that generates the non
linearity can be found by investigating the voltage conversion ratios of PWM
converters in Table 1.1 for dc converters, in Table 1.2 for threephase convert
ers, and duty ratios shown in Equation (1.1). The conventional PWM with a
constant slope carrier produces a linear relationship between the duty ratio
and the control signal as shown in Equation (1.1). When the duty ratio is used
7
to control nonlinear converters that have a nonlinear relationship between the
duty ratio and the output voltage, as shown in Table 1.1 and Table 1.2, the
output voltage is proven to be a nonlinear function of the control voltage.
The nonlinear problem of PWM converters has been solved mainly by
the smallsignal linearization technique of negative feedback control.
Recently, largesignal PWM linearization techniques were proposed in [10]
and [11]. As an alternative linearization technique, the largesignal PWM lin
earization features an openloop, steadystate linear controltooutput rela
tionship, regardless of operating conditions, leading to simple and stable
control circuit design. Moreover, this technique has better line voltage regula
tion not only for the linear converters, but also for the nonlinear converters
that are difficult for the feedforward control [12].
The largesignal PWM linearization techniques in [10, 11] can success
fully solve the nonlinear problem for dcdc converters and singlephase invert
ers, in which the PWM controller deals only with a single control variable.
However, threephase converters or multiphase converters have more than
one control variable. Therefore, the firstorder PWM linearization is limited in
threephase converters or multiphase converters. Nevertheless, the idea of
the largesignal linearization is a useful concept that could be extended to the
threephase converters, thus motivating the present research and leading to
the following objectives of the thesis:
* a general way to synthesize the highorder linearizing PWM for bal
anced threephase converters.
8
* a simple analog highorder linearizing PWM prototype circuit without
multipliers/dividers.
* a circuitoriented analysis technique for balanced threephase convert
ers.
* model and analysis of highorder linearizing PWM modulator.
* simulation and experimental verification.
This dissertation is organized as follows. Chapter 2 characterizes the
lowfrequency property of the PWM switch and reviews the switchingfunction
averaging (SFA) technique. The derivation of the SFA statespace equations of
a threephase converter is presented. Components of balanced threephase
converters are modeled in the ofb coordinates, by which the timevariant
threephase converter can be graphically transformed into a timeinvariant
equivalent circuit for steadystate and dynamic analyses.
Chapter 3 reviews PWM techniques for dc and threephase converters,
in which largesignal linearization is emphasized. Two popular PWM tech
niques, sinusoidal PWM (SPWM) and spacevector modulation (SVM), are dis
cussed in details in this chapter.
Chapter 4 identifies the nonlinear problem in threephase converters.
Two largesignal linearization techniques for threephase PWM converters
are proposed in this chapter. One technique uses several firstorder linearizing
PWM circuits to synthesize duty ratios for the switches in the converter indi
vidually. It involves multipliers/dividers to compute the inputs to the integra
tors. The other technique employs the proposed highorder LPWM circuit to
9
solve SFA equations. The inputs to the highorder LPWM circuit are linear
functions of the control and input voltages. Therefore, no multipliers/dividers
are required in the circuit, making analog implementation simple.
Chapter 5 focuses on the analysis of the LPWMcontrolled converters.
The largesignal and smallsignal models of the LPWM are derived in this
chapter. The time delay caused by sampling effects in the highorder LPWM is
also investigated.
Chapter 6 concentrates on implementation and experimentation of the
proposed analog highorder linearizing PWM. The experimental circuits and
results are presented in this chapter.
Chapter 7 consists of the summary and conclusion of this dissertation.
CHAPTER 2
MODELING AND ANALYSIS OF THREEPHASE CONVERTERS
This chapter presents the modeling and analysis techniques for three
phase PWM converters. These techniques are important for the synthesis and
analysis of the linearizing pulsewidth modulation and threephase PWM con
verters.
This chapter consists of six sections. The first section characterizes the
lowfrequency property of the PWM switch with switchingfunction averaging
(SFA) technique. The derivation of the SFA statespace equations of a three
phase converter is presented. The second section transforms the SFA state
space equations into an equivalent circuit that is used for fast simulation. The
third section reviews the abcofb transformation that is applied to the time
variant equivalent circuit to remove time dependency. The fourth section pre
sents the graphical models for all components of the threephase converter in
the ofb coordinates. This section also demonstrates how to construct the time
invariant equivalent circuit of a threephase converter in the ofb coordinates.
The fifth section solves the ofb equivalent circuit graphically for the steady
state analysis. The sixth section derives the smallsignal equivalent circuit by
perturbing the control and input variables in the steadystate ofb equivalent
10
11
circuit. With the help of smallsignal equivalent circuit, the controltooutput
transfer function of the converter can be easily found graphically.
Although the boost inverter is used as an example to demonstrate the
whole procedure, the analysis and modeling techniques in this chapter can be
applied to any other threephase PWM converter. To simplify explanation, it is
assumed throughout the thesis that the components are ideal and the
switches are lossless and fourquadrant.
2.1 Derivation of StateSpace Equations of PWM Converters
2.1.1 SwitchingFunction Averaging Model of PWM Switch
To analyze the steadystate and dynamic performance of a PWM con
verter, which contains reactive components, the statespace equations must be
presented. There are many approaches to derive the statespace equations for
PWM converters, among which the statespace averaging technique [13, 14] is
the most popular. This approach requires the identification of the switched
networks and the derivation of the statespace equations for all switched net
works that is easy to do in dc converters because of the small number of
switched networks. However, a threephase converter usually has a large
number of switched topologies, and with the increase of phase numbers, the
number of switched topologies will increase rapidly. For a given PWM method,
the switched networks in one switching cycle can be different from those in
another cycle. Moreover, different PWM schemes generate different switched
12
networks. Therefore, application of the statespace averaging technique to
analyze threephase converters is tedious.
Without probing into topological details, the switchingfunction averag
ing method [9, 15] treats the switch as a component in the same way as we
treat other linear components by defining the PWM switch model. This allows
derivation of statespace equations simply by inspection and application of
definitions of circuit elements, Kirchhoff's laws, and other electrical princi
ples. The switch model is derived by characterizing its low frequency property,
as explained in this section.
The singlepolemultiplethrow (SPMT) switch shown in Figure 21 is
one of the fundamental building elements in the PWM converters. The pole is
usually connected to the inductor, and the throw is connected to either the
voltage sources or the capacitors. The SPMT switch is reduced to a singlepole
doublethrow (SPDT) switch in a dc converter; it may be SPDT or a single
poletriplethrow (SPTT) switch in most threephase converters.
I p Pole 1
i li2 I ik i*M
d*ll d*12 d*1k d*1
v*1 v*2 v*k V*M
Figure 21 A singlepolemultiplethrow switch.
13
The operation of the throw k in Figure 21 is specified and modulated
by the switching function d*lk, as shown in Figure 22, where the asterisk *
denotes the instantaneous switching function. The function is one when the
throw is closed and zero when the throw is open. A switching function defined
this way can always be assigned to any throw in the converter without prior
knowledge of modulation strategy or sequence of switched topologies. There
fore, this switch model allows derivation of statespace equations of a PWM
converter without specifying modulation strategy. After the statespace equa
tions are derived, they can be used for any PWM strategy to do a specific anal
ysis [9].
In Figure 21, v*lv*M are throw voltages, i*ii*M are throw currents,
i*p is the pole current, and v*p is the pole voltage. The asterisk * denotes the
exact value. In switching functions shown in Figure 22 dildlM stand for duty
ratios of switching functions. Ts is the switching period.
1
d*11 0
 dllTs 1
d*12 0
d*lk 0 [ T
"dlkTs 1
SdlMTsO
Figure 22 Switching functions of SPMT switches.
14
To avoid short circuit, no two throws turn on at the same time, and all
switching functions must add up to one at any instant to avoid open circuit. In
other words, one and only one of the switching functions of the SPMT switch is
one at any instant. This can be expressed as follows:
M * (2.1)
Sdlk = 1
k=l
It is obvious that there are only M1 independent switching functions of the
onepoleMthrow switch.
At any moment, only one throw is connected to the pole. Therefore, the
throw current equals pole current during its connection with the pole:
i k = dlki P (2.2)
Over one switching cycle, the pole is connected to the throws one by one;
thus, the pole voltage v*p is just a linear combination of the products of throw
voltages and corresponding switching functions:
M
Vp = , dlkV k (2.3)
k= 1
The SPMT switches are switched at a high frequency. The voltages or
currents connected with the pole and throws, either dc or sinusoidal ac, are
varying slowly, relative to the switching frequency. Therefore, over one switch
ing cycle, the terminal voltages and branch currents of the pole and throws
can be assumed as constant. The duty ratio dlk (without asterisk), which is
the average of the switching function d*lk over the switching period T., is
15
modulated at a frequency sufficiently slower than the switching frequency.
Therefore, the duty ratio is also assumed as constant over the switching cycle.
In the analysis and modeling of switchedmode converters, attention
usually is restricted to lowfrequency components of voltages and currents.
The highfrequency components (also called ripples) are designed to be small
and can be neglected due to the combination of fast switching and proper
placement of filter corner frequencies. Therefore, the exact value with the
asterisk in the previous switchingfunction equations can be, approximately,
replaced by their lowfrequency value for analysis and modeling of lowfre
quency components. This modeling technique is called switchingfunction
averaging herein. The duty ratios in Equation (2.1) are then replaced by their
averaged values:
M
Sdlk = 1 (2.4)
k=l
The pole voltage and the throw currents in Equations (2.2) and (2.3) are
replaced by their averaged values:
M (2.5)
VP = dlkVk
k= 1
ik = dkip (2.6)
All values in Equations (2.4)  (2.6) vary slowly relative to the switching
frequency; thus, they characterize the lowfrequency properties of the SPMT
switch shown in Figure 21. With these equations, the SPMT switches can be
treated as components in the way we treat other components. The derivation
16
of statespace equations of a PWM converter becomes routine and can be done
using statespace concept [16], definitions of circuit elements, Kirchhoffs
laws, and other electrical principles.
2.1.2 Derivation of StateSpace Equations
Since the SPMT switch in the converter has been modeled as a compo
nent by the switchingfunction averaging technique, there is no need to iden
tify the switched topologies. Statespace equations are derived simply by
following the procedures described in ref 16. The only attention is to identify
the SPMT switches in the converter. A threephase boost inverter is used here
to demonstrate how to identify the SPMT switches and how to derive state
space equations for PWM converters. Since the statespace equations of the
switchedmode converter are derived by averaging the switching functions of
the switch, they may be called switchingfunctionaveraging statespace equa
tions (SFA statespace equations) [9] in this thesis.
A threephase boost inverter is illustrated in Figure 23. Since we know
that the pole of the SPMT switch usually is connected with inductors and
throw is connected with voltage sources or capacitors, it is easy to find that
there are two singlepoletriplethrow (SPTT) switches in the threephase
boost inverter. The SPTT switch on the top consists of the switches Sil, S12,
and S13 and is characterized by dll, d12, and d13. The SPTT switch on the bot
tom is grouped by S21, S22, and S23 and is characterized by d21, d22, and d23.
Duty ratios d11 d23 correspond to the switching functions of switches S11 S23,
17
Sll/ S1/0 W R R R
pole vp2
l a a la a a l l a
, v, and v. The first pole voltage is vp, and the second pole voltage is vb
i13 iF vc
= dllVa+dl2vb+d3V (2.7)
21 d122V, + 23
s ï¿½/ s ï¿½/ Sï¿½  C  C C
S21 S22 S23 T T T
pole 2 Vp2
Figure 23 A threephase boost inverter with two SPTT switches.
respectively, and they are modulated at a frequency sufficiently lower than the
switching frequency.
The states of the inverter are inductor current iL and capacitor voltages
Va, Vb, and vc. The first pole voltage is Vpl, and the second pole voltage is Vp2
Based on the SFA model of the SPMT switch in Equations (2.4)  (2.6), Vpl and
Vp2 can be expressed as the linear combination of capacitor voltages and duty
ratios of the switches:
vpl = dllva+d12vb+dl3vc (2.7)
vp2 = d21 va + d22vb + d23vc (2.8)
The voltage across the inductor can be obtained by application of Kirchhoff's
voltage law:
diL
Ldt = V  (v  Vp2) (2.9)
dt g9(pI~V2
18
Combination of Equations (2.7)  (2.9) yields
diL
Ldt = V a bb  dcc (2.10)
where da, db, and d, are effective duty ratios:
da = d  d21 db = d12  d22 dc = dl3d23 (2.11)
Since the current through the switch is the product of the inductor cur
rent and the duty ratio of the switch according to Equations (2.4)  (2.6), the
capacitor currents can be derived by applying Kirchhoff's current law as fol
lows:
dva 2va  vb  vc 2va  vb  Vc
C d= (dll d1)iL 3R = daiL 3R (2.12)
dvb 2vbva vc 2vb va vc
Cd = (d12 d22)iL 3 = dbiL 3 (2.13)
dvc 2v,  Vb  v 2v,  Vb  va
C = (d13d23)iL 3 a = dciL 3 a (2.14)
dt 3R cL 3R
Equations (2.10) and (2.12)  (2.14) are called SFA statespace equations of the
threephase boost inverter. Although they are derived for the boost inverter,
the switch model and the derivation procedure are general to other PWM con
verters.
The SFA statespace equations of the PWM converter are derived with
out knowledge of any PWM strategy and thus are general to any PWM modu
lation scheme: continuous sinusoidal PWM, spacevector modulating, and so
forth. Once a specific PWM modulation technique is applied to the converter,
the switching patterns and duty ratios in the SFA equations are known. For
19
the same PWM converter, a different PWM strategy leads to different coeffi
cients in the SFA equations and state solutions.
The SFA statespace equations shown in Equations (2.10) and (2.12) 
(2.14) are derived in the stationary reference frame or abc coordinates, in
which all the state variables and coefficients in the equations are timevari
ant. Obviously, solving the timevariant statespace equations is very tedious
and difficult. Therefore, they are transformed into the ofb coordinates to
remove the time dependencies from the statespace matrices [15] by the abc
ofb transformation [17]. These coordinates consist of an 0sequence phasor, a
forward(rotating) phasor, and a backward(rotating) phasor. After the trans
formation, the SFA statespace equations become timeinvariant, and the
threephase boost inverter can be analyzed by solving the statespace equa
tions in the ofb coordinates. For a balanced threephase system, the equations
containing the 0sequence, forward, and backward phasors are completely
decoupled. The steadystate backward phasors are directly related to the volt
age and current phasors in the circuit. Unfortunately, the steadystate and
dynamic analysis of converters [15], based on ofb statespace equations, con
tain intensive algebraic calculation and matrix manipulation. In addition, the
equationoriented model of the converter is not intuitive to computer simula
tion.
In contrast, circuitoriented techniques [18, 19, 20, 21] are preferred for
handanalysis/calculation and computer simulation. Such circuitoriented or
"graphical" techniques not only produce the averaged equivalent circuit
20
of a PWM converter expeditiously, but also result in a model that is
insightful and amenable to implementation in standard circuit simula
tors.
In the following section, the SFA statespace equations shown in Equa
tion (2.10) and (2.12)  (2.14) are transformed into an equivalent circuit in the
abc coordinates using the PWM Switch Model described in refs 18 and 19.
This equivalent circuit is useful in fast simulation and prediction of various
waveforms in the converter even though it is a timevariant circuit. Following
this section, the thesis provides a technique that transforms a timevariant
threephase converter into a timeinvariant equivalent circuit in the ofb coor
dinates. With the help of the ofb equivalent circuit, the steadystate and
dynamic analysis of the threephase converter becomes much easier.
2.2 Equivalent Circuit in the ABC Coordinates
According to the PWMSwitchModel technique [18], the PWM switch
can be modeled as a dc transformer that is a standard component in the simu
lator (such as Saber). The turns ratio is the duty ratio of the switching signal
of the PWM switch. This technique is used in dc converters [18], but its idea
can be extended to threephase PWM converters or other PWM converters.
Therefore, the SFA statespace equations of the PWM converter derived in the
previous section can be transformed into an equivalent circuit using several dc
transformers. This equivalent circuit is constructed by appropriate connec
tions between the dc transformers and other components. The connections are
21
determined by the SFA statespace equations. The turns ratios of those dc
transformers in the equivalent circuit are effective duty ratios in the SFA
equations [22].
As an example, the threephase boost inverter, as shown in Figure 23,
is used to demonstrate the derivation of the equivalent circuit from the SFA
statespace equations. The SFA equations of the threephase boost inverter in
Section 2.1.3 are organized and rewritten as follows:
diL
L = V  dV  dVb  dcV (2.15)
dv 2va vb  Vc
C dt= daiL  3R (2.16)
dt 3R
dvb 2Vb  V  V
C = db 3R (2.17)
dvc 2vc  vb  va
dC dciL (2.18)
dt 3R
A dc transformer is shown in Figure 24. Its turns ratio is determined
by the duty ratio of the switching signal of the PWM switch. The duty ratios
da, db, and d, in the SFA equations shown in Equations (2.15)  (2.18) can be
modeled by the dc transformer shown in Figure 24. The connection relation
ships between the transformers and other components in the boost inverter is
defined by Equations (2.15)  (2.18). The resulting equivalent circuit of the
threephase inverter is shown in Figure 25. If the capacitor voltages va, vb,
and vc are reflected from the output side to the input side, one can easily find
that the inductor voltage of the equivalent circuit in Figure 25 is the same as
22
+ 1 d:1 d
i di
+ * * +
dva v
av
Figure 24 The dc transformer with the duty ratio of d.
L d1:1
iL + R
II c V
d2:1
V 1C vb
S+ R
I C Vc
Figure 25 The equivalent circuit of the threephase boost
inverter in the abc coordinates.
that in Equation (2.15). The capacitor currents in the equivalent circuit also
are found to be the same as those in Equations (2.16)  (2.18). Therefore, the
equivalent circuit exactly represents the lowfrequency properties of the
threephase boost inverter. Because there are no real switches in the equiva
lent circuit, the simulation of this circuit is expedited and memory space of the
computer is also greatly saved. The simulation results of the equivalent cir
cuit are the lowfrequency components of the voltages and currents in the
23
inverter that are sufficient for us to predict various waveforms and design the
inverter.
To appreciate how fast and accurate the equivalent circuit is, the circuit
in Figure 25 is simulated in Saber. The simulation results are compared with
the realtime simulation of the threephase boost inverter. Supposing that the
PWM method applied to the threephase boost inverter is continuous SPWM,
one choice for duty ratios is
D1 +_Dsin( 1 Dm sin(0
Kdl 3+ 11 3ddlO 3 3 d)
D
1 Dm . 2 Dm ( 2
+ sin( d sin(0d +
3 3 a3 33
2D
[d] sin(ot)
2t(00 Hz), R = 10 Ohm, C = 100 eF. The realtime simulation result is shown inare
S26 a t tm 0 m . T
3 Dd 23
The simulated inverter has the following parameters: DM = 0.9, Vg = 200 V, Q =
2n7(100 Hz), R = 10 Ohm, C = 100 gYF The realtime simulation result is shown in
Figure 26, and the time for 40 ms simulation is 30 seconds. The results of the
simulation with the equivalent circuit is shown in Figure 27, and the time for
40 ms simulation is only 0.4 seconds.
24
300 Va Vb Vc
150
0
150
300
0 10 20 30 40
Figure 26 The realtime simulation of the threephase boost
inverter.
300 va vb Vc
150
0
150
300 , tms
0 10 20 30 40
Figure 27 The simulation of the threephase boost inverter
with the equivalent circuit
Both simulations give the same output voltage:
Va = 262Z320 (2.21)
25
The only difference between the two simulation results is that the realtime
simulation contains the highfrequency ripple, but the equivalent circuit sim
ulation has no ripple. The equivalent circuit produces exactly the lowfre
quency components of the output voltages. One interesting result obtained by
both simulations is that the output voltage has a 320 phase shift from the con
trol voltage, and the amplitude is also different from the value (222 V), pre
dicted by the conversion ratio in Table 1.2 of Chapter 1, as shown in Figure 2
8. This interesting result can be easily predicted by the steadystate analysis
of the ofb equivalent circuit of the threephase boost inverter, which will be
presented in the next section.
300 262 V
(V)0     .. ..  
150 va vb vc30 dia1/3)
150
150
300 21
0 5 10 15 20
Figure 28 The realtime simulation results showing the phase
shift and amplitude of the output voltages of the
threephase boost inverter.
The equivalent circuit in Figure 25 is derived in the abc coordinates
that is a timevariant circuit. Although it is effective for the fast simulation,
26
the steadystate analysis, especially the dynamic analysis with the timevari
ant equivalent circuit, is tedious and difficult. Therefore, it must be trans
formed into the ofb coordinates or odq coordinates to remove the time
dependency. Since abcodq transformation leads to two coupled subcircuits
[21], the resulting equivalent circuit is not convenient for analysis. The pro
posed timeinvariant equivalent circuit in this thesis, however, is derived in
the ofb coordinates, in which two subcircuits are completely decoupled, mak
ing the analysis much easier [22] and allowing one to write down answers by
inspection.
2.3 ABCOFB Transformation
The abcofb transformation matrix T transforms a timevarying vector
Xabc in the stationary (abc) coordinates into a timeinvariant complex vector
ofb in the rotating (ofb) coordinates according to
Xabc = Txfb (2.22)
ofb = T1xabc (2.23)
where, for a balanced threephase system with positive phase sequence,
x xcos(ex)
xabc = xb = xcos(6x  27/3) (2.24)
xc xcos(9x + 2c/3)
where
27
O(t) = to ()dTOx4 (2.25)
where o is the instantaneous frequency;
jOT ejOT
1 e e '
1 i(OT ) i(OT + )
T =  Te 3 T 3 (2.26)
1( 2% . 2x1
1 1 1
_ 1 eJT J(OT ) eJT++
=7 1 e e (2.27)
JOT eJ(OT 3 ) e_ T 3
e e e
where
OT(t) = OO(T)d'r4T (2.28)
Note that T1 = (T*)T (the conjugate transpose matrix of T),
 0
x
o J3 (x r)
Xof= = 2 e (2.29)
2
where xo is the zerosequence component, xf is the forward (rotating) phasor,
and xbw is the backward (rotating) phasor. Both Ox and OT are the initial
28
phases. Note that Xf and xbw are complex conjugates and constant (dc) under
steady state.
2.4 Equivalent Circuit in the OFB Coordinates
2.4.1 Models of ThreePhase Components in the OFB Coordinates
A threephase converter consists of resistors, inductors, capacitors,
sources, and switches. Their models in the ofb coordinates are obtained by
applying abcofb transformation and retaining Kirchhoff's voltage and current
laws to their connectivity, that is, after transformation, circuit topology is the
same as before. In the following analysis, R is the resistor matrix, L is the
inductor matrix, C is the capacitor matrix, and I is the 3x3 identity matrix:
R = IR L = IL C = IC (2.30)
Voltage sources. For the set of abc voltage sources in Figure 210(a),
application of Equations (2.22)  (2.28) yields the set of ofb voltage sources in
Figure 210(b). The ofb voltages/currents are found from the abc voltages/cur
rents by Equation (2.29).
Resistors. For the set of abc resistors shown in Figure 210(c),
VRabc = RiRabc (2.31)
application of Equations (2.22)  (2.28) to (2.31) yields
Rofb = RiRofb (2.32)
The ofb resistor set is thus as shown in Figure 210(d).
Inductors. For the set of abc inductors in Figure 210(e),
29
r  I r
Ssa iso o i iRa I +Ra
VRb
Isa V _1 Ib +Vb
SR
i Vbw+ VV b Lbw
i I+v Iï¿½ Vsbw IIs Ib +VRcI
Vsbw R
'Sc +C  sbw +f . I'RC +VRc
II I I R I
L J L J L   J
(a) abc voltage source set (b) ofb voltage source set (c) abc resistor set
r o  + Co
Ro +VRLa I+VLa  r 
I L LO
V iff I I VL
IRf +VRf bI+Lb  I f vLf
0"I 0 +L/(  L Io
I R I L IL jIoL
SI VRbw IiLc I+VLc I iLbw I VLbw
Itbw lV l +n .lI, fy lI+YY_.AAA ~ I
I R I I L I I L jo)L I
L   J L  J L J
d) ofb resistor set (e) abc inductor set (f) job inductor set
r "
iCar a 0iCo +vC"
I I
'c iCf + VCf ]
icb I + cb I I
It. I__ _ _ I_ , 
L    
(h) ofb capacitor set
Figure 29 Graphical models of voltage sources, resistors, induc
tors, and capacitors in the abc coordinates and the ofb
coordinates.
30
diLabc
Labc = L dt (2.33)
application of Equations (2.22)  (2.28) to (2.33) yields
d(TiLofb) diLofb
1 ___Lofb_ _ YLofb dT _
VLofb = L dt T T + dt t Lofb
(2.34)
SdiLofb ( IdT 2
= L + T Li
dt dt) Lofb
where
dT 0 0 0
dT
S dt L = 0jcoL 0 (2.35)
0 jCL
The ofb inductor set is thus as shown in Figure 210(f). The ofb "inductor" is a
real dynamic inductor L in series with an imaginary static resistor +joL.
Capacitors. The circuit models for the capacitors are the duals of those
for the inductors and are shown in Figure 210(g)  l(h).
Threephase singlepoledoublethrow (SPDT) switches. The SPDT
switches shown in Figure 210(a) are commonly found, for example, in the
buck inverter and boost rectifier [9]. Their lowfrequency model in the abc
coordinates is shown in Figure 210(b), where one choice for the duty ratios is
D
d = m is 2e K d 1  d (2.36)
b1 2+ 2 cos d 3 b2 bI
Ld D dc2 dc
L J 1 m 2x IK\
+ cos
22 2 3
where
31
Od(t) = co(T))Tdt  d (2.37)
The pole voltages and the throw currents can be expressed as
pa a l
pb = pabc = db1 ts = dabc, 1vts (2.38)
V dc1
pa
its =[d db1 dc1] ipb = Tabc, lipabc (2.39)
where dabcW is the transpose matrix of dabc. Note that the voltage reference
node of the proceeding equations is assumed to be vt..
Application of Equations (2.22)  (2.28) to (2.38) and (2.39) yields
* T
Vpofb = dofb, 1ts its =(dofb, ) pofb (2.40)
where (d*fb, 1)T is the conjugate transpose of dfb, 1;
[ 3D e+i(OdOT) d3D e d T)DT (2.41)
dofb, 1 = Dem T d (2.41)
The ofb model for the threephase SPDT switches is, as shown in Figure 2
10(c). Note that the variables for the transformer in the ofb coordinates are
generally complex. For a complex transformer, such as the one whose turns
ratio is l:dbwl in Figure 210(c), the transformation relationships are
Vpbw = dbwlvts itsbw =bw d pbw (2.42)
where d*bwl is the conjugate of dbwl.
32
its
vt+  11t 1
t+ + :dal l:dbl l:dc
K papa Vtsa
^ I dai /db1 / d<1 / + . b jQ^
Vts I . Vpb
I 21pb v
SIda2 ..  pc
\ db2 dc lpc Vt
t_ .
V (a) (b)
its 1:dol 1:df l:dbwl
Vts po p I pf+ pbw
Vts V f Vpbw
Vt
(c)
Figure 210 (a) Threephase SPDT switches; (b) switch model in
abc coordinates; (c) switch model in ofb coordinates.
Threephase singlepoletriplethrow (SPTT) switches. The SPTT
switches shown in Figure 211(a) are commonly found in, e.g., the boost
inverter and buck rectifier [9]. Their lowfrequency model in the abc coordi
nates is shown in Figure 211(b), where one choice for the duty ratios is
1 m 1 m
F 3 3 d 3 3
la2 2a D
lb 1 _ O3 sd 2b  (3 cos(Od (2.43)
Sl 1 Dm 271 1 m ( 2
+ Cos Od + ) 1 Dcos( )
3 3 3ef3 3 3s
The effective duty ratios are
33
2D
d d d 3 C
a la 2a
vb l~ db d] [ 3)b(4
1ta
c c c 2D
_cos0dï¿½ 6+
tb = tabc= ips = dabcips (2.46)
!3 Ldc
The voltage reference node of the proceeding equations is assumed to be the
common node of the threephase voltages. Application of Equations (2.22) 
(2.28) to (2.45) and (2.46) yields
,T
v = (dfb) Vtofb itofb = dofbips (2.47)
where (d*ofb)T is the conjugate transpose matrix of dofb;
dofb = d0 M e Dm (2.48)
The ofb model for the threephase SPTT switches is thus as shown in Figure
211(c). Before leaving this section, it is worth noting that, unlike the dq
34
transformation, the ofb transformation results in decoupled zerosequence,
forward, and backward components subcircuits.
SPTT switch ips da:1 ita
S% 4 +.
+   ' Vta
dia dib di,/ ita db itb
S Vta Vps
Vps itb Vtb Vt
itc dc:tc itc
d2a d2b d2 +
 tc
(a) (b)
ips d*:1 to
+Ito
d*f: 1 itf
(c) d*bw: itbw
I1 Vtbw
Figure 211 (a) Threephase SPTT switches; (b) switch model in
the abc coordinates; (c) switch model in the ofb coordi
nates.
2.4.2 Derivation of Equivalent Circuit in the OFB Coordinates
The equivalent circuit for a balanced threephase PWM converter can
be constructed graphically in the ofb coordinates just by replacing each set of
threephase switches by appropriately connected ofb transformers, and each
set of threephase components by the corresponding ofb component models.
The resulting ofb equivalent circuit is timeinvariant, in which the forward
35
component and backward component are totally decoupled. Therefore, the
analysis of threephase converters with the ofb equivalent circuit is easy.
To construct the ofb equivalent circuit, we need to identify dc and ac
components in the abc coordinates. The ac components are replaced by their
graphical models in the ofb coordinates, and dc components remain in the ofb
equivalent circuit. As a result, the threephase boost inverter is divided into
five parts, as shown in Figure 212. Part one and part two are in the dc side of
the inverter, including the dc voltage source and the inductor. In steadystate
conditions, the inductor current is dc. Therefore, it is not necessary to trans
form the voltage source and inductor. Parts three, four and five are in the ac
side of the inverter and include timevariant switches, capacitors, and resis
tors.
SPart 3 Part 4
2 1 I I I I
I I I
I I R
I I
I T Part 1
S I I IC C C
S211f 22 S23 I I
FiIre 1 I=rionn Part 5
L  J L   J
Figure 212 Partitioning the threephase boost inverter.
36
De:1 if Vf
+I
I/jnC
Vf C R
Vg +T
De:1 ibw Vbw
JICI+ " 1/jC s
Vbw C R
Figure 213 The equivalent circuit of the threephase boost
inverter in the ofb coordinates.
The ac components are transformed to their ofb models in the ofb equiv
alent circuit of the inverter. The resulting timeinvariant equivalent circuit in
ofb coordinates is shown in Figure 213. Since zerosequence is zero in the bal
anced threephase converter, the zero sequence circuit is excluded from Figure
213. The transformer turns ratios df and dbw in the ofb circuit are time
invariant, and they have the same value when Od = T = 0 in Equation (2.48)
D
so that D = Dbw = , which is represented by De in Figure 213:
D
De = D = Db (2.49)
2.5 Graphical SteadyState Analysis
To analyze the threephase boost inverter in the ofb coordinates under
steadystate condition, replace all the inductors by short circuits and all the
37
capacitors by open circuits in the ofb equivalent circuit shown in Figure 213,
the resulting steadystate equivalent circuit is shown in Figure 214.
L De:l If Vf
+I
Vf c 1/iJC R
De:1 Ibw Vbw
lVbw C l/jac R
Figure 214 The steadystate equivalent circuit of the three
phase boost inverter in the ofb coordinate.
Reflecting the resistors (real and complex) in the secondary of the
transformer to the primary, the circuit in Figure 214 becomes a simple circuit
shown in Figure 215. Two conjugate resistors in Figure 215 form a voltage
IL +
+R
D DeljoRC
g + 2 R
.D R
D eV bw el + joRC
Figure 215 A simple circuit to solve the steadystate output volt
age and inductor current.
divider; thus the backward voltage and inductor current can be obtained eas
ily. The backward phasor is given by
38
Vbw := V (I  j p(2.50)
2De pCO
that is the same as that derived from the equationoriented method [9]. Sub
stituting Equations (2.49) into (2.50) and applying Equation (2.29), the phasor
of the output voltage va can be found as
V
Va = (1  jsRC) (2.51)
Dm
The inductor current obtained from Figure 215 is a dc current, which is
3V
L g (1 + (RC)) (2.52)
2D R
m
For Dm = 0.9, Vg = 200 V, Q = 2t(100 Hz), R = 10 Ohm, C = 100 gF,
V = 262Z32 . This predicted output voltage agrees well with that obtained
from realtime simulation, as is evident in Figure 28. Note that the reactive
elements appears in the steadystate variables, introducing a righthalfplane
zero. This righthalfplane zero causes some phase shift to the output voltage.
In order to reduce the phase shift,
Q << p (2.53)
2.6 Graphical SmallSignal Analysis
As shown in Figure 213, the two transformers in the ofb equivalent cir
cuit are the same. They can be combined into one transformer with the turns
ratio De, as shown in Figure 216. The transformer in the equivalent circuit
39
can be modeled as a voltagecontrol voltage source and a currentcontrol cur
rent source, as shown in Figure 217(a).
iL De: 1
C ï¿½ R Vbw
V + l/j
I/jC +
Vf
Figure 216 The equivalent circuit of the threephase boost
inverter in the ofb coordinates.
Vpade
i ip, I De:l p
c del: P c   lp
+ I  + + I* *l +
T DT
Vca v icde Vpa ca e pa
(a) (b) a
Figure 217 (a) The largesignal model of the dc transformer; (b)
its smallsignal model.
Application of smallsignal perturbation to the capacitor voltage vpa,
the inductor current ic, and the control variable de in the circuit yields
de = D, + , ic = Ic + i , vpa = Vpa + pa (2.54)
40
where the caret implies smallsignal perturbations. Neglect of the steady
state and secondorder terms then leads to a smallsignal equivalent circuit of
the transformer, as shown in Figure 217(b). It consists of a dc transformer
and two dependent sources that are controlled by the duty ratio. De, Ic, and
Vpa in the capital letter are dc values derived from the steadystate analysis of
the inverter. Replacing the transformer in Figure 216 by its smallsignal cir
cuit in Figure 217 yields the smallsignal equivalent circuit of the inverter, as
shown in Figure 218, where every variable is replaced by its smallsignal
value with the head "A"
De:
+ C R
S V e Vbw 
e
+
VfC  R
Figure 218 The smallsignal equivalent circuit of the three
phase boost inverter.
Let ig = 0, the real part of controltooutput transfer function of the
inverter can be solved from the smallsignal circuit in Figure 218, which
is given by
( s s
S V = (2.55)
de 2D2 D(s)
d, 2De
41
the imaginary part is given by
Di Vg O) (2.56)
^e 2D2COp D(s)
where
D e lf2I 2LCR L LC 2 LC2R 3
D(s) =1+ RC+ 2 +  +s + 2 s (2.57)
2 2De
2D 2D2 R D 2s D
2D2 R 1
1+ 2 z1 = 1 + m (2.58)
1+ C\pOP
P)
P = (2.59)
C RC
If the design allows
<< 1 (2.60)
where the LC corner is located at
D
0oo = J2 D (2.61)
it suffices to approximate the poles by
S 1+s 1 S_2)
D(s) l +   f1 +  (2.62)
COPA Q coo 00
where
2D2R
Q = (2.63)
moL
42
From Equation (2.62), the poles of the threephase boost inverter consist of a
real pole and complex poles. Like the dc boost converter, the bandwidth is
affected by the duty ratio De.
The controltooutput transfer function of the inverter can be found
from Equations (2.55) and (2.56), which is given by
Vm V9 (O ( z OZ
G(s) =2 o  1 + (2.64)
^le 2D2 D D(s)
Letting de = 0, the inputtobackward phasor transfer function of the inverter
can be found in the circuit in Figure 218, which is given by
Vbw P+1 oP J__ Q
b  p (2.65)
D9 2De D(s)
The audiosusceptibility, that is, the inputtooutput transfer function, is
solved from Equation (2.65)
H(s)  Vm  2 + 2l
H(s) = om 1 +) (2.66)
9 2De p D(s)
The transfer functions graphically derived from the smallsignal equivalent
circuit in Figure 218 completely agree with those derived by the equationori
ented method [9]. However, the graphical derivation is much simpler than the
manipulation of statespace equations.
In conclusion, among existing modeling techniques, the switchingfunc
tion averaging is the easiest technique to model threephase converters. The
equivalent circuit of the converter in the abc coordinates constructed from the
43
SFA statespace equations expedites the simulation. The equivalent circuit of
the threephase converters in the ofb coordinates is constructed graphically by
replacing sets of threephase components with appropriately connected ofb
components. With the help of the ofb equivalent circuit, the steadystate and
smallsignal analyses of the threephase converters can be worked out graphi
cally, which is proven to be easier than the equationoriented method.
CHAPTER 3
REVIEW OF PULSEWIDTH MODULATION
This chapter reviews the existing pulsewidth modulation (PWM) tech
niques for both dc converters and threephase converters, in which largesig
nal linearization is emphasized. Two popular PWM methods for threephase
converters, continuous sinusoidal PWM (SPWM) and spacevector modulation
(SVM), are discussed in detail.
3.1 Pulsewidth Modulation for DC Converters
The conventional PWM with a constantslope carrier is the most popu
lar in dc or singlephase converters, but it gives rise to undesirable nonlinear
relationship between the output and control voltage in some topologies. Some
linearizing PWM techniques (LPWM) have been proposed for linearization of
dc or singlephase converters [10, 2334]. In these PWM techniques, the slope
of the carrier signal is not constant. Thus, the duty ratio generated from the
LPWM is a nonlinear function of the input and control voltages that may can
cel out the nonlinear controltooutput relationship of the converter and make
the output voltage to track the control signal linearly.
44
45
3.1.1 OneCycle Control
The onecycle control [10, 2328] has been widely used in various dc or
singlephase PWM converters. When the onecycle control was proposed in ref
10, the aim was to make the output voltage of the dc buck converter and Cuk
converter follow the control voltage tightly without being disturbed by input
voltage. It was subsequently proven that this control method can be easily
used in other topologies, controls, and applications [23]  [28]. For instance, it
can make the input current track the sinusoidal input voltage, allowing unit
power factor to be obtained [25]. The basic concept of the onecycle control is to
force the average of the switchedvariable, such as the diode voltage in the
buck converter, to be proportional to the control variable in each switching
cycle. Therefore, a onecycle controller can make the output voltage propor
tional to the control voltage, that is, transform a switching power converter
into a linear power amplifier in a largesignal sense. The onecycle controller
developed in ref 24 is a generalized circuit that can be used by any dc or sin
glephase PWM converter. In addition to the largesignal linearization of
PWM converters, the onecycle controller has some advantages over the con
ventional PWM techniques, such as the switching loss compensation, good line
voltage regulation, and stable and simple control circuits.
3.1.2 FeedForward Pulsewidth Modulation
The feedforward control is mostly used in the linear buck converter or
buckderived converters to reduce source disturbance on the output voltage,
46
where the slope of the ramp signal in the modulator varies with the input volt
age. Its application in the nonlinear PWM converters is published in ref 12, in
which good line voltage regulation is obtained for linear and nonlinear con
verters. However, the controltooutput gain is zero for the boost converter and
nonlinear for "quadratic" converters [11].
The feedforward control is adapted to a pulsewidth modulation [11]; it
is called feedforward PWM (FFPWM). With the FFPWM, any linear or non
linear PWM converter can be linearized. The steadystate controltooutput
relationship of the converter becomes linear regardless of operating condi
tions. The FFPWM not only implements largesignal linearization of PWM
converters, but also reduces the source disturbance on the output voltage of
the converter. The FFPWM has no stability problems and no effects on con
verter output impedance. If tight output voltage regulation is required, a
smallsignal voltage feedback can be used with less difficulty and with
improved response compared with the conventional PWM modulation.
Although the technique in ref [11] does not provide a general modulator cir
cuit as the onecycle controller, it provides us with a general way to synthesize
the largesignal linearizing PWM circuit.
3.1.3 PeakCurrent Mode Control
The peakcurrent mode control is widely used in dc or singlephase con
verters [3537], in which the peak inductor current always equals the refer
ence current, regardless of all other operating conditions. This control method
47
may be considered as a largesignal linearizing PWM in terms of linearization
of the inductor current, such as the input current of the acdc converter. The
output voltage, however, is still controlled by a nonlinear controltooutput
relationship. Therefore, this control method needs an extra voltage feedback
loop to linearize the output voltage and keep it stable. The current mode con
trol has inherent advantages, such as fast dynamic response, automatic cur
rent protection, and so forth.
It is important to note that the carrier signals used in the above PWM
methods are not constant, but they vary with the control signal from one
switching cycle to another. They are herein called the PWM with a varying
slope carrier. The duty ratio generated from them is a nonlinear function of
the control signal. This is different from the conventional PWM with a con
stant slope carrier. It is also worth noting that the slope of the carrier signal of
PWM with varyingslope carrier is constant, that is, a straight line even
though the slope rate changes from one switching cycle to another. The carrier
used in refs 31  33, however, is nonlinear, which could be the exponential
function.
The LPWM modulators previously discussed are general PWM methods
suitable for all dc or singlephase PWM converter topologies, including linear
and nonlinear converters. The LPWM circuits can be implemented by simple
analog circuits, usually integrators and comparators. Although the PWM
methods [3841] are also able to implement largesignal linearization of the
48
PWM converters, they involve more sophisticated analog circuits, such as mul
tipliers/dividers.
3.2 Pulsewidth Modulation for ThreePhase Converters
Threephase PWM converters are employed in many areas of today's
power industries, including active filtering [2], UPS [3], VAR compensation
[4], power generation [5], motor drives [6, 7]. Compared with dc PWM convert
ers, threephase converters face more requirements, such as harmonics, bal
ance/unbalance systems, and so forth. Moreover, they need more sophisticated
control and drive circuits. Undoubtedly, linearization in PWM modulation will
bring benefits, such as easier control, lower harmonic distortion, and source
disturbance rejection, to the threephase PWM converters and help achieving
the stringent application requirements.
Many PWM schemes for threephase PWM converters have been pub
lished and applied in various power applications [4258]. They can be classi
fied into seven categories: sinusoidal, spacevector modulation, selective
harmonicelimination, optimal, current control, direct amplitude control,
and sigmadelta modulation.
3.2.1 Sinusoidal PWM
Sinusoidal PWM technique (SPWM) [42] is based on the principle of
comparing a triangular carrier signal with a sinusoidal reference. The imple
mentation of the technique with analog circuits is simple and can produce
49
very good sinusoidal waveforms. In recent years, much effort has been made
toward digitization of the SPWM [4346]. Online computation of instants of
intersection of the triangular carrier and sinusoidal reference waveforms is
not possible because no closedform solution is available for intersection
instants. Therefore, the reference sinusoidal waveforms have been replaced by
trapezoidal [43], stepped [44], or triangular waveforms [45]. The carrierbased
SPWM technique has disadvantages, such as attenuation of the fundamental
component and large switching losses. Most of all, the slope of the highfre
quency carrier in the PWM is constant and the duty ratios are linear functions
of the reference signals. Therefore, the SPWM is unable to implement linear
ization of nonlinear PWM converters.
3.2.2 SpaceVector Modulation
Spacevector modulation [49] (SVM) can utilize most of the power
source and reduce switching losses, which makes it the most popular PWM
technique in threephase converters. The SVM technique generates PWM sig
nals by averaging the three switchingstate vectors to equal the reference vec
tor over each switching cycle. Since the SVM involves a significant amount of
computation to determine the commutation instants of the switches, it is usu
ally implemented by digital signal processor (DSP) or microprocessor [50]. The
clock speed of the DSP or microprocessors, however, could impede the progress
of PWM toward higher frequency. Analog implementation is an alternative to
DSP for highspeed SVM. As with sinusoidal pulsewidth modulation, the SVM
50
can be implemented by comparing a sixstep control signal, generated from
the reference voltage, with a constantslope carrier signal [1]. Such implemen
tation, however, gives rise to nonlinear relationships between the control and
output voltages, preventing the output voltages from tracking the control sig
nals.
3.2.3 Optimal PWM
The optimal PWM technique [51] produces the switching pattern based
on optimization of some performance criteria. The number and positions of the
pulses or notches within each switching cycle are selected according to these
criteria, which could be harmonic loss, torque pulsation, or load currents.
They are precalculated and stored in memory for use in real time. Thus, com
putation power from a microprocessor is needed to synthesize the correct
switching patterns.
3.2.4 CurrentControlled PWM
The currentcontrolled PWM technique [52] is intended to make the
output current track the reference current. In this technique, the output cur
rents with superimposing ripples are fed back and compared with hysteresis
levels placed around the reference signal to determine the switching fre
quency. As the ripple is regulated within the hysteresis band, the average out
put follows the average reference. Three independent controllers are needed to
control three phase legs separately in this scheme; each controller has its own
51
switching frequency related to its output. Although it has good dynamic per
formance, this technique suffers from lowfrequency harmonics and high
switching losses.
3.2.5 SelectiveHarmonicElimination PWM
The selectiveharmonicelimination PWM technique [53] formulates a
waveform that is chopped M times and possesses odd quarterwave symmetry,
and contains the information about where the pulse starts or ends. Therefore,
any M harmonics can be nullified by solution of the corresponding M simulta
neous transcendental equations, which need extensive numerical calculation.
This technique is intended to attack the harmonics by suppressing an arbi
trary number of them in the output spectrum.
3.2.6 SigmaDelta Modulation
Sigmadelta modulation [5457] consists of a hysteresis comparator and
an integrator. The integrator estimates the reference voltage from the modu
lated PWM signal by low pass filter averaging. The estimated voltage is com
pared with the actual reference voltage through a hysteresis comparator to
generate the error signal, which is quantized to form the PWM signal. There
fore, the output voltage, which is equivalent to the average of the modulated
PWM signals, is able to follow the reference voltage within the hysteresis
band. Sigmadelta modulation suffers from the problem of variable frequency
and filter stability problems at high frequencies [56]. Although attempts have
52
been made to solve the variable frequency problem, they increase the complex
ity of the control circuit.
3.2.7 Direct Amplitude Control
The direct amplitude control [58] can make fundamental amplitude of
the output voltage directly follow the reference voltage. Using Fourier analy
sis, the algorithm is to equalize the subamplitude of the output voltage with
the subamplitude of the reference voltage for a complete fundamental cycle.
This technique involves a significant computation; thus, it usually is imple
mented by DSP or microprocessor.
Among the above PWM techniques, the SPWM and SVM are the most
popular in various threephase converters. However, due to the constantslope
carrier, both PWM methods can produce a nonlinear relationship between the
control and output voltages. This results in the output voltage failing to track
the reference voltage linearly. Nevertheless, it can be shown that both SPWM
and SVM can be developed into the linearizing PWM (LPWM) through the
proposed largesignal linearization technique in this thesis. In the following
sections, the conventional SPWM and SVM are discussed and synthesized, so
that the proposed linearizing PWM can be better appreciated.
A threephase boost inverter, shown in Figure 31, is used as an exam
ple to demonstrate how to synthesize the conventional SPWM and SVM.
The statespace equations of the inverter were given by Equations (2.15) 
(2.18) in Chapter 2 and repeated here:
53
L r        SPTT switch
S12 S3 R R R
Sll S12 S13
via Va
+ ib
Vg Vb
le  Vc
vc
S21 S22 / S23
   TL
/    SPTT switch
Figure 31 A threephase boost inverter.
diL
dt (3.1) 
L = Vg  (d=  d2)va  (dl2  d22)vb  (d13  d23)vc (3.1)
dv, 2va  vb  vc
Cd = (d,  d21 )iL 3R (3.2)
dvb 2vb  Va  vc
C = (d12  d22)iL  (3.3)
dt 3R
dv, 2vc  va  vb
C = (d13  d23)iL  3 (3.4)
dt3R
In the above equations, Va, vb and vc are balanced threephase voltages. Their
frequency and amplitude are known from the specifications:
va = Vmsin(o)t) Vb = Vmsin(O)t1200) vc = Vmsin(oct+ 1200) (3.5)
Duty ratios in the statespace equations dll  d23 are unknown, which will be
synthesized from Equations (3.1)  (3.4).
For a balanced threephase system, Equation (3.2) can be expressed as
54
dv Va
C = (dlld2l)iL  (3.6)
Multiplying va on both sides yields
dv v
VaC = (dll d2l)iLVa (3.7)
The same procedure is applied to Equations (3.3) and (3.4) yields
dvb Vb
VbC = (dl2d Lvb (3.8)
dv v
VcC  = (dl3d23)iLvc  (3.9)
Under steady state, the inductor current is assumed as dc; thus, diL/dt
in Equation (3.1) is zero. Substituting Equations (3.7)  (3.9) into (3.1) yields
V2
VgIL = 3 (3.10)
V2
IL = 3 (3.11)
2RVg
It is obvious that Equation (3.10) is the conservation of power. The input
inductor current is dc, the value is determined by Equation (3.11). Once IL is
obtained, duty ratios in Equations (3.7)  (3.9) can be solved from
dva va
C +
dt R
(dl d2) = j= da (3.12)
IL
dvb Vb
dt R
(d12 d22) = = db (3.13)
55
dvc vc
C +
dt R
(d13  d23)  dc (3.14)
According to Equations (3.12)  (3.14), one can find that the effective duty
ratios da, db and dc are sinusoidal. Since the number of unknowns dll d23 in
Equations (3.12)  (3.14) are more than the number of equations, we have
more freedom to decide duty ratios, leading to many PWM techniques. A dif
ferent modulation technique gives a different solution.
3.3 Synthesis of Continuous Sinusoidal Pulsewidth Modulation
In the continuous sinusoidal PWM (SPWM), the duty ratios dll  d23
are continuous sinusoidal functions. In general, the duty ratio of each switch
consists of a dc offset and a sinusoidal modulation. For a balanced threephase
converter, duty ratios d1l  d23 could be
1 m.
+ sin(cot)
d 3 3
312 = 3+ sin ) (3.15)
d1 D+sin Cot+
21 D
S22_1 sin 3 (3.16)
L 23i 1 DM si 27c
1 3 3 +
1 m.m 2x
d 3
56
where
Dm 1 (3.17)
The dc offset in Equations (3.15) and (3.16) is to keep the duty ratios positive.
Equation (3.15) represents the sinusoidal modulation function for the top (sin
glepoletriplethrow) SPTT switches shown in Figure 31. Equation (3.16)
represents the sinusoidal modulation function for the bottom SPTT switches
shown in Figure 31. The amplitudes of the sinusoidal modulation function for
the same switch group must be the same to constitute balanced threephase
sinusoids. The amplitudes and phases of sinusoidal modulation function for
different switch groups could be different, as long as two switch groups are
topologically independent. Note that the duty ratios in Equation (3.16) have
the same amplitude and oppose phase from the duty ratios in Equation (3.15)
that results in the optimal effective duty ratios:
2D
sm(Gt)
Sda d 1 22 D s
12 22 3 s(t  (3.18)
c dl13 23 2D
Lsinsm t +2
that is related only to the continuous sinusoidal modulation techniques.
The amplitude Dm and the phase of duty ratio da in Equation (3.18) can
be obtained from Equation (3.12), which is given by
I = 3 Vm 1 + (RC) (3.19)
m 2 RIL
57
Zd, = tan (wRC) (3.20)
where IL is determined by Equation (3.11).
Combining Equations (3.11) and (3.19), one can find the amplitude of the out
put voltage is
V  = + (cRC)2 (3.21)
D
3.4 Synthesis of SpaceVector Modulation
The balanced threephase voltages va, vb, and vc are shown in Figure 3
2. In spacevector modulation (SVM), the phase voltages are divided into six
segments, and each segment occupies 600. In each segment, one SPTT switch
in Figure 31 is permanently attached to one of the three capacitors as the
other sweeps through all three. The position of the stationary switch, as well
as the sweeping ones, are determined by sixstep sequence.
In the first segment, 00600, vb < va and vb < vc. Let d22 = 1, and d21 =
d23 = 0, i. e., S22 is on, S21 and S23 are off all the time during this segment.
The switches, Sil, S12, and S13, are switched at high switching frequency. The
corresponding duty ratios, dll, d13, and d12, are determined by Equations
(3.12)  (3.14), respectively. Taking advantage of d22 = 1 and d21 = d23 = 0, then
Equations (3.12)  (3.14) become
dva v
C + da (3.22)
dt R
d,, = d, (3.22)
58
Va Vb Vc
I \ Iv / v\ vI
0 60 120ï¿½ 1800 2400 3000 3600
Figure 32 The balanced threephase voltages.
dvb Vb
dt R
(d121) = t =db (3.23)
IL
dv v
C +
dt R
d13 R d (3.24)
From Equations (3.22) and (3.24), it can be found that d1l and d13 are sinuso
ids, and d12 is the sinusoid with the dc offset. In summary, during 00  600, the
duty ratios for the switches are
dll = Dmsin(ot) (3.25)
d13 = Dmsin (t+ (3.26)
d12 = 1+ Dmsin(o t ) (3.27)
d21 = d23 = 0 d22 = 1 (3.28)
59
where the amplitude Dm and the phase of the duty ratio d1l can be obtained
from Equation (3.22) as the following:
VmJl + (oRC)2
Id111 = Dm = 1 + (3.29)
Zda = tg (owRC) (3.30)
The effective duty ratios are
 D sin(cot)
da d 11 d 22 D iot
9m
db = d12d22 D= si tj3 (3.31)
Sd 13 23 D sin cot + 2)
Combination of Equations (3.11) and (3.29) yields
Vm = Z 1 + (o RC)2 (3.32)
In fact, the threephase boost inverter works like a dc boost converter.
When S12 is on, the inductor gets energy from the source. When either S11 or
S13 is on, the inductor transfers the energy to the load. Since only one SPTT is
switched at high frequency and the other is switched at low frequency, the
SVM has less switching loss than continuous SPWM.
The duty ratios dll  d23 for the six segments are listed in Table 3.1, and
their waveforms over one period are shown in Figure 33. It is obvious that the
duty ratio functions in the SVM are piecewise sinusoidal and have sixstep
symmetry.
60
I ,
d12
1] ^TA
d13
o I I
I d22
SI d23
0 1
00 600 1200 1800 2400 3000 3600
Figure 33 Duty ratios for spacevector modulation.
Table 3.1 Duty ratios for the threephase boost Inverter with the SVM.
STEP dl1 dl2 d13 d21 d22 d23
1 Dsin(ot) l + Dsin(ot  2) D sinOt + 2f 0 1 0
m m 3 m 3
2 1 0 0 lDsin(wt) Dmfsin (ot_2) Dmsin(ot +)
3 Dmsin((ot) Dmsin(ot) 1 +Dmsin ft+ ) 0 0
0 1 0 D sin(0t) 1  D sin (Ot Dmsin ct+
5 1+ Dmsint(Of) Dmsinot  Dsin ot+ 0 0
6 1 0 0 Dsin(ot) Dmsin(ot)2 1 Dsin (ot+)2
CHAPTER 4
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR
This chapter investigates the feasibility of largesignal linearization of
threephase PWM converters by analog linearizing pulsewidth modulator
(LPWM). The study shows that threephase PWM converters have nonlinear
relationships between the control and output voltages when they are con
trolled by the conventional analog SPWM or SVM modulators. Some sophisti
cated analog circuits may employ analog multipliers/dividers to compute the
switching instants for threephase converters to implement linearization.
However, the complexity of the resulting circuitry makes them impractical.
The firstorder linearizing PWM circuit uses integrators to compute commuta
tion instants to linearize controltooutput relationship in dc or singlephase
converters. They can also be used to control threephase converters, but, as
indicated in this chapter, the inputs to the integrators are nonlinear function
of control voltages, resulting in use of analog multipliers/dividers.
A highorder linearizing PWM modulator is developed in this chapter.
It is able to make output voltages of threephase PWM converters track con
trol signals linearly even in the nonlinear topologies. Instead of multipliers/
dividers, the highorder linearizing PWM uses only integrators with the reset,
and sample/hold to compute the switching instants for the switches. The
62
63
inputs to the integrators are just linear functions of the control and state vari
ables.
In the first section of this chapter, the firstorder LPWM is reviewed,
which is helpful to understand the concept of largesignal linearization and
analog implementation of the LPWM. The nonlinear problem, caused by the
conventional PWM modulator in threephase PWM converters, is identified in
the second section. A general way to linearize PWM converters is discussed in
the third section. Implementation of the LPWM modulator by firstorder
LPWM circuits for a threephase inverter is given in the fourth section. The
fifth section presents the highorder LPWM that linearizes threephase con
verters with simple analog circuits. The techniques to synthesize a highorder
LPWM and eliminate multipliers/dividers in the LPWM circuit of threephase
converters is discussed. An analog implementation of the highorder LPWM
for a threephase converter is derived and simulated.
4.1 FirstOrder Linearizing Pulsewidth Modulator
The carrier signal in the conventional PWM modulator has the fixed
frequency and constant slope. The duty ratio of switching signals generated by
the conventional PWM is directly proportional to the control signal. The car
rier signal in the linearizing PWM (LPWM) has the fixed frequency, but vary
ing slope. The duty ratio generated from the LPWM is a nonlinear function of
the control signal and input voltage.
64
switch Vm(t)
u(t) v(t)
+ K
 vM(t)
rese v(t) (t)T
clock IIL T
(a) (b)
Figure 41 (a) A firstorder linearizing PWM; (b) its operation
waveform.
The firstorder LPWM modulator usually consists of a resetable inte
grator and a comparator, as shown in Figure 41(a). Its operation waveform is
illustrated in Figure 41(b). When the clock signal comes, the output signal vp
of the comparator becomes high, turning on the switch in the converter. At the
same time, the integrator starts to integrate the input signal u(t). When the
integrator output reaches comparator input v(t), the output pulse drops to low
and turns the switch off, as shown in Figure 41(b).
It is supposed that the clock is sufficiently fast so that the function u(t)
and v(t) can be assumed as constant over each switching cycle. Therefore, the
amplitude of the ramp generated by the integrator is
rT
v 1 = u(t)dt = u(t) (4.1)
so
The slope of the carrier ramp varies with u(t), which is
65
Slope (t) (4.2)
T
s
The average value of u(t) over one switching cycle is given by
d(t)T,
f u(t)dt = u(t)d(t) (4.3)
so
0
which equals the comparator input v(t):
u(t)d(t) = v(t) (4.4)
In most of de converters, the relationship between the output voltage
and the duty ratio can be expressed in the form of Equation (4.4). For example,
the average output voltage Vo of a dcdc boost converter is the function of duty
ratio and the input voltage:
V
V= (4.5)
1D
Transformed into the form of Equation (4.4), then (4.5) becomes
VoD = Vo  V, (4.6)
If the boost converter is controlled by the firstorder LPWM shown in Figure
41(a), and let v(t) = Vc  Vg and u(t) = Vc, the duty ratio can be given as
V  V
D = c g (4.7)
Vc
Substitution of Equation (4.7) to (4.5) yields
Vo = Vc (4.8)
Note that the average output voltage Vo of a nonlinear boost converter can
track the control voltage linearly, as indicated in Equation (4.8).
66
Some dc converters have quadratic duty ratios in the controltooutput
relationship [11] [24]. They can still be linearized by the LPWM, as shown in
Figure 41, but adding one more integrator and gain block to it. This is
because
SDT, T
uD2 = u on)2 2 J 1 u(c)dt dt (4.9)
(so 0 Ts0 r
4.2 Nonlinear Problem in ThreePhase Converters
The nonlinear relationship between the output and control voltages
exists in most threephase PWM converters that are controlled by conven
tional PWM modulations, such as sinusoidal PWM (SPWM) and spacevector
modulation (SVM). The highfrequency carrier signal in the conventional
SPWM and SVM has a constant slope. The duty ratios of switching signals
generated by these PWMs are proportional to the control voltages, which are
not able to cancel the nonlinear dutyratiotooutput relationship of the con
verters. As a result, output voltages are not able to track the control signals
linearly. In the balanced threephase converters, the output waveform is sinu
soidal, not affected by this nonlinear controltooutput relationship. The
amplitude, however, is affected by the nonlinear controltooutput relation
ship.
As an example, a balanced threephase boost inverter, as shown in Fig
ure 42, is used to demonstrate the nonlinear problem in threephase convert
ers. It is controlled by the conventional SPWM. The input voltage Vg is dc, the
67
threephase output voltages va, vb, and vc are purely sinusoidal. The control
voltages Vcntla and Vcntlb into the SPWM are sinusoidal waveforms with the
amplitude Dn:
1 D
Vcntla =  + sin(t) (4.10)
1 Dm
Vcntlb = + 3 sin(ot 1200) (4.11)
The de offset in the control voltages is needed to guarantee the duty ratio pos
itive.
The control voltages are compared with the constantslope carrier in
the SPWM. The duty ratios of the resulting PWM signals are proportional to
5 mH
S11/ S12/ 13, :10 10 110
va Ohm Ohm Ohm
Vg 200V b 
S21 SI S22 $2 S23
St S1t4 S13T S21 S2 3 Dm
/ SPWM Modulator Vcntla
/ v  Vcntlb
Vtri ____
Figure 42 The threephase boost inverter controlled by the
conventional SPWM modulator.
68
the control voltages in Equations (4.10) and (4.11). According to the theory of
the balanced threephase inverter in Chapter 2 and the SPWM in Chapter 3,
the output voltages remain balanced sinusoidal waveforms, but their ampli
tude becomes inversely proportional to the amplitude of the control voltage:
V
V = 1 +(oRC)2 (4.12)
DM
m
Therefore, the output voltages are not able to track the control voltage linearly
in the boost converter. This nonlinear relationship is verified by the simula
tion results. The amplitudes of the output voltages for different control volt
ages for shown in Figure 43.
V
1200 [m
1000 Q = 27(60Hz)
R = 10 Ohm
C = 100 gF
800  V= 200 V
600
400
200%2 04 Dm 0.6 0'8 1.0
Figure 43 The amplitude of the output voltages versus the
amplitude of control voltages for the ideal case.
69
4.3 LargeSignal Linearization of PWM Converters
A typical PWM converter controlled by the LPWM is shown in Figure
44. The PWM converter could have single or multiple input/output variables.
The variables at the input/output side could be dc or ac. For example, in
threephase inverters, the input is dc voltage and the output are threephase
voltages, in which the output voltages are controlled variables. In threephase
rectifiers, the inputs are threephase voltages and the output is dc voltage,
where both output dc voltage and the input currents are controlled variables.
In order to simplify the explanation of largesignal linearization technique
presented in this thesis, we consider only the output voltage vo as the con
trolled variable. The objective is to make the output voltage vo track the con
trol voltage vc linearly through the LPWM.
vg(ac, dc)  S E v(ac, dc)
 POWER STAGE
ig (ac, dc) _ : io (ac, dc)
....... d(ac, dc)
: HIGHORDER vc (ac, dc)
L ,LPWM
LPWCl ic (ac, dc)
Clock
Figure 44 The converter controlled by the highorder LPWM.
70
In PWM converters, the output voltage, vo, is controlled by duty ratio d:
vo = f(d, vg) (4.13)
which is the nonlinear function of the duty ratio in most of PWM converters.
When it is controlled by the conventional PWM, the duty ratio is proportional
to the control voltage.
d = vc (4.14)
The resulting output voltage will be the nonlinear function of the control volt
age:
o = f(Vc, Vg) (4.15)
However, the largesignal LPWM in Figure 44 is synthesized by (4.13).
It is able to solve (4.13) and find duty ratio as the function of the output volt
age and input voltage:
d = f(vo,, g) (4.16)
The output voltage thus equals the control voltage:
vo = vc (4.17)
According to (4.17), the output voltage of the nonlinear PWM converter con
trolled by the LPWM is able to track the control voltage linearly. The nonlin
ear controltooutput relationship is completely eliminated without using any
feedback loop.
In general, the task of the LPWM controller is to obtain the duty ratio
by solving modulation Equation (4.16), which could be done either by digital
signal processors (DSP) or by analog circuits. Even though only analog imple
71
mentation of the LPWM is discussed in this thesis, it is intended to parallel
the recent advances in analog firstorder LPWM techniques for dc or sin
glephase converters [11, 24].
Analog implementation of the LPWM could be done by the conventional
PWM and nonlinear modulation function in Equation (4.16). To synthesize the
control voltage given by Equation (4.16), multipliers/dividers or other sophis
ticated circuits must be used. As a result, the complexity of the resulting cir
cuitry makes them impractical.
To avoid complicated circuits such as multipliers/dividers, the integra
tor (with reset) are used by the firstorder LPWM to solve modulation Equa
tion (4.16) in dc or singlephase converters [11, 24], as shown in Section 4.1 of
this chapter. The firstorder LPWMs can be used for some threephase con
verters, as long as the modulation equation does not have nonlinear terms of
control signals [59, 60].
For most threephase converters, a modulation equation (4.16) usually
contains some nonlinear terms of control signals. The synthesis of the LPWM
with firstorder modulators, therefore, will involve multipliers/dividers. The
highorder LPWM technique developed in this thesis is able to eliminate the
nonlinear terms of control voltages in the modulation equation (4.16). The
resulting circuitry, called highorder LPWM, contains integrators with reset
and hold, and also comparators. The inputs to integrators are just linear func
tions of control voltages. Therefore, the highhigh LPWM is simple and easy to
use.
72
In the following sections, a general procedure is presented to use
firstorder LPWM circuits to synthesize the LPWM for threephase converters.
Although it may end up with using multipliers/dividers in the LPWM circuit,
the synthesis procedure is still helpful to understand the linearization of
threephase converters and use firstorder LPWM modulators in threephase
converters.
4.4 Linearization by FirstOrder LPWM
The first step to linearize threephase converters by firstorder LPWM
circuits is to find the SFA equations of the converter. The derivation of the
SFA equations of a PWM converter is discussed in Chapter 2. For a PWM con
verter with M independent switches, the duty ratios of the switching signals
for these M switches are defined as:
dT = [dl, d2 ..., dM] (4.18)
After solving SFA equations of the converter, each variable in Equation (4.18)
can be expressed as a function of the output voltage vo and input voltage vg:
Pi(vg,vr)
d = ) (4.19)
Ql(vg, Vr)
dM = ( Vr) (4.20)
QMg(vg, r)
where the output voltage vo is replaced by the reference voltage vr For exam
ple, the duty ratios of the threephase boost inverter shown in Figure 31 of
Chapter 3, dll  d13, are functions of input voltage Vg and output voltages
73
determined by Equations (3.16), (3.19), and (3.20) in Chapter 3. When con
trolled by the duty ratios shown in Equations (4.19)  (4.20), the output volt
age should equal to the reference voltage. It is worth noting that the duty
ratios expressed by Equations (4.19)  (4.20) are only dependent on the refer
ence voltage and input voltage, and they are not coupled with each other.
Therefore, they can be synthesized individually by M firstorder LPMW cir
cuits.
To synthesize the duty ratios using the firstorder LPWM circuits,
transform (4.23)  (4.24) into the following forms:
Ql(vin, Vr)di = PI(vin, Vr) (4.21)
QM(vin, vr)dM = PM(vin, Vr) (4.22)
Assume that the duty ratios d1  dM are uniquely determined by the
input voltage and the reference voltages of the converter in each of the above
equations. Additionally, it is assumed that the switching frequency is suffi
ciently high, and the input voltage and the reference voltage vary slowly, so
that the input voltages and the reference voltages can be treated as constant
during each switching cycle. This is true because the sinusoidal signal, which
is needed to be synthesized in most power applications, is usually 60 Hz, but
the switching frequency could be as high as several KHz to several hundred
KHz. Various losses in the converter are neglected to simplify the analysis.
These losses can be compensated by the feedback circuit in practice. With the
above assumptions, (4.21)  (4.22) may be transformed into the integration
forms:
74
TdI
1 Qi(V, v,)dt = P,(vg, Vr) (4.23)
S 0
so
T,dM
Sf QM(V,, vr)dt = PM(Vg, Vr) (4.24)
so
0
Each of these integration equations can be implemented by a firstorder
LPWM circuit with a resetable integrator and one comparator, as shown in
Figure 45. The operation waveform can be referred to Figure 41(b).
n ' I
R v
Rese P
ClockLLL.
Figure 45 The firstorder LPWM circuit to synthesize one
of the duty ratios.
When the clock signal is coming, the Q function is integrated, and the
output of the integrator is compared with the P function. During this time, the
PWM signal vp is high, turning on the switch in the converter. When the out
put of the integrator ramps up to the P function, the PWM signal vp becomes
low, turning off the switch. The resulting duty ratio of PWM signal Vp is
75
P(V g, Vr)
d= (Vg V (4.25)
S Q(v, Vr)
The P function and Q function in the LPWM circuit are functions of the
input and reference voltages, which can be synthesized from the input and ref
erence voltages by operational circuits, such as adders/subtractors, and multi
pliers/dividers, as shown in Figure 46.
As an example, consider the largesignal linearization of a threephase
boost inverter, as shown in Figure 47. This converter consists of six switches
I Multiplier
Vg Divider
Adder
Subtractor Q
Vr
Figure 46 P function and Q function generator.
(two singlepoletriplethrow switches), but only four of these switches are
independent. This is because
d1 +dl2+dl3 = 1 (4.26)
d21 + d22 + d23 = 1 (4.27)
The sinusoidal PWM (SPWM) technique discussed in Chapter 3 is
applied in the inverter. One of choices of duty ratios is
76
D
1 m
S + sin(cot)
d 3D 3
3IDm 2r (4.28)
12 3+ sin(cot (428)
11 D
Dm
d 33
21 Dm
d2 1=  sin wt 21 (4.29)
3 3 3 ot )
d23 D
 1 m .sin ot+ 2
L    SPTT
iRL "R R JR
dll / d12 /( dl3
Va
Vb
__ LLJb "c
d21 d2 d2 d23 C C  C
ST
Figure 47 The threephase boost inverter.
The SFA statespace equations derived from the threephase inverter are
diL
L = V,  (dl  d21)v  (dl2  d22)Vb  (d13  d23)vc (4.30)
dt Vg 1 0v 2(dib 1323V
77
dva 2v  vb  v
Cd = (d d2l)iL 3R c (4.31)
dvb 2Vb Vavc
Cd = (dl2  d22z)L  3 c (4.32)
Cdt = 3R (4.32)
Under steadystate condition, Substitution of Equations (4.28) and
(4.29) into (4.30)  (4.32) and application of a simple algebra manipulation,
they yields
1
dllVb  d12a (b Va) (4.33)
di(Va Vc)+ dl2(Vb  V) c (4.34)
From Equations (4.33) and (4.34), duty ratios d1l and dl2 can be found as the
nonlinear functions of the input and output voltages:
1 2 2 2 Vg
(Va + Vb + vb ) + Va
d = (4.35)
d1 2 2 2
Va + Vb + Vc
1 2 2 2 Vg
a + b + c)+ Vb(4.36)
d12 = 2 2 2
va + vb + vc
From Equations (4.28) and (4.29), we can find that
2 2
d21 =  d11 2 = 23d12 (4.37)
then, duty ratios d21 and d22 are
1 2 2 2 Vp
3(Va + Vb + V ) Va
d21 = 2 2 2 (4.38)
Va + Vb + Vc
78
1 2 2 2 V2
(a + vb + )2b
d22 2 2 2 (4.39)
va + Vb + Vc
The analog circuit to solve Equations (4.35), (4.36), (4.38), and (4.39) is shown
in Figure 48, in which six multipliers and four firstorder LPWM circuits are
used.
The threephase boost inverter shown in Figure 42 is simulated with
the LPWM circuit shown in Figure 48 in Saber. The control voltage is
va = Vmsin(Qt) with Vm = 262 V and Q = 2n7(60Hz). The LPWM circuit is
implemented by four firstorder LPWM circuits. The simulation results of out
put and control voltages as shown in Figure 49 imply that the output voltages
are able to track the control voltages linearly. The difference in the amplitude
and in the phase of the output voltages originates from the reactive compo
nents.
In summary, the threephase converter can be linearized by firstorder
LPWM circuits. The switching instants of the switches are determined by
integrators in the LPWM. The input signals of the PWM circuit, called the P
function and Q function, are normally the nonlinear functions of the input and
output voltage in threephase converters. Analog implementation of these
nonlinear function involves multipliers/dividers, making it complicated and
not practical. However, if the P function and Q function are linear functions of
input and control voltages, the LPWM modulator for threephase converter
can be implemented by firstorder LPWM circuits without using multipliers or
dividers [59, 60].
79
Vav Va2
V2 1/3 + d21
Vc  I clock
1/2
P  r d22
XOR
f 23 2
1/2 1/3 d2clck+d22
1/3 d1 l
clock
1/2
XO
1/2 1/3 cl ck dll+dl2
Figure 48 The LPWM implemented by the firstorder LPMW
circuits for a threephase boost inverter.
80
300 von vnh Vr 280 V
200
Va Vb Vc
100.
100
200
t(ms)
17.5 20 22.5 25 27.5 30 32.5 35
Figure 49 The simulation results of output and control voltages
of the threephase boost inverter controlled by
firstorder LPWM circuits. va = Vmsin(Mt) with Vm=
262 V and 0 = 27t(60Hz).
4.5 Linearization by HighOrder LPWM
The main problem for firstorder LPWM circuits to linearize
threephase converters is that multipliers/dividers may be employed to syn
thesize the inputs to the integrators. The problem is solved by a technique
presented in this section. With the help of this technique, the SFA equations of
a threephase converter are reduced into a set of SFA equations that have only
one unknown duty ratio in each of them, and they have coefficients of linear
functions of the control voltages. Different from the firstorder LPWM imple
mentation, this technique does not need to find the expressions of duty ratios,
81
and P and Q functions, and it synthesizes the LPWM directly from the
reduced SFA equations. The resulting LPWM circuit is called highorder
LPWM because it uses more than one integrators to get one duty ratio. The
analog highorder LPWM modulator is developed for a general PWM con
verter in this section. It employs only integrators (with reset and hold) to com
pute the commutation instants of the switches. The inputs to the integrators
and comparators are linear functions of the control and input voltages. The
synthesis procedure of the highorder LPWM is demonstrated through a
threephase boost inverter. The modulator, together with the inverter, is simu
lated in Saber. The result shows that the output voltages can track the control
voltages linearly, and the highorder LPWM modulator is simple and easy to
use.
The synthesis of the highorder LPWM is based on the steadystate SFA
equations of the PWM converter, which are just linear functions of state vari
ables and duty ratios of the switches, as described in Chapter 2. In the
steadystate condition, the derivative terms in statespace equations are zero.
As an example, the steadystate SFA equations of a PWM converter with two
independent duty ratios are given by
alldll+al2dl2 = k1 (4.40)
a21d=l +a22d2 = k2 (4.41)
where coefficients all  a22, k1 and k2 are related to control and input voltages.
For the LPWM modulator, they are reference and input voltages.
82
As we know, the duty ratios d1l and d12 are slowly varying sinusoidal
signals. When the switching frequency is sufficiently high, the value of the
duty ratio in the current switching cycle can be assumed equal to the value in
the last cycle:
n n
dII _dln" (4.42)
d2n  d12n (4.43)
where the superscript n stands for the current cycle, n1 for the last cycle.
Substitution of Equations (4.42) and (4.43) into (4.40) and (4.41) yields:
al2d2 = kalldn (4.44)
A =1 (4.44)
a21dll = k2 a22d2n1 (4.45)
21 = (4245)
nI nI
where a22d12 and alld11 are sampled and held during the previous
switching cycle. They are available to solve d12 and dll, respectively, during
the current cycle. Since the switching frequency is assumed sufficiently high,
and the control and input voltages vary slowly, all the coefficients in Equa
n1 n1
tions (4.44) and (4.45), including a22dn2 and alld , can be treated as con
stant.
Obviously, if dll and d12 are solved directly from (4.40) and (4.41), their
expressions are nonlinear functions of all  a22, kl, and k2, namely, control
voltages. Using integrators to solve these functions would involve nonlinear
inputs to the integrators and multipliers/dividers in the resulting LPWM cir
cuit, as shown in Figure 48. In contrast, the coefficients in Equations (4.44)
and (4.45), all  a22, kl, and k2, are linear functions of control voltages. Using
83
integrators to solve these equations involves only linear inputs to the integra
tors. Thus, the resulting circuitry would not require multipliers/dividers to
synthesize the nonlinear inputs to the modulator circuits, making analog
implementation much easier.
In order to use analog circuits to synthesize Equations (4.44) and (4.45),
these equations are transformed into the integration forms:
Td12
SJ al2dt = klalld1l (4.46)
0
Td,
so
a21dt = k2a22dn12 (4.47)
The duty ratio d1l can be obtained by comparing the integration of a21 with
k2  a2dn1 through an integrator and a comparator. The duty ratio dl2
can be solved in the same way. However, to do so, a22d12 and a 1dn on
the left side of the equations should be available. Note that
Sf alldt=alldll (4.48)
so
Td12
S a22dt=a22d12 (4.49)
so
Then these sampled terms can be implemented by the integrator with reset
and hold, as shown in Figure 410. The integrator starts to integrate all after
reset by the RESET signal. At the moment dllTs, the integration is stopped by
the HOLD signal. The output of the integrator is held at the value of alldl,
84
v0 RESET
INTEGRATOR
HOLD
d, ls
Salldll
RESET HOLD al
al 0 t
0 Ts
Figure 410 The integrator with reset and hold.
which will be available for the next switching cycle. The signal a22d12 can be
generated in the same way.
From Equations (4.46)  (4.49) and the integrator shown in Figure 410,
the highorder LPWM circuit can be synthesized, as shown in Figure 411. Its
operation waveforms are shown in Figure 412. When the clock signal comes,
the bottom integrators, #1 and #2, start to integrate their input signals all
and a21. As soon as integrator output vo2, the integration of a21, reaches
n1
k2 a22d2 , the comparator produces a pulse S1 with the duty ratio of
dl . This pulse resets the top integrators, #3 and #4, and provides the HOLD
signal for integrator #1. Thus, the integrator output vol is held at the value of
a11dl7 , which will be used to solve a2. After reset, two integrators #3 and
#4 on the top start to integrate a12 and a22. As soon as integrator output Vo4,
the integration of a2, reaches k1  alldl , the comparator generates a pulse
S2 with the duty ratio of d12. This pulse provides the HOLD signal for #3
85
al vo4
#4S2 dl
2 n1d12
Vo3
a22 Rf S/H
a,3 cntl 
k  21 +One Sht
v Reset I
a21 R 2
P#2 n
Vol
Clock all R H
k I
k1
Figure 411 The highorder Linearizing PWM.
integrator. The integrator output vo3 is thus held at the value of a22dn2, which
will be used to solve dll in the next cycle. The reset signal for the top integra
tors is generated by a oneshot circuit. To prevent changes in the integrator
output vo3 from affecting the solution of duty ratio dll, an extra sample/hold
circuit is added to the PWM circuit. The AND gate in the circuit is used to dis
able the top comparator when solving for the duty ratio d11.
As an example, a threephase boost inverter, as shown in Figure 47, is
controlled and linearized by the highorder LPWM circuit in Figure 411. The
spacevector modulation (SVM) discussed in Chapter 3 is applied in this con
verter. In the SVM, the threephase voltages va, vb, and v, are divided into six
86
Clock
ol0 alldl
Vo2 k2 a22dn21
Reset
Vo3 a 22d 12
Vo4 klalld l
Si
d T_
S2
12 Tds
Ts
Figure 412 Waveforms of the highorder LPWM.
segments as shown in Figure 413. In each segment, one SPTT switch in Fig
ure 47 is permanently attached to one of the three capacitors as the other
sweeps through all three. The position of the stationary switch as well as the
sweeping ones are determined by a sixstepped sequence.
In the first segment, 00600, vb < va and vb < vc. Let d22 = 1, and d21 =
d23 = 0, that is, S22 is on while S21 and S23 are off all the time during this seg
ment. The switches, S11, 812 and 813, are switched at high switching fre
quency. In steady state, the averaged statespace equations are
87
:va v Vc
SSegment: \ Segment: \ Segment:
*II \ IV * \ VI
.Segment \ / Segment \ / Segment
I : I : : V
0ï¿½ 600 1200 1800 2400 3000 3600
Figure 413 Balanced threephase voltages.
0 = Vg  (dl  d2)a  (d2  d22b  (d3  d23)v (4.50)
0 = (d i d2L 2a Vb  (4.51)
3R
0 =(2Vb Vac (4.52)
0 = (d12d2iL 3R(4.52)
During the first segment,
d21 = d23 = 0 (4.53)
d22 = 1 (4.54)
Substituting Equations (4.53) and (4.54) into (4.50)  (4.52) and applying sim
ple algebra, the steadystate equations for the first segment can be trans
formed as
Vg = dllVab+dl3Vcb (4.55)
(2d11 + dl3)Vcb = (2d13 + dll)Vab (4.56)
From Equations (4.55) and (4.56), duty ratios dll, d13, and d12 can be
solved by the highorder LPWM. Note that the output voltages will track the
88
reference voltages Va, vb, and vc linearly when the boost inverter is controlled
by the LPWM to solve Equations (4.55) and (4.56). In other words, the inverter
would have lowdistortion sinusoidal waveforms at the output, and nonlinear
ity of the boost type inverter is eliminated.
If the steadystate equations for the six segments are listed, one can see
that they have the same forms as Equations (4.55) and (4.56). Thus, they can
be expressed as a general form as follows:
Vg = dxvx + dyvy (4.57)
(2dx + dy)vy = (2dy + dx)v, (4.58)
dz = 1  dx  dy (4.59)
where Vg is the dc input voltage of the inverter.
The coefficients vx and vy in Equations (4.57) and (4.58) are the refer
ence voltage signals to the LPWM. They are sixstepped piecewise sinusoidal
linetoline voltages, as shown in Figure 414. Within different sixstepped
segments, vx and Vy takes different linetoline voltages, as shown in Table 4.1,
which are synthesized from continuous threephase reference signals Va, vb,
and vc. The outputs of the modulator are PWM signals with duty ratios dx, dy
and dz. For each segment of the SVM, dx, dy, and dz, are assigned to three
switches of the inverter based on Table 4.1. During the first segment, for
instance, dx = dll, dy = d13, dz = d12. The positions of dx and dy over one complete
period are shown in Table 4.1.
89
vx
00 600 1200 1800 2400 3000 3600
Figure 414 Sixstepped reference voltage signals to the LPWM.
Table 4.1 Sixstep reference voltages and duty ratios.
vx Vy dx dy dz "1" "0" "0"
Seg. I Vab Vcb d1l d13 d12 d22 d23 d21
Seg. II vac Vab d23 d22 d21 dll d12 d13
Seg. III Vbc Vac d12 dll d13 d23 d21 d22
Seg. IV Vba Vbc d21 d23 d22 d12 d13 d1l
Seg. V Vca Vba d13 d12 dll d21 d22 d23
Seg. VI Vcb Vca d22 d21 d23 d13 dll d12
In order to use the proposed highorder LPWM, it is necessary to trans
form the preceding equations into the following forms:
dnv = V dn v (4.60)
dxy = (dnl,+ (2v Vy) (4.61)
90
Comparing Equations (4.60) and (4.61) with (4.44) and (4.45), it is not difficult
to find
a12 = Vy (4.62)
a22 = (4.63)
a21 = (4.64)
all = 2vx V (4.65)
ki = Vg (4.66)
k2 = d 1 v (4.67)
Replacing the inputs of the highorder LPWM in Figure 411 with the
sixstep reference voltages in Equations (4.62)  (4.66), the highorder LPWM
for the threephase boost inverter is then synthesized, as shown in Figure
415. It is worth noting that the inputs to the integrators are linear functions
of the reference and input voltages, and analog implementation of which
involves only adders/subtractors, as shown in Figure 415.
The threephase boost inverter, as shown in Figure 42, is simulated
with the proposed highorder LPWM circuit shown in Figure 415 in Saber.
The control voltage is va = Vmsin(Qt) with Vm = 262 V and Q = 27n(60Hz). The
inputs of the LPWM circuit in Figure 415 are the sixstep reference signals vx
and vy, which are generated from va, vb, and ve. The circuits that generate
sixstep reference signals are not shown here, but they can be easily built in
Saber by some analog switches and some comparators. The waveforms of vx
91
v vo4
R H, 4
4 Y
I 1v . s/I i['v._ i
Vx H SH 1Al/2
cntl
One Sh )t
+ v r V Reset h i Sz
R H
l#2 Sx
2Vxvy ol ^
RJ H
II 1, 1#1)
Clock Vg +
Figure 415 The highorder LPWM for threephase boost inverter.
and Vy are shown in Figure 414. The outputs of the LPWM circuits are the
PWM signal dx, dy, and dz. They will be assigned to the six switches S11  S23,
according to Table 4.1, by the encoding circuits. This circuit can be imple
mented, according to Table 4.1, by logic circuits. One of the simple encoding
circuits is given in Chapter 6. The simulation results shown in Figure 416 are
the output and control voltages. Output voltages in Figure 416 are supposed
to equal control voltages according to the theory of the LPWM synthesis. How
ever, the simulation shows that the amplitude of output voltages is a little
higher than that of control voltages, and there is a phase shift between output
and control voltages. This phenomenon is because of the reactive components
in the power converter, which will be explained in detail in Chapter 5. The
92
300  vo Vh vnr 280 V
100
100
200
300 , t(ms
17.5 20 22.5 25 27.5 30 32.5 35
Figure 416 The simulation results of output and control voltages
of the threephase boost inverter controlled by the
highorder LPWM circuit. va = Vmsin(f2t) with V =
262 V and Q = 2tn(60Hz).
simulation results for the integrator outputs and the outputs of the highorder
LPWM circuit are shown in Figure 417.
Over the one sinusoidal cycle, the switching signals for six switches in
the inverter are shown in Figure 418. It shows that each switch in the
inverter operates at high frequency and low frequency alternatively. The duty
ratios d1l  d23 for six switches can be obtained by taking the average of the
switching functions in the Saber. The results are shown in Figure 419. The
duty ratios of the SVM, solved by the highorder LPWM, are piecewise sinuso
ids, similar with the piecewise sinusoidal modulation waveforms described for
the conventional SVM method in Chapter 3. However, their amplitudes are
93
different. The modulation amplitude of the conventional SVM is proportional
to the control voltage, where as the amplitude of the duty ratios generated by
LPWM is a nonlinear function of the control and input voltages.
In conclusion, most threephase PWM converters have nonlinear con
troltooutput relationships that make the output voltages unable to track the
control voltages linearly when they are controlled by the conventional PWM
modulator. The firstorder LPWM modulator can be used to linearize the
threephase converter, but it may involve multipliers/dividers to synthesize
the inputs to the integrators, as long as there are nonlinear terms of control
voltages in the expressions of duty ratios.
The technique presented in this chapter is able to reduce the SFA equa
tions of the converter into a set of SFA equations that have only one unknown
duty ratio in each of them. The coefficients of these SFA equations are just lin
ear functions of the control voltages. The PWM circuit synthesized from these
SFA equations, called highorder LPWM, uses only integrators (with reset and
hold) to compute switching instants of the switches. The inputs to integrators
are just linear functions of control and input voltages. A highorder LPWM is
synthesized and simulated for a threephase boost inverter. The results show
that the output voltages can track the control voltage linearly. The control cir
cuit is simple and easy to use.
The synthesis technique of the highorder LPWM modulator is devel
oped for a threephase boost inverter here, but it may be extended to all the
threephase converters or multiphase PWM converters.

Full Text 
98
5.1 Analysis of HighOrder Linearizing PWM
5.1.1 Modeling the HigrhOrder LPWM
A balanced threephase boost inverter with the highorder LPWM is
shown in Figure 51. According to discussions in Section 4.5 of Chapter 4, the
modulation equations to synthesize the highorder LPWM modulator are
given by
Vgr = dnVab + d\3Vcb
(2 dn+du)vcb = (2 dl3 + dn)vab
d\2 ~ ^~d\\ ~d\3
(5.1)
(5.2)
(5.3)
where d^, d]^, and d12 are duty ratios of the PWM signals for the switches
Sn, S13, and S12, respectively; vcb and v^ are linetoline voltages obtained
from the control voltages va, v^, and vc; and Vgj. is sampled from input voltage
Figure 51 The LPWM controlled threephase boost inverter.
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115
Comparing Equations (5.60) and (5.61), one can find that the modulator is
able to follow the step inputs very quickly, and it goes into steadystate condi
tion within one switching cycle after step response.
In summary, the highorder LPWM is modeled and analyzed for a
threephase boost inverter. The sampling effects in the modulator contribute a
pole that is determined by the switching frequency. This LPWM provides a
wide bandwidth for the transfer function of the modulator. The modulator is
able to react very quickly to the step input, and it goes into steadystate condi
tion within one switching cycle according to step response measured in the
simulation. For this reason, the delay caused by the sampling effects can be
neglected when we analyze and design the highorder LPWM.
149
Since two inductors formed by signal path and its return path are on the same
core and have the same turns, commonmode currents are forced to be equal.
The same commonmode filters can be added to control voltage lines, and
input power lines of the inverter. The system with commonmode filters is
shown in Figure 640(c). Inductor values of commonmode filters used in the
experiment are also shown in Figure 640(c).
Components used in the experiment are listed in Table 6.4. Output volt
ages of the inverter are 60 Hz sinusoids, thus, AC capacitors are used in the
output of the inverter. The package of MOSFETs and diodes are TO247 that
is easy to be mounted on heat sinks by metal screw because there is no metal
contact around the hole of the package. Analog switch HI20 IS is very fast
switch with ton = 30 nS and t0ff = 40 nS. It is used to reset capacitor in integra
tors of the LPWM at high switching frequency. Analog switch CD4051 is
slower than HI201S. It is used to generate sixstep reference voltages. Opera
tional amplifier LM833 has low distortion (0.002%), low offset voltage (0.3
mV), thus, it is very suitable for reference voltage circuit. It also has a wide
bandwidth (15 MHz) without increasing external components or decreasing
stability, thus, it is suitable for integrators in the LPWM. LM311 is used as
zerocross detector in the reference circuit shown in Figure 61 and voltage
comparator in the LPWM shown in Figure 615. This devices are much less
prone to spurious oscillations. Offset balancing and strobe capability makes
LM311 easy to use in the experiment. High speed and current output of
ICL7667 make it very suitable to drive MOSFET. In the experiment, two
12
networks. Therefore, application of the statespace averaging technique to
analyze threephase converters is tedious.
Without probing into topological details, the switchingfunction averag
ing method [9, 15] treats the switch as a component in the same way as we
treat other linear components by defining the PWM switch model. This allows
derivation of statespace equations simply by inspection and application of
definitions of circuit elements, Kirchhoffs laws, and other electrical princi
ples. The switch model is derived by characterizing its low frequency property,
as explained in this section.
The singlepolemultiplethrow (SPMT) switch shown in Figure 21 is
one of the fundamental building elements in the PWM converters. The pole is
usually connected to the inductor, and the throw is connected to either the
voltage sources or the capacitors. The SPMT switch is reduced to a singlepole
doublethrow (SPDT) switch in a dc converter; it may be SPDT or a single
poletriplethrow (SPTT) switch in most threephase converters.
v*
p
1
i*
1 p
r
Pole 1
i*l
ri*2
i*k
r K l
4 *
rx M
\
\
\...
\
d*n
d 12
d*ik
d*iM
V*!
V*r
v*k
V*M
Figure 21 A singlepolemultiplethrow switch.
138
Figure 626 The output voltage, control voltage, input voltage,
and inductor current. A = 100, K = 25.9 KHz, fs = 24.5
KHz, R = 22 Q, and C = 50 pF.
0 = tan (coRC) = 22.5 (6.13)
The amplitude of the output voltage is shown to 230 V/2 in the measurement.
It is quite close to the theoretical value given by Equation (6.11):
Vom = AVmyJl + ((RC)2 = 121 V (6.14)
where Vm is the amplitude of the control signal, 2.12 V/2 in the measurement.
The difference between the measurement and theoretical value are due to
power losses. The inductor current is dc with some ripples, and its average
value is 6.84 A. The measured value and theoretical value are listed in Table
6.3 for comparison.
CHAPTER 7
SUMMARY AND CONCLUSION
This study shows that the threephase PWM converters have nonlinear
relationships between the control and output voltages when they are con
trolled by the conventional analog SPWM or SVM modulators. Some sophisti
cated analog circuits may employ analog multipliers/dividers to compute the
switching instants for threephase converters to implement linearization.
However, the complexity of the resulting circuitry makes them impractical.
The firstorder linearizing PWM circuit uses integrators to compute commuta
tion instants to linearize the controltooutput relationship for dcdc convert
ers or singlephase inverters. The firstorder LPWM can also be used to
control threephase converters. However, as indicated in this thesis, the inputs
to the integrators of the firstorder LPWMs could be nonlinear functions of
control voltages and must use analog multipliers/dividers.
A highorder linearizing PWM is developed in this thesis. It is able to
make the output voltages of the threephase PWM converter track the control
signals linearly even in the nonlinear topologies. Instead of multipliers/divid
ers, the highorder LPWM uses only integrators with the reset, and sample/
hold to compute the switching instants for the switches. The inputs to the
integrators are linear functions of the control and state variables. The key to
151
137
Tek antilB 25.OkS/s 295 Acqs
[ T ]
Figure 624 The threephase output voltages of the inverter with
R = 35 Q
Tek ana 25.OkS/s
7 Acqs
T
Figure 625 The threephase output voltages of the inverter
with R = 22 Q
77
dva
 (dn ~ d2X)iL 
2va Vb~Vc
(4.31)
dt
3 R
dvb
 (<7]2 d22)ii 
2vb~VaVc
(4.32)
dt
3 R
Under steadystate condition, Substitution of Equations (4.28) and
(4.29) into (4.30) (4.32) and application of a simple algebra manipulation,
they yields
dnvb~dnva = \(Vb~Va)
(4.33)
dn(vavc) + du(vbvc) = T"
' 12v t> c' 2 c
(4.34)
From Equations (4.33) and (4.34), duty ratios d^ and d12 can be found as the
nonlinear functions of the input and output voltages:
d ii
d\2 ~
1.2 2 2.
(v + Vl + V ) I V
3 V a b c> 2 a
2 2 2
vfl + vb + vc
1,2 2 2.
^Va + Vb + Vc) + YVb
2 2 2
V + Vl + V
a b c
From Equations (4.28) and (4.29), we can find that
2
d2\ = 3~^11 ^22 = a ^12
2
3
then, duty ratios d2i and d22 are
~(U + vt + U) V
a 0 c' 2
1,2 2 2.
^21
2 2 2
V + Vl + V
a b c
(4.35)
(4.36)
(4.37)
(4.38)
141
the prototype circuit is shown in Figure 632. The pictures of the highorder
LPWM modulator is shown in Figure 634. The top views of the reference cir
cuit and gatedrive logic are shown in Figure 633. The top view of the three
phase boost inverter is shown in Figure 635.
Figure 629 The efficiency of the prototype threephase
boost inverter.
Tek afiHH 5. OOMS/s 268 Acqs
[ T. ]
vgs
(S21)
Vds
(S2i)
25 May 2000
13:29:58
Figure 630 The switching waveforms of a MOSFET in
the inverter.
91
Figure 415 The highorder LPWM for threephase boost inverter.
and vy are shown in Figure 414. The outputs of the LPWM circuits are the
PWM signal dx, dy, and dz. They will be assigned to the six switches S23,
according to Table 4.1, by the encoding circuits. This circuit can be imple
mented, according to Table 4.1, by logic circuits. One of the simple encoding
circuits is given in Chapter 6. The simulation results shown in Figure 416 are
the output and control voltages. Output voltages in Figure 416 are supposed
to equal control voltages according to the theory of the LPWM synthesis. How
ever, the simulation shows that the amplitude of output voltages is a little
higher than that of control voltages, and there is a phase shift between output
and control voltages. This phenomenon is because of the reactive components
in the power converter, which will be explained in detail in Chapter 5. The
93
different. The modulation amplitude of the conventional SVM is proportional
to the control voltage, where as the amplitude of the duty ratios generated by
LPWM is a nonlinear function of the control and input voltages.
In conclusion, most threephase PWM converters have nonlinear con
troltooutput relationships that make the output voltages unable to track the
control voltages linearly when they are controlled by the conventional PWM
modulator. The firstorder LPWM modulator can be used to linearize the
threephase converter, but it may involve multipliers/dividers to synthesize
the inputs to the integrators, as long as there are nonlinear terms of control
voltages in the expressions of duty ratios.
The technique presented in this chapter is able to reduce the SFA equa
tions of the converter into a set of SFA equations that have only one unknown
duty ratio in each of them. The coefficients of these SFA equations are just lin
ear functions of the control voltages. The PWM circuit synthesized from these
SFA equations, called highorder LPWM, uses only integrators (with reset and
hold) to compute switching instants of the switches. The inputs to integrators
are just linear functions of control and input voltages. A highorder LPWM is
synthesized and simulated for a threephase boost inverter. The results show
that the output voltages can track the control voltage linearly. The control cir
cuit is simple and easy to use.
The synthesis technique of the highorder LPWM modulator is devel
oped for a threephase boost inverter here, but it may be extended to all the
threephase converters or multiphase PWM converters.
123
obtained from Sa Sc by logic circuits, as shown in Figure 66. The experimen
tal waveforms of sixstep signals are shown in Figure 68.
The highorder LPWM employs the discontinuous reference voltage, vj_
and v2, and input voltage Vg to determine the PWM signals S1; S2, and S3, as
shown in Figure 6l(a). vb and v2 are piecewise linetoline reference voltages,
the values of which are listed in Table 6.1 for the six steps of the SVM.
Table 6.1 The reference voltages vb and v2 for six steps.
Steps
V1
v2
I
vab
vcb
II
vac
vab
III
vbc
vac
IV
vba
vbc
V
vca
vba
VI
vcb
vca
Table 6.2 The logic table of analog multiplexer CD4051.
C(Sc)
B(Sb)
A (Sa)
Output (vb)
Output (v2)
0
0
1
vac
vab
0
1
0
Vba
Vbc
0
1
1
vbc
vac
1
0
0
Vcb
vca
1
0
1
vab
vcb
1
1
0
vca
Vba
76
dl2
dl3
1 D
1 m .
 + sin(coi)
D
 +
m
. . 2n
sin cot
D
 +
m
, 271
sin cor +
d2\
d22
d23_
1 D
1 m . .
3 ~sin(^
1 Dm ( 2k
 rsin cor
3 3 v 3
1 f 2k
 sin cor + 
(4.28)
(4.29)
The SFA statespace equations derived from the threephase inverter are
diL
L = Vg(dud2l)va(dnd22)Vb(dl3d23)vc
(4.30)
51
switching frequency related to its output. Although it has good dynamic per
formance, this technique suffers from lowfrequency harmonics and high
switching losses.
3.2.5 SelectiveHarmonicElimination PWM
The selectiveharmonicelimination PWM technique [53] formulates a
waveform that is chopped M times and possesses odd quarterwave symmetry,
and contains the information about where the pulse starts or ends. Therefore,
any M harmonics can be nullified by solution of the corresponding M simulta
neous transcendental equations, which need extensive numerical calculation.
This technique is intended to attack the harmonics by suppressing an arbi
trary number of them in the output spectrum.
3.2.6 SigmaDelta Modulation
Sigmadelta modulation [5457] consists of a hysteresis comparator and
an integrator. The integrator estimates the reference voltage from the modu
lated PWM signal by low pass filter averaging. The estimated voltage is com
pared with the actual reference voltage through a hysteresis comparator to
generate the error signal, which is quantized to form the PWM signal. There
fore, the output voltage, which is equivalent to the average of the modulated
PWM signals, is able to follow the reference voltage within the hysteresis
band. Sigmadelta modulation suffers from the problem of variable frequency
and filter stability problems at high frequencies [56]. Although attempts have
119
as the reference signal and generates balanced threephase voltages vj, and vc
from it. In balanced threephase, lags va by 120; vc leads va by 120. Thus,
vc can be generated by a leading phaseshift circuit, and v^ is obtained just by
adding va and vc, as shown in Figure 62.
The phaseshift circuit in Figure 62 consists of one capacitor and three
resistors. The resistors Rj and R2 decide the voltage gain that is one when
they are equal. The resistor Rp and capacitor C determines the amount of
phaseshift that is given by the following equation:
head = <61)
Given = 4.7 pF, co = 27c(60Hz), Rp = 325.8 Q, vc leads va by
Â§lead = 120 (6>2)
For a balanced threephase system, v^ can be obtained by
Vb = ~(Va + Vc) (63)
that is done by an inverted adder in Figure 62. The experimental waveforms
are shown in Figure 63.
The threephase linetoline voltages are used to synthesize the sixstep
reference voltages vj_ and v2 in the SVM. They are simply generated by sub
tracting two linetoneutral voltages, as shown in Figure 64. Their experi
mental waveforms are shown in Figure 65.
The sixstep signals Sgl Sg6 are used to assign the PWM signals pro
duced by the LPWM to the six switches in the boost inverter, as shown in Fig
46
where the slope of the ramp signal in the modulator varies with the input volt
age. Its application in the nonlinear PWM converters is published in ref 12, in
which good line voltage regulation is obtained for linear and nonlinear con
verters. However, the controltooutput gain is zero for the boost converter and
nonlinear for quadratic converters [11].
The feedforward control is adapted to a pulsewidth modulation [11]; it
is called feedforward PWM (FFPWM). With the FFPWM, any linear or non
linear PWM converter can be linearized. The steadystate controltooutput
relationship of the converter becomes linear regardless of operating condi
tions. The FFPWM not only implements largesignal linearization of PWM
converters, but also reduces the source disturbance on the output voltage of
the converter. The FFPWM has no stability problems and no effects on con
verter output impedance. If tight output voltage regulation is required, a
smallsignal voltage feedback can be used with less difficulty and with
improved response compared with the conventional PWM modulation.
Although the technique in ref [11] does not provide a general modulator cir
cuit as the onecycle controller, it provides us with a general way to synthesize
the largesignal linearizing PWM circuit.
3.1.3 PeakCurrent Mode Control
The peakcurrent mode control is widely used in dc or singlephase con
verters [3537], in which the peak inductor current always equals the refer
ence current, regardless of all other operating conditions. This control method
121
Figure 65 The experimental waveforms of threephase line
line reference voltages.
128
hold circuit in Figure 611 is implemented by one capacitor and one OpAmp
as shown in Figure 614 (b). When the sampling signal Scnti is available,
switch HI201S is turned on. Capacitor C is charged to input voltage vs. After
the switch is turned off, this value will be held until next sampling signal is
available. The prototype of the highorder LPWM is shown in Figure 615. The
reset signal for the integrator #1 and #2 comes from the clock signal. Instead
of the oneshot circuit to generate the reset signal for the integrators #3 and
#4, the prototype circuit uses Clock Sx to generate the reset signal for them.
When there is a clock signal, the integrators #1 and #2 are reset; the
value of integrator #3 is sampled by this signal and held in the S/H circuit for
the calculation of PWM signal After the clock signal, two integrators on
the bottom start to integrate their input signals until the output of integrator
#2 reaches the input at the + pin of the comparator. At this moment, the
PWM signal S]^ goes to zero, and the output of integrator #1 is held for the cal
culation of the PWM signal S2. During the period is on, the two integrators
on the top are kept reset, and S2 is zero. As soon as Sj becomes zero, these
integrators start to integrate their inputs, and S2 becomes one. When the out
put of integrator #4 equals the input of the comparator, the PWM signal S2
goes to zero, and the output of integrator #3 is held for next calculation. The
PWM signal S3 is generated by Sj S2
The operation of the highorder LPWM is verified by the experimental
results of the prototype circuit. Figure 616 shows the outputs of integrators in
the highorder LPWM modulator; input and output waveforms of comparators
112
The bandwidth of the highorder LPWM modulator is obtained from
Equation (5.54):
f3clB
9_ 1
87t(sin(0 + 120) sin(0 120))TS
(5.57)
The bandwidth is a function of sampling period Ts and angle 0. At 0 = 0, the
bandwidth is the smallest, and the bandwidth is given by
f * = hi (158)
For fs = 24 KHz, f 3dB = 2.9 KHz. Given Vg = 5 V, Vm = 10 V, fs = 24 KHz, A = 1,
the frequency response of Km of the highorder LPWM is shown in Figure 55.
The 3dB bandwidth is shown around 2.9 KHz.
30
40
50
60
201og  Km
dB(V)
no sampling
with sampling
Figure 55 The frequency response of the highorder LPWM.
94
(V) 5
O
26.6 26.62 26.64 26.66 26.68 26.7
Figure 417 The simulation results for integrator outputs and
the LPWM outputs.
26
the steadystate analysis, especially the dynamic analysis with the timevari
ant equivalent circuit, is tedious and difficult. Therefore, it must be trans
formed into the ofb coordinates or odq coordinates to remove the time
dependency. Since abcodq transformation leads to two coupled subcircuits
[21], the resulting equivalent circuit is not convenient for analysis. The pro
posed timeinvariant equivalent circuit in this thesis, however, is derived in
the ofb coordinates, in which two subcircuits are completely decoupled, mak
ing the analysis much easier [22] and allowing one to write down answers by
inspection.
2.3 ABCOFB Transformation
The abcofb transformation matrix T transforms a timevarying vector
Xgbg in the stationary (abc) coordinates into a timeinvariant complex vector
in the rotating (ofb) coordinates according to
xabc ofb
(2.22)
X abc
(2.23)
where, for a balanced threephase system with positive phase sequence,
X
a
*cos(0p
x abc
xb
=
xcos(0^27t:/3)
X
c
xcos(0^. + 2tu/3)
(2.24)
where
73
determined by Equations (3.16), (3.19), and (3.20) in Chapter 3. When con
trolled by the duty ratios shown in Equations (4.19) (4.20), the output volt
age should equal to the reference voltage. It is worth noting that the duty
ratios expressed by Equations (4.19) (4.20) are only dependent on the refer
ence voltage and input voltage, and they are not coupled with each other.
Therefore, they can be synthesized individually by M firstorder LPMW cir
cuits.
To synthesize the duty ratios using the firstorder LPWM circuits,
transform (4.23) (4.24) into the following forms:
Q\(vin vr)d\ = Pl(vin>vr) (4'21)
Qm^u^Mm = PMVinVr) (4'22>
Assume that the duty ratios d^ djyj; are uniquely determined by the
input voltage and the reference voltages of the converter in each of the above
equations. Additionally, it is assumed that the switching frequency is suffi
ciently high, and the input voltage and the reference voltage vary slowly, so
that the input voltages and the reference voltages can be treated as constant
during each switching cycle. This is true because the sinusoidal signal, which
is needed to be synthesized in most power applications, is usually 60 Hz, but
the switching frequency could be as high as several KHz to several hundred
KHz. Various losses in the converter are neglected to simplify the analysis.
These losses can be compensated by the feedback circuit in practice. With the
above assumptions, (4.21) (4.22) may be transformed into the integration
forms:
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FILES
CHAPTER 6
IMPLEMENTATION AND EXPERIMENTAL RESULTS OF
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR
A highorder linearizing pulsewidth modulator (LPWM) is constructed
to control a prototype threephase boost inverter in this chapter. The whole
test system is shown in Figure 6l(a). The ideal switch in Figure 6l(a) is
implemented by a MOSFET and a diode, as shown in Figure 61(b). A passive
snubber consisting of a capacitor and a resistor is put in parallel with the
MOSFET to reduce di/dt noise and protect MOSFET from overheat. The PWM
algorithm used for the inverter is spacevector modulation (SVM), also called
sixstep modulation, that is discussed in Section 3.4 of Chapter 3 and Section
4.5 of Chapter 4.
The reference circuit in Figure 61(a) generates balanced threephase
voltages from a singlephase signal va. It also provides two sixstep reference
voltages Vi and v2 for the highorder LPWM, and sixstep signals Sgl Sg6 for
gatedrive logic. The highorder LPWM in Figure 6l(a) synthesizes PWM sig
nals, Si S3, from the sixstep reference v and v2, and input voltage Vg. It
provides the PWM signals Si S3 to gatedrive logic. The gatedrive logic in
Figure 61(a) assigns three PWM signals Sj S3 to the six switches in the
inverter based on the sixstep signals Sgl Sg6. Isolation circuits, MOSFET
drivers, and floating power supplies are not included for simplicity.
116
47
may be considered as a largesignal linearizing PWM in terms of linearization
of the inductor current, such as the input current of the acdc converter. The
output voltage, however, is still controlled by a nonlinear controltooutput
relationship. Therefore, this control method needs an extra voltage feedback
loop to linearize the output voltage and keep it stable. The current mode con
trol has inherent advantages, such as fast dynamic response, automatic cur
rent protection, and so forth.
It is important to note that the carrier signals used in the above PWM
methods are not constant, but they vary with the control signal from one
switching cycle to another. They are herein called the PWM with a varying
slope carrier. The duty ratio generated from them is a nonlinear function of
the control signal. This is different from the conventional PWM with a con
stant slope carrier. It is also worth noting that the slope of the carrier signal of
PWM with varyingslope carrier is constant, that is, a straight line even
though the slope rate changes from one switching cycle to another. The carrier
used in refs 31 33, however, is nonlinear, which could be the exponential
function.
The LPWM modulators previously discussed are general PWM methods
suitable for all dc or singlephase PWM converter topologies, including linear
and nonlinear converters. The LPWM circuits can be implemented by simple
analog circuits, usually integrators and comparators. Although the PWM
methods [3841] are also able to implement largesignal linearization of the
21
determined by the SFA statespace equations. The turns ratios of those dc
transformers in the equivalent circuit are effective duty ratios in the SFA
equations [22],
As an example, the threephase boost inverter, as shown in Figure 23,
is used to demonstrate the derivation of the equivalent circuit from the SFA
statespace equations. The SFA equations of the threephase boost inverter in
Section 2.1.3 are organized and rewritten as follows:
II
*JI
3 l^3
V8daV
a~dbVbdcVc
(2.15)
dv
c
= daiL~
2va~Vb~Vc
(2.16)
dt
3 R
dvb
= db^L ~~
2vb ~Va~Vc
(2.17)
dt
3 R
o I
>
1
t.
1
u
II
2vc~vbva
(2.18)
~ dt
3 R
A dc transformer is shown in Figure 24. Its turns ratio is determined
by the duty ratio of the switching signal of the PWM switch. The duty ratios
da, d^, and dc in the SFA equations shown in Equations (2.15) (2.18) can be
modeled by the dc transformer shown in Figure 24. The connection relation
ships between the transformers and other components in the boost inverter is
defined by Equations (2.15) (2.18). The resulting equivalent circuit of the
threephase inverter is shown in Figure 25. If the capacitor voltages va, v^,
and vc are reflected from the output side to the input side, one can easily find
that the inductor voltage of the equivalent circuit in Figure 25 is the same as
25
The only difference between the two simulation results is that the realtime
simulation contains the highfrequency ripple, but the equivalent circuit sim
ulation has no ripple. The equivalent circuit produces exactly the lowfre
quency components of the output voltages. One interesting result obtained by
both simulations is that the output voltage has a 32 phase shift from the con
trol voltage, and the amplitude is also different from the value (222 V), pre
dicted by the conversion ratio in Table 1.2 of Chapter 1, as shown in Figure 2
8. This interesting result can be easily predicted by the steadystate analysis
of the ofb equivalent circuit of the threephase boost inverter, which will be
presented in the next section.
Figure 28 The realtime simulation results showing the phase
shift and amplitude of the output voltages of the
threephase boost inverter.
The equivalent circuit in Figure 25 is derived in the abc coordinates
that is a timevariant circuit. Although it is effective for the fast simulation,
79
Figure 48 The LPWM implemented by the firstorder LPMW
circuits for a threephase boost inverter.
144
(a)
Figure 634 (a) Top view of the reference circuit; (b) top view
of the gatedrive logic.
140
The highorder LPWM modulator has good line voltage regulation. This
is because any change of input voltage Vg will make duty ratio change in the
same direction through the LPWM modulator. Therefore, the output voltage,
inversely proportional to the duty ratio, can be kept constant. To test the line
voltage regulation of the modulator, control voltage is kept constant; the out
put voltage is measured for different input voltages. The curve of output volt
age versus input voltage is shown in Figure 628. It is tested under the
condition of Vmpk =2.28 V and A = 53. When input voltage is varied from 45 V
to 85 V, the steadystate line voltage regulation is in the range of 7%.
The efficiency of the inverter is measured, as shown in Figure 629. The
switching waveform Vgs and Vs of one MOSFET are shown in Figure 630.
The voltage across the inductor is shown in Figure 631. Finally, the picture of
150
140
130
120
110
100
Figure 628 The input voltage regulation of the high order
LPWM.
Vopk(V)

XX
X
X
X
X

ideal value
x x measurement



Vg(V).
40 50 60 70 80 90
78
^22
1,2 2 2,
(V + Vl + V ) Vl
2' a b c> 2
2 2 2
a b c
(4.39)
The analog circuit to solve Equations (4.35), (4.36), (4.38), and (4.39) is shown
in Figure 48, in which six multipliers and four firstorder LPWM circuits are
used.
The threephase boost inverter shown in Figure 42 is simulated with
the LPWM circuit shown in Figure 48 in Saber. The control voltage is
va = Vwsin(Qr) with Vm = 262 V and Q = 2k(60Hz). The LPWM circuit is
implemented by four firstorder LPWM circuits. The simulation results of out
put and control voltages as shown in Figure 49 imply that the output voltages
are able to track the control voltages linearly. The difference in the amplitude
and in the phase of the output voltages originates from the reactive compo
nents.
In summary, the threephase converter can be linearized by firstorder
LPWM circuits. The switching instants of the switches are determined by
integrators in the LPWM. The input signals of the PWM circuit, called the P
function and Q function, are normally the nonlinear functions of the input and
output voltage in threephase converters. Analog implementation of these
nonlinear function involves multipliers/dividers, making it complicated and
not practical. However, if the P function and Q function are linear functions of
input and control voltages, the LPWM modulator for threephase converter
can be implemented by firstorder LPWM circuits without using multipliers or
dividers [59, 60].
82
As we know, the duty ratios d1;L and d12 are slowly varying sinusoidal
signals. When the switching frequency is sufficiently high, the value of the
duty ratio in the current switching cycle can be assumed equal to the value in
the last cycle:
j n~ 1 j
d ii =n
(4.42)
> n 1 n
12 =12
(4.43)
where the superscript n stands for the current cycle, n1 for the last cycle.
Substitution of Equations (4.42) and (4.43) into (4.40) and (4.41) yields:
,n i jn ~1
a1212 ~ ^1 ~ allll
(4.44)
7 ,n 1
a2laU K2 ~ a22a\2
(4.45)
where a22d2 and z, 1 are sampled and held during the previous
switching cycle. They are available to solve d12 and d11; respectively, during
the current cycle. Since the switching frequency is assumed sufficiently high,
and the control and input voltages vary slowly, all the coefficients in Equa
tions (4.44) and (4.45), including
#22^ 12 1 and aud"n 1, can be treated as con
stant.
Obviously, if dn and d12 are solved directly from (4.40) and (4.41), their
expressions are nonlinear functions of a^ a22, k^, and k2, namely, control
voltages. Using integrators to solve these functions would involve nonlinear
inputs to the integrators and multipliers/dividers in the resulting LPWM cir
cuit, as shown in Figure 48. In contrast, the coefficients in Equations (4.44)
and (4.45), an a22, kj, and k2, are linear functions of control voltages. Using
Ill
. = 3 = 3 Vg
m p, 9 2 2
V 2ABV;nTsP(s)
(5.51)
dm 3 A
^ 2B2V,nTsP(S)
(5.52)
where
5 = sin(0 + 120) sin(0 120)
P(s) = s + A 1
4527\
(5.53)
(5.54)
From the smallsignal analysis, we can find that the sampling effects
contribute a pole in the transfer functions of the highorder LPWM. The pole
is related to the sampling frequency. Increase of the switching frequency will
move away the pole and make the bandwidth wider. At s = 0, Km and Kg are
found as the follows:
K
2 V,
\s = 0
(5.55)
K
2 A
= 0
v 3V
g m
(5.56)
Comparing the above equations with Equations (5.16) and (5.17), one can find
that the gain Km and Kg, derived without considering the sampling effects, are
the dc gains of the transfer function of the highorder LPWM with sampling
effects. When the sampling effects are involved, these gains reduce with the
frequency.
38
2D.
(2.50)
that is the same as that derived from the equationoriented method [9]. Sub
stituting Equations (2.49) into (2.50) and applying Equation (2.29), the phasor
of the output voltage va can be found as
Va = (! jCIRC)
(2.51)
The inductor current obtained from Figure 215 is a dc current, which is
3 V 2
iL = f(i + (a*c) )
2D R
m
(2.52)
For Dm = 0.9, Vg = 200 V, Q = 2n(100 Hz), R = 10 Ohm, C = 100 iF,
V = 262Z32. This predicted output voltage agrees well with that obtained
from realtime simulation, as is evident in Figure 28. Note that the reactive
elements appears in the steadystate variables, introducing a righthalfplane
zero. This righthalfplane zero causes some phase shift to the output voltage.
In order to reduce the phase shift,
Q (p (2.53)
2.6 Graphical SmallSignal Analysis
As shown in Figure 213, the two transformers in the ofb equivalent cir
cuit are the same. They can be combined into one transformer with the turns
ratio De, as shown in Figure 216. The transformer in the equivalent circuit
120
Tek lifiliH 25.0k5/S 549 Acqs
E T ]
C2 Freq
59.665 HZ
C2 Ampl
2.08 V
20 May 2000
22:32:03
Figure 63 The experimental waveforms of threephase refer
ence voltages.
Figure 64 The circuit to generate threephase linetoline ref
erence voltages.
160
tors, 1986 Annual Meeting Record of IEEE Industry Applications
Society, New Orleans, 1986, pp. 244251.
[50] BongHwan Kwon, Byungduk Min, A Fully SoftwareControlled
PWM Rectifier with Current Link, IEEE Transactions on Indus
trial Electronics, vol. 40, no. 3, July 1993, pp. 255363.
[51] G. S. Buja, G. B. Indri, Optimal Pulsewidth Modulation for Feed
ing AC Motors, IEEE Transactions on Industry Applications, vol.
IA11, January/February 1977, pp. 3844.
[52] A. B. Plunkett, A Current Controlled PWM Transistor Inverter
Drive, IEEE Industry Applications Society Record, 1979, pp. 785
792.
[53] J. B. Casteel, R. G. Hoft, Optimum PWM Waveforms of Micropro
cessor Controlled Inverter, IEEE Power Electronics Specialists
Conference Record, Syracuse, 1978, pp. 243250.
[54] P. D. Ziogas, The Delta Modulation Techniques in Static PWM
Inverters, IEEE Transactions on Industry Applications, vol. IA
11, March/April 1981, pp. 199204.
[55] G. Joos, P. D. Ziogas, D. Vincenti, A Model Reference Adaptive
PWM Technique, IEEE Power Electronics Specialists Conference
Record, Milwaukee, 1989, pp. 695703.
[56] A. H. Chowdhury, A. Mansoor, M. A. Choudhury, M. A. Rahman,
Online Improved Inverter Waveform by Variable Step Delta Mod
ulation, IEEE Power Electronics Specialists Conference Record,
Taipei, 1994, pp. 143148.
[57] Jose R. Espinoza, Geza Joos, Phoivos D. Ziogas, A General
Purpose Voltage Regulated Currentsource Inverter Power
Supply, Proceedings of IEEE Applied Power Electronics Confer
ence, San Diego, 1993, pp. 778784.
[58] T. J. Liang, Direct Amplitude Control Algorithm for Microcom
puterbased Pulsewidth Modulation Inverter, IEEE Power Elec
tronics Specialists Conference Record, Atlanta, 1995, pp. 319325.
[59] Chongming Qiao, Keyue M. Smedley, A General ThreePhase PFC
Controller Part I. for Rectifiers with a ParallelConnected Dual
Boost Topology, Proceedings of the 1999 IEEE Industry Applica
tions Conference, vol. 4, Phoenix, 1999, pp. 25042511.
106
Hl(s) = H(s) + KgG(s)
(5.30)
To simplify the derivation, Kg in (5.17) is expressed by De and Vg
(5.31)
Substituting (5.29) and (5.31) into (5.30), then Hl can be found as
(5.32)
The results in Equation (5.32) show that the inputtooutput transfer function
Hl is ideally zero at dc when the threephase boost inverter is controlled by
the highorder LPWM. Compared with the threephase boost inverter without
the LPWM control, as shown in Equation (5.28), the audiosusceptibility of the
LPWMcontrolled threephase boost inverter, as shown in (5.32), is signifi
cantly reduced.
To appreciate this improvement, we use LPWM and conventional PMW
to control threephase boost inverter separately, and compare the amplitudes
of output voltages in both cases. The simulation results are shown in Figure 5
4. According to simulation results, the steadystate amplitude of output volt
ages, controlled by conventional PWM, would vary with the input voltage;
however, the steadystate amplitude of output voltages, controlled by the
LPWM, would not be affected by the input voltage.
124
The relationship between steps and reference voltages in Table 6.1 can be
implemented by analog multiplexers. The logic table of multiplexers is shown
in Table 6.2, in which A, B, and C are input states of the multiplexer. The cir
cuit implementation is show in Figure 69. The experimental waveforms of
sixstep voltages Vj and v2 are shown in Figure 610.
vac vba vbc ^cb ^ab vca^aSj) Sc SnSv, ^ab ^bc vac ^ca ^cb
23456 ABC
CD4051
OUTPUT
vl
ABC 1 2 3 4 56
16
CD4051
OUTPUT 8
,+5 V
d= 0.1 gF
v2
Figure 69 Analog multiplexers to generate reference voltages
Vl and v2.
The highorder LPWM simulated in Section 4.5 of Chapter 4 is re
drawn in Figure 611. The integrator shown in Figure 611 is implemented by
an OpAmp and two analog switches in the prototype circuit, as shown in Fig
ure 612. The analog switch HI20 Is is on when the control signal is low; it is
off when the control signal is high. Thus, one inverter is needed to invert the
control signal so that the switch is on when the external control signal is high;
it is off when the external control signal is low. The operation waveforms of
this integrator are shown in Figure 613. At the beginning of switching cycle,
capacitor C is discharged by reset signal. The output voltage v0 of the integra
45
3.1.1 OneCycle Control
The onecycle control [10, 2328] has been widely used in various dc or
singlephase PWM converters. When the onecycle control was proposed in ref
10, the aim was to make the output voltage of the dc buck converter and Cuk
converter follow the control voltage tightly without being disturbed by input
voltage. It was subsequently proven that this control method can be easily
used in other topologies, controls, and applications [23] [28]. For instance, it
can make the input current track the sinusoidal input voltage, allowing unit
power factor to be obtained [25]. The basic concept of the onecycle control is to
force the average of the switchedvariable, such as the diode voltage in the
buck converter, to be proportional to the control variable in each switching
cycle. Therefore, a onecycle controller can make the output voltage propor
tional to the control voltage, that is, transform a switching power converter
into a linear power amplifier in a largesignal sense. The onecycle controller
developed in ref 24 is a generalized circuit that can be used by any dc or sin
glephase PWM converter. In addition to the largesignal linearization of
PWM converters, the onecycle controller has some advantages over the con
ventional PWM techniques, such as the switching loss compensation, good line
voltage regulation, and stable and simple control circuits.
3.1.2 FeedForward Pulsewidth Modulation
The feedforward control is mostly used in the linear buck converter or
buckderived converters to reduce source disturbance on the output voltage,
18
Combination of Equations (2.7) (2.9) yields
L77 = VSdaVadbVbdcVc (2.10)
where da, d^, and dc are effective duty ratios:
da = ~d2\ db = dn~d22 dc = d\2d22 (2.11)
Since the current through the switch is the product of the inductor cur
rent and the duty ratio of the switch according to Equations (2.4) (2.6), the
capacitor currents can be derived by applying Kirchhoffs current law as fol
lows:
dv
C 
(d\i ~ d2i)iL 
2va~Vb~Vc
= dalL
2va~Vb~Vc
(2.12)
dt
3 R
3 R
rdvb
{d i2 d22)ib
2vb~Va~Vc
~ db^L~
2vb ~Va~Vc
(2.13)
dt
3 R
3 R
dvc
C 
(d13 d22)iL
2vc ~Vb~Va
= dciL
2vc~VbVa
(2.14)
dt
3 R
3 R
Equations (2.10) and (2.12) (2.14) are called SFA statespace equations of the
threephase boost inverter. Although they are derived for the boost inverter,
the switch model and the derivation procedure are general to other PWM con
verters.
The SFA statespace equations of the PWM converter are derived with
out knowledge of any PWM strategy and thus are general to any PWM modu
lation scheme: continuous sinusoidal PWM, spacevector modulating, and so
forth. Once a specific PWM modulation technique is applied to the converter,
the switching patterns and duty ratios in the SFA equations are known. For
31
6d(i) = j'm(T)xx*d
(2.37)
The pole voltages and the throw currents can be expressed as
V
d i
pa
a 1
Vpb
V pabc
db\
V
d i
L pc\
cl
v. = d , v,
ts abe, 1 ts
(2.38)
lts
d i d i i d i
a 1 >1 cl
pa
[pb
pc
T
~ dabc, \ ipabc
(2.39)
rn
where dabc is the transpose matrix of dat,c. Note that the voltage reference
node of the proceeding equations is assumed to be vt_.
Application of Equations (2.22) (2.28) to (2.38) and (2.39) yields
T
V p ofb dofb, 1 Vts
*ts ^dofb,0 *pofb
!(: m
where (d j) is the conjugate transpose of d0ft, ^
(2.40)
_ Sn Sn MdtpIr
4 mc
, _rj3 J3n 
d ri i D c
ofb, 1 _ 2 4 ^
(2.41)
The ofb model for the threephase SPDT switches is, as shown in Figure 2
10(c). Note that the variables for the transformer in the ofb coordinates are
generally complex. For a complex transformer, such as the one whose turns
ratio is l:dbwl in Figure 210(c), the transformation relationships are
pbw dbw\vts ltsbw abwllpb
= d,
w
(2.42)
where d*kwl is the conjugate of dbwl.
55
(3.14)
According to Equations (3.12) (3.14), one can find that the effective duty
ratios da, d^ and dc are sinusoidal. Since the number of unknowns d1;L d23 in
Equations (3.12) (3.14) are more than the number of equations, we have
more freedom to decide duty ratios, leading to many PWM techniques. A dif
ferent modulation technique gives a different solution.
3.3 Synthesis of Continuous Sinusoidal Pulsewidth Modulation
In the continuous sinusoidal PWM (SPWM), the duty ratios d;Q d23
are continuous sinusoidal functions. In general, the duty ratio of each switch
consists of a dc offset and a sinusoidal modulation. For a balanced threephase
converter, duty ratios d^ d23 could be
(3.15)
(3.16)
85
Figure 411 The highorder Linearizing PWM.
integrator. The integrator output vo3 is thus held at the value of a22d"2, which
will be used to solve d1;L in the next cycle. The reset signal for the top integra
tors is generated by a oneshot circuit. To prevent changes in the integrator
output vo3 from affecting the solution of duty ratio d11; an extra sample/hold
circuit is added to the PWM circuit. The AND gate in the circuit is used to dis
able the top comparator when solving for the duty ratio d;Q.
As an example, a threephase boost inverter, as shown in Figure 47, is
controlled and linearized by the highorder LPWM circuit in Figure 411. The
spacevector modulation (SVM) discussed in Chapter 3 is applied in this con
verter. In the SVM, the threephase voltages va, v*,, and vc are divided into six
127
At the moment t = dTs, the input signal Vj is disconnected from the integrator
by the Hold signal, and the output voltage v0 will be held at
v
O
dh
RCVi
dTsKvt
(6.5)
o
where vÂ¡ is assumed constant during each switching cycle, and K is the gain of
the integrator. If time constant RC is designed equal to the switch period Ts,
then,
= dv;
(6.6)
The inputs to the integrators of the LPWM are sixstep voltages Vj and
v2, linear functions of control voltages generated by analog multiplexers in
Figure 69. The input 2vilv2 is the linear combination of vx and v2 that can be
implemented by a simple subtractor, as shown in Figure 614 (a). The sample/
Figure 614 (a) The subtractor circuit; (b) sample and hold.
CHAPTER 2
MODELING AND ANALYSIS OF THREEPHASE CONVERTERS
This chapter presents the modeling and analysis techniques for three
phase PWM converters. These techniques are important for the synthesis and
analysis of the linearizing pulsewidth modulation and threephase PWM con
verters.
This chapter consists of six sections. The first section characterizes the
lowfrequency property of the PWM switch with switchingfunction averaging
(SFA) technique. The derivation of the SFA statespace equations of a three
phase converter is presented. The second section transforms the SFA state
space equations into an equivalent circuit that is used for fast simulation. The
third section reviews the abcofb transformation that is applied to the time
variant equivalent circuit to remove time dependency. The fourth section pre
sents the graphical models for all components of the threephase converter in
the ofb coordinates. This section also demonstrates how to construct the time
invariant equivalent circuit of a threephase converter in the ofb coordinates.
The fifth section solves the ofb equivalent circuit graphically for the steady
state analysis. The sixth section derives the smallsignal equivalent circuit by
perturbing the control and input variables in the steadystate ofb equivalent
10
154
The highorder LPWM and the analysis techniques in this thesis are
developed for the balanced threephase converter. How to deal with the unbal
anced threephase or multiphase systems is still confusing. The largesignal
and smallsignal models of the converter and modulator are derived in this
thesis, but how to use these models to implement closeloop control still needs
more time and effort. The prototype modulator is implemented by discrete
analog components that consume a relatively large board area and need time
to make it work. It is hoped that the modulator can be integrated in the
future. This thesis is just a small step in a long journey. More challenging
work lies ahead requiring more time, effort, and innovative mind.
107
200
(V)
150
Vg = 200 V, 1 = 27t(100 Hz),
R = 10 Ohm, C = 100 pF
(a)
input voltage Vg with step disturbance
100
Figure 54 (a) input voltage; (b) amplitudes of output voltages
controlled by LPWM, and by conventional PWM.
5.2 Sampling Effects in HighOrder LPWM
5.2.4 Modeling the HighOrder LPWM with Sampling Effects
This section discusses modeling of the highorder LPWM that includes
sampling effects. Considering the sampling effects, the modulation equations
to synthesize the highorder LPWM modulator are given by
dl3 Vcb = Vgr~d 11 Vab
,n 1 ,n 1 N i ,n .
dUVcb = 2(/ll Vab) +odl32v ab~vcb)
1 ,n
2C
dn= ^ dii ~ di3
(5.33)
(5.34)
(5.35)
With the above modulation equations, the synthesis of the highorder LPWM
by analog circuits will not involve multipliers/dividers that have been dis
cussed in Section 4.5 of Chapter 4. In modulation equations (5.33) (5.35), Vgp
va)3, and vcb are known. The task of modeling the highorder LPWM is to solve
92
Figure 416 The simulation results of output and control voltages
of the threephase boost inverter controlled by the
highorder LPWM circuit. va = Vmsin(Clt) with Vm =
262 V and Q = 27t(60Hz).
simulation results for the integrator outputs and the outputs of the highorder
LPWM circuit are shown in Figure 417.
Over the one sinusoidal cycle, the switching signals for six switches in
the inverter are shown in Figure 418. It shows that each switch in the
inverter operates at high frequency and low frequency alternatively. The duty
ratios d1;L d23 for six switches can be obtained by taking the average of the
switching functions in the Saber. The results are shown in Figure 419. The
duty ratios of the SVM, solved by the highorder LPWM, are piecewise sinuso
ids, similar with the piecewise sinusoidal modulation waveforms described for
the conventional SVM method in Chapter 3. However, their amplitudes are
113
To appreciate how quick the highorder LPWM is, this LPWM is tested
in Saber for the step response. The test circuit is shown in Figure 56. As
shown in Figure 56, at 0 =0, the amplitude of the reference signals jump
from 10 V to 20 V, where the output of the modulator S2, and S3 are mea
sured and shown in Figure 57. During the switching cycle before 0 =0, the
amplitude of control voltages is 10 V, resulting in the amplitude of duty ratios
Figure 56 The circuit to test the step response of the high
order modulator. Vm = 10 V, Vg = 5 V, fs = 24 KHz.
Dm = 0.333 according to Equation (5.48). The measured effective duty ratios of
Sj, S2, and S3 are 0.0065, 0.29, and 0.295, respectively. According to
4+4+4 = \i>l <559>
The amplitude of effective duty ratios can be extracted as:
103
Vg/A
Â£
Lio
fiAVm ~ iW
Vn
+
D
2 R
D ev f S"cl yo)/?C
D
2 i?
D eV bw I
(b)
e 1 + jcdRC
V
m
Figure 52 (a) The steadystate ofb equivalent circuit of the
LPWMcontrolled threephase boost inverter; (b)
its simplified circuit.
The output voltage phasor V0 is given by
Vo = VJljQRC) (5.23)
From Equation (5.23), one can easily find that the amplitude of the output
voltages is given by
Vom = AVmJl + (uRC)2 (5.24)
which is the same as Equation (5.20). The output voltages have a phase shift
from the control voltages that is determined by
Zv0fl = tan\(Â£>RC) (5.25)
As a conclusion, the highorder LPWM can make the output voltages of
the threephase boost inverter track the control voltages linearly, independent
of the operating condition of the inverter, as shown in Equation (5.24). The
CHAPTER 3
REVIEW OF PULSEWIDTH MODULATION
This chapter reviews the existing pulsewidth modulation (PWM) tech
niques for both dc converters and threephase converters, in which largesig
nal linearization is emphasized. Two popular PWM methods for threephase
converters, continuous sinusoidal PWM (SPWM) and spacevector modulation
(SVM), are discussed in detail.
3.1 Pulsewidth Modulation for DC Converters
The conventional PWM with a constantslope carrier is the most popu
lar in dc or singlephase converters, but it gives rise to undesirable nonlinear
relationship between the output and control voltage in some topologies. Some
linearizing PWM techniques (LPWM) have been proposed for linearization of
dc or singlephase converters [10, 2334]. In these PWM techniques, the slope
of the carrier signal is not constant. Thus, the duty ratio generated from the
LPWM is a nonlinear function of the input and control voltages that may can
cel out the nonlinear controltooutput relationship of the converter and make
the output voltage to track the control signal linearly.
44
48
PWM converters, they involve more sophisticated analog circuits, such as mul
tipliers/dividers.
3.2 Pulsewidth Modulation for ThreePhase Converters
Threephase PWM converters are employed in many areas of todays
power industries, including active filtering [2], UPS [3], VAR compensation
[4], power generation [5], motor drives [6, 7], Compared with dc PWM convert
ers, threephase converters face more requirements, such as harmonics, bal
ance/unbalance systems, and so forth. Moreover, they need more sophisticated
control and drive circuits. Undoubtedly, linearization in PWM modulation will
bring benefits, such as easier control, lower harmonic distortion, and source
disturbance rejection, to the threephase PWM converters and help achieving
the stringent application requirements.
Many PWM schemes for threephase PWM converters have been pub
lished and applied in various power applications [4258]. They can be classi
fied into seven categories: sinusoidal, spacevector modulation, selective
harmonicelimination, optimal, current control, direct amplitude control,
and sigmadelta modulation.
3.2.1 Sinusoidal PWM
Sinusoidal PWM technique (SPWM) [42] is based on the principle of
comparing a triangular carrier signal with a sinusoidal reference. The imple
mentation of the technique with analog circuits is simple and can produce
150
devices inside one package of ICL7667 are paralleled to increase its driving
capability.
Table 6.4 Components used in the experiment.
Component
Type
Inductor
3 mH Magnet Tek C59U
Capacitor
Mallory (50 lF/370 Vac/60 Hz)
MOSFET
IRF360 (400 V, 20 A, 0.2 Â£2)
Diode
MUR3060 (600 V, 30 A, 1.7 V)
Inverter power source
Sorenson DCR15015A
OP AMP
LM833
Comparator
LM311
Analog switch
HI20 IS
Analog multiplexer
CD4051
MOSFET driver
ICL7667
Optocoupler
6N137
Digital multiplexer
74LS151
Inverter
74LS04
AND gate
74LS08
OR gate
74LS32
Voltage regulator
UA7805
Voltage regulator
UA7905
142
Tek arrnu s.ooms/s 2ss Acqs
Figure 631 The waveform of the voltage across the inductor and
gate signal of one MOSFET.
highorder
LPWM
inductor
heat sink
and fan
power
stage
Figure 632 The picture of the prototype circuit.
146
Figure 636 (a) Layout with a big loop; (b) improved layout with
small loop.
good layout
Figure 637 (a) Layout with a common ground line; (b) improved
layout without ground sharing.
part would be shared by other parts. The better layout is shown in Figure 6
37(b), in which power paths are connected with ground by separate lines. To
prevent power noise caused by high di/dt and parasitic L or dv/dt and parasitic
C from contaminating control signal, we need to separate power and signal
paths/grounds. Signal ground is connected to power ground is shown in Figure
638(a), in which parasitic L would introduce voltage spike into control signal.
2
UTILITY LINES
OR
CONVERTERS
HIGHFREQUENCY
CARRIER
POWER STAGE
(switches, inductors,
and capacitors)
CUSTOMER
LOADS
MODULATOR
(analog, or DSP
circuits)
CONTROL
SIGNALS
Figure 11 A basic switching power converter.
The modulator can be implemented by analog and digital means,
depending on the requirements, complexity, and costs in converter design. The
digital modulator is used mostly in threephase converters since it has more
computation capability. However, when the switching frequency is increased
by size and weight requirements, the digital modulator will be limited by its
clock speed. Meanwhile, when the reference voltage does not change smoothly,
the sample/hold circuit with the digital modulator would be restrained by res
olution. In contrast, the analog modulator is much faster, and it can handle
any frequency, limited only by the capability of power stage [1].
A conventional pulsewidth modulator (PWM), as shown in Figure 12,
consists of a comparator, a ramp carrier signal vrmp, and a control signal vc.
The carrier signal provides highswitching frequency to the control signal and
to the switches in the converter. The control signal is followed by the con
trolled variables, such as output voltages. In the conventional PWM, the car
rier signal has a constant slope. The control signal is compared with the
143
Figure 633 (a) Top view of the highorder LPWM modulator;
(b) bottom view of the modulator.
109
,n 1 .
Substitution of du in Equation (5.33) into (5.42) yields
,n + 1 2 sr ,n 2 .
13 Vc> = T(2vcbVab) + dn(VabVcbVab)
(5.43)
,n 2
By adding duvcb to both sides of Equation (5.43) and employing the assump
tion in Equation (5.39), the continuous differential equation for duty ratio d13
can be found as follows:
d x (VabVcb Vab Vcb) T/ (2vcb Vab)
~Ad\3) = 13 + V~
dt
2 T
Vcb^s
gr 2
2vibT,
(5.44)
dn ~ ^ ~dn ~~ d\3
(5.45)
dn, di3? and di2 can be solved from Equations (5.40), (5.44), and (5.45). Note
that these differential equations have included sampling effects.
5.2,5 SteadyState Analysis
Under steadystate condition, letting the derivative in Equation (5.40)
be zero yields:
d Xsr
11
2vab ~ Vcb
3V
gr
2 V,
2 ,2
vab + vcb~vabvcb
2 ,2
vab+vcb~vabvcb
3AV,
sin0
(5.46)
di3 is obtained from Equation (5.44) by letting the derivative be zero:
d 
a 13
2vab~Vcb
3E
gr
2 ,2
Vab + VcbVabVcb
2 ,2
Vab + Vcb~VabVcb
 (547)
where v2ab + v2cb vabvcb = ^ V2m in the balanced threephase voltages. The effec
tive duty ratios can then be expressed as
87
0 Vg(dud2\)va(dn d22)vb (d13 d23)vc
(4.50)
2v vfc v
0 (dn d2X)iL
(4.51)
2v> y. v_
0 (d12 d22) ^
(4.52)
During the first segment,
d2\ ~ d23 = ^
(4.53)
d22 = ^
(4.54)
Substituting Equations (4.53) and (4.54) into (4.50) (4.52) and applying sim
ple algebra, the steadystate equations for the first segment can be trans
formed as
Vg = d\\vab + d\3vcb
(4.55)
(2 dn+dl3)vcb = (2 dl3 + dn)vab
(4.56)
From Equations (4.55) and (4.56), duty ratios d11; d13, and d12 can be
solved by the highorder LPWM. Note that the output voltages will track the
75
d
p
Pi?g, vr)
Q(vg, vr)
(4.25)
The P function and Q function in the LPWM circuit are functions of the
input and reference voltages, which can be synthesized from the input and ref
erence voltages by operational circuits, such as adders/subtractors, and multi
pliers/dividers, as shown in Figure 46.
As an example, consider the largesignal linearization of a threephase
boost inverter, as shown in Figure 47. This converter consists of six switches
v
g
Multiplier
Divider
Adder
Subtractor
P
Q
Figure 46 P function and Q function generator.
(two singlepoletriplethrow switches), but only four of these switches are
independent. This is because
dii +d\2 + dn 1
(4.26)
d2i 4* d22 "4* ^23 1
(4.27)
The sinusoidal PWM (SPWM) technique discussed in Chapter 3 is
applied in the inverter. One of choices of duty ratios is
17
Figure 23 A threephase boost inverter with two SPTT switches.
respectively, and they are modulated at a frequency sufficiently lower than the
switching frequency.
The states of the inverter are inductor current l and capacitor voltages
va, vb, and vc. The first pole voltage is vpl, and the second pole voltage is vp2.
Based on the SFA model of the SPMT switch in Equations (2.4) (2.6), vpl and
vp2 can be expressed as the linear combination of capacitor voltages and duty
ratios of the switches:
V = dnva + dnvb + dnvc
(2.7)
Vp2 = d2\va + d22vb + d23vc
(2.8)
The voltage across the inductor can be obtained by application of Kirchhoffs
voltage law:
= V
(2.9)
58
Figure 32 The balanced threephase voltages.
(d\2 1) 
dv, v.
C^ + 
dt R
= du
(3.23)
dv vr
Cc + c
, dt R ,
d\3 = 7 = c
* J
(3.24)
From Equations (3.22) and (3.24), it can be found that d1;L and d13 are sinuso
ids, and d12 is the sinusoid with the dc offset. In summary, during 0 60, the
duty ratios for the switches are
d ii = Dm sin(coi)
(3.25)
d 13 = Dmsin
(,+f
(3.26)
d\2 = 1 + DrnS'm
d2\ ~ d23 ~
f CO t
2k
d 22 i
(3.27)
(3.28)
Table 3.1 Duty ratios for the threephase boost Inverter with the SVM.
STEP
dn
Cl
N>
d13
d2i
d22
d23
1
D sin(oor)
m v '
1 + D sin f cor
m v 3 J
D sin wi +
m \ 3 J
0
1
0
2
i
0
0
1 D sinicot)
m
. f 2n\
m { 3 J
D sinfcof + ^1
m V 3 J
3
D sin(coi)
m v '
D sin cor
m { 3 )
1 + D sinfcor +
m { 3 J
0
0
1
4
0
1
0
D sin(cot)
m v J
1 D sinicoi^l
m { 3 )
D sinfco + 1
my 3 J
5
1 + D sin(coi)
m v '
n ( In']
D sin coi
m { 3 )
D sinico? + ^]
m V 3 )
i
0
0
6
i
0
0
D sin(coi)
m v '
Dmsin(?y)
1 r, ( t 2n\
1 D sin cot +
m{ 3 J
50
can be implemented by comparing a sixstep control signal, generated from
the reference voltage, with a constantslope carrier signal [1], Such implemen
tation, however, gives rise to nonlinear relationships between the control and
output voltages, preventing the output voltages from tracking the control sig
nals.
3.2.3 Optimal PWM
The optimal PWM technique [51] produces the switching pattern based
on optimization of some performance criteria. The number and positions of the
pulses or notches within each switching cycle are selected according to these
criteria, which could be harmonic loss, torque pulsation, or load currents.
They are precalculated and stored in memory for use in real time. Thus, com
putation power from a microprocessor is needed to synthesize the correct
switching patterns.
3.2.4 CurrentControlled PWM
The currentcontrolled PWM technique [52] is intended to make the
output current track the reference current. In this technique, the output cur
rents with superimposing ripples are fed back and compared with hysteresis
levels placed around the reference signal to determine the switching fre
quency. As the ripple is regulated within the hysteresis band, the average out
put follows the average reference. Three independent controllers are needed to
control three phase legs separately in this scheme; each controller has its own
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR
FOR THREEPHASE POWER CONVERTERS
By
Jun Chen
August 2000
Chairman: Dr. Khai D. T. Ngo
Major Department: Electrical and Computer Engineering
The feasibility of using an analog pulsewidth modulator (PWM) to lin
earize balanced threephase converters is investigated in this dissertation.
Prototype circuits, models, and analysis techniques are developed.
Most balanced threephase PWM converters that are controlled by the
conventional analog PWMs have nonlinear relationships between the control
and output voltages/currents. This study shows that these nonlinear relation
ships can be linearized by an analog highorder linearizing pulsewidth modu
lator (LPWM) that makes the output voltage track the control voltage linearly.
Instead of multipliers/dividers, the highorder LPWM uses only integrators
with the reset, and sample/holds to compute the switching instants for the
switches in the converter. The inputs to the integrators of the LPWM are just
linear functions of the control and state variables, but are nonlinear functions
v
29
Vca
1 + Sa 
Asa
0+
t
hb
o
+Ab
i
he
O*
1 +vsc _
L J
(a) abc voltage source set
L J
(b) ofb voltage source set
i_ j
(c) abc resistor set
L J
d) ofb resistor set
La +VLa
I L
*Lb I +VLb "
miVYi
I L
i]Lc 1 Wv\ 
o I
, L
L J
(e) abc inductor set
JLo i vLo
1
lLf Â¡ vLf
WV^
L jcoL
^Lbw I vLbw
yyy.
I L jcoL
i
(f) ofb inductor set
i !"+VCa
JCa 1 +
oH
\\c*
Cb I + yo 
0K
I
JCc  +
o I
vCc
 IIC'
L J
(g) abc capacitor set
Co 1 + 
'Co
Cl 1
Â¡a Â¡ +
1
1
1
Cbw 1
C11
1/jcoC
Mat
+ vCbw
1
1
1/jcoC
VvV
(h) ofb capacitor set
Figure 29 Graphical models of voltage sources, resistors, induc
tors, and capacitors in the abc coordinates and the ofb
coordinates.
145
Figure 635 Top view of the threephase boost inverter.
6.3 Practical Issues in Experiment
To make the test system work, several practical issues need to be con
sidered, including component layout and commonmode noise. In component
layout, a big current loop should be avoided because the big loop is associated
with increased inductance. When there is di/dt noise around this loop, it will
introduce a voltage determined by Ldi/dt. This voltage generates a spike in
the signal, and it could make false operation in circuits. This situation is illus
trated in Figure 636(a). A better layout is shown in Figure 636(b). To prevent
signals from interfering each other, we should connect power paths of different
components to ground separately. A layout for several signals sharing a power
path is illustrated in Figure 637(a). In this layout, a noise associated with one
when other analog PWMs, such as feedforward PWMs and onecycle control
lers, are used.
The analog highorder LPWM is synthesized from switchingfunction
averaging (SFA) equations of the threephase PWM converter. Thanks to the
SFA model of the PWM switch, the derivation of SFA statespace equations of
the converter is simply done by inspection and application of definition of cir
cuit elements, Kirchhoffs law, and other electrical principles without probing
into topological details of the converter. The set of SFA equations can be trans
formed into an equivalent circuit in the stationary coordinates to make simu
lation more efficient.
In order to analyze threephase converters that are controlled by the
LPWM or other pulsewidth modulation techniques, all threephase component
models in the rotating coordinates, including PWM switches, sources, and pas
sive components, are developed. After threephase components are replaced
by their models in the rotating coordinates, the timevariant threephase cir
cuit is transformed into a timeinvariant equivalent circuit that makes analy
sis and design much easier. The model of the highorder PWM is also
developed. It is useful to analyze the LPWMcontrolled converter and evaluate
time delay caused by sampling effects.
The synthesis and analysis theories of the highorder LPWM are veri
fied by a 1 KW prototype of a threephase boost inverter. Both simulation and
experimental results agree with the analysis. The experimental results show
that the control circuit is simple, and the output voltages of the inverter can
vi
95
Hi]
X
Lo
ni
x'
Lo
T3
t(ms)
0 2 4 6 8 1 1.2 1.4 1.6
Figure 418 The switching signals of six switches in the inverter
over the complete sinusoidal period.
72
In the following sections, a general procedure is presented to use
firstorder LPWM circuits to synthesize the LPWM for threephase converters.
Although it may end up with using multipliers/dividers in the LPWM circuit,
the synthesis procedure is still helpful to understand the linearization of
threephase converters and use firstorder LPWM modulators in threephase
converters.
4.4 Linearization bv FirstOrder LPWM
The first step to linearize threephase converters by firstorder LPWM
circuits is to find the SFA equations of the converter. The derivation of the
SFA equations of a PWM converter is discussed in Chapter 2. For a PWM con
verter with M independent switches, the duty ratios of the switching signals
for these M switches are defined as:
T
d = [/j, c?2> > d
(4.18)
After solving SFA equations of the converter, each variable in Equation (4.18)
can be expressed as a function of the output voltage vQ and input voltage vg:
(4.19)
pM^vr
Qm(VS Vr)
(4.20)
where the output voltage vG is replaced by the reference voltage vr For exam
ple, the duty ratios of the threephase boost inverter shown in Figure 31 of
Chapter 3, d^ d13, are functions of input voltage Vg and output voltages
40
where the caret implies smallsignal perturbations. Neglect of the steady
state and secondorder terms then leads to a smallsignal equivalent circuit of
the transformer, as shown in Figure 217(b). It consists of a dc transformer
and two dependent sources that are controlled by the duty ratio. De, Ic, and
Vpa in the capital letter are dc values derived from the steadystate analysis of
the inverter. Replacing the transformer in Figure 216 by its smallsignal cir
cuit in Figure 217 yields the smallsignal equivalent circuit of the inverter, as
shown in Figure 218, where every variable is replaced by its smallsignal
value with the head A.
Figure 218 The smallsignal equivalent circuit of the three
phase boost inverter.
Let v = 0, the real part of controltooutput transfer function of the
6
inverter can be solved from the smallsignal circuit in Figure 218, which
is given by
de 2 D] D
(2.55)
132
Tek HftliV 1 O.OMS/s 682 Acqs
[ T ]
51
52
53
20 May 2000
23:31:54
Figure 619 The experimental waveforms of the outputs of high
order LPWM.
84
Figure 410 The integrator with reset and hold.
which will be available for the next switching cycle. The signal a22d12 can be
generated in the same way.
From Equations (4.46) (4.49) and the integrator shown in Figure 410,
the highorder LPWM circuit can be synthesized, as shown in Figure 411. Its
operation waveforms are shown in Figure 412. When the clock signal comes,
the bottom integrators, #1 and #2, start to integrate their input signals
and a2l. As soon as integrator output vo2, the integration of a21 > reaches
k2 a22^12 > comparator produces a pulse S]^ with the duty ratio of
d'\{. This pulse resets the top integrators, #3 and #4, and provides the HOLD
signal for integrator #1. Thus, the integrator output vol is held at the value of
a\\d'\\ > which will be used to solve al2. After reset, two integrators #3 and
#4 on the top start to integrate a12 and a22. As soon as integrator output vo4,
the integration of a12, reaches kÂ¡ axxd'\x, the comparator generates a pulse
S2 with the duty ratio of d'\2. This pulse provides the HOLD signal for #3
129
Figure 615 The prototype of the analog highorder LPWM for
the threephase boost inverter.
3
Figure 12 A conventional PWM.
carrier signal through the comparator. The output pulse vp (also called switch
ing function), generated by the comparator, is used to drive the switch in the
converter. It has the duty ratio of
(1.1)
where Vm is the amplitude of the ramp vrmp.
The output signal vp determines the switching patterns of the con
verter. The controlled variable of the converter, such as the output voltage, is
the function of the converter input and duty ratio of vp that is determined by
Equation (1.1). Therefore, the controlled variable of the converter can be regu
lated by adjusting the control signal.
The power stage in Figure 11 could be the dcdc converter in the dc
power conversion or the threephase converter in threephase ac power con
version. The most popular dc converters are shown in Figure 13. The conver
sion ratios between the output and input voltages are listed in Table 1.1. The
singlephase converters and dc converters with the transformer isolation are
CHAPTER 1
INTRODUCTION
With the development of highspeed, highpower semiconductors, the
switching power converter has gradually replaced linear power amplifiers to
become the main power conversion product on the market. The switching
power converter not only provides more efficient power conversion than the
linear power converter, but also has more flexible control capability that
allows the converter to meet various power demands and requirements.
Therefore, research on switching power converters has received much atten
tion. A major research issue is the linearization of switching power converters
that makes the controlled variable track the control signal and improve the
performance of the converter.
A basic switching power converter consists of two sections, as shown in
Figure 11. The first section is called the power stage that usually consists of
semiconductor switches and energy storage components. The power stage
receives the unregulated energy from the utility power line or power convert
ers and provides the regulated energy to customer loads. The second section is
the modulator that provides control signals to the power stage.
1
96
Figure 419 The simulation results of the duty ratios for the six
switches in the inverter.
110
d
a
1 ~
1
i i
db
=
dl2 d22
d
c
13/23
3AV,
3 a y,
3 a y.
sin
sin
sin(0)
e^l
(5.48)
By comparing Equation (5.12) and Equation (5.48), a conclusion can be
made. Under steadystate condition, the effective duty ratios that involve
sampling effects are shown to be the same as those without sampling effects.
5.2.6 SmallSignal Analysis
This section discusses the dynamics of the highorder LPWM intro
duced by sampling. The dynamic study is restricted only to the smallsignal
sense. The angle 0 of threephase voltages in Equations (5.40) (5.45) is
assumed constant, thus, the coefficients in these equations become constant.
The smallsignal analysis will predict the response of the modulator to small
perturbations around a quiescent operating point. Let the input and output of
the modulator consist of a steadystate and a perturbed component:
Vm = Vm + Vm dm = Dm + ^ ^ gr = Vgr + Vgr (549)
where caret implies smallsignal perturbations. Substitution of Equation
(5.49) into Equation (5.40) and neglect of the steadystate and secondorder
terms then yield the following response for the duty ratio dn in the Laplace
domain:
dM = Kmvm + Kgvg
(5.50)
16
of statespace equations of a PWM converter becomes routine and can be done
using statespace concept [16], definitions of circuit elements, Kirchhoffs
laws, and other electrical principles.
2.1.2 Derivation of StateSpace Equations
Since the SPMT switch in the converter has been modeled as a compo
nent by the switchingfunction averaging technique, there is no need to iden
tify the switched topologies. Statespace equations are derived simply by
following the procedures described in ref 16. The only attention is to identify
the SPMT switches in the converter. A threephase boost inverter is used here
to demonstrate how to identify the SPMT switches and how to derive state
space equations for PWM converters. Since the statespace equations of the
switchedmode converter are derived by averaging the switching functions of
the switch, they may be called switchingfunctionaveraging statespace equa
tions (SFA statespace equations) [9] in this thesis.
A threephase boost inverter is illustrated in Figure 23. Since we know
that the pole of the SPMT switch usually is connected with inductors and
throw is connected with voltage sources or capacitors, it is easy to find that
there are two singlepoletriplethrow (SPTT) switches in the threephase
boost inverter. The SPTT switch on the top consists of the switches S11? S12,
and S13 and is characterized by d11? d12, and d13 The SPTT switch on the bot
tom is grouped by S21, S22, and S23 and is characterized by d21, d22, and d23.
Duty ratios d^ d23 correspond to the switching functions of switches S1;L S23,
70
In PWM converters, the output voltage, vG is controlled by duty ratio d:
v0 = f(d,vg) (4.13)
which is the nonlinear function of the duty ratio in most of PWM converters.
When it is controlled by the conventional PWM, the duty ratio is proportional
to the control voltage.
d = vc (4.14)
The resulting output voltage will be the nonlinear function of the control volt
age:
vo = f(vcvg) (415)
However, the largesignal LPWM in Figure 44 is synthesized by (4.13).
It is able to solve (4.13) and find duty ratio as the function of the output volt
age and input voltage:
d = vg) (4.16)
The output voltage thus equals the control voltage:
v0 = vc (4.17)
According to (4.17), the output voltage of the nonlinear PWM converter con
trolled by the LPWM is able to track the control voltage linearly. The nonlin
ear controltooutput relationship is completely eliminated without using any
feedback loop.
In general, the task of the LPWM controller is to obtain the duty ratio
by solving modulation Equation (4.16), which could be done either by digital
signal processors (DSP) or by analog circuits. Even though only analog imple
152
implement the highorder LPWM modulator is the assumption that the duty
ratio in the current switching cycle approximately equals that in the previous
cycle. This assumption is true because the control signal is much slower than
the switching frequency in practice. The analog highorder LPWM is simple
and easy to use.
The general procedure of the synthesis of the highorder LPWM modu
lator for balanced threephase converters is developed. The modulator is syn
thesized from the switchingfunction averaging (SFA) equations of the three
phase PWM converter. In this thesis, the derivation of SFA statespace equa
tions is simply done by inspection and application of definition of circuit ele
ments, Kirchhoffs law and other electrical principles without probing into
topological details of the converter. The set of SFA equations can be trans
formed into an equivalent circuit in the abc coordinates to make simulation
more efficient.
The circuitoriented analysis technique is developed for balanced three
phase PWM converters. All the threephase components, including PWM
switches, sources, and passive components, are modeled in the ofb coordinates.
After threephase components are replaced by the ofb models, the timevari
ant threephase circuit is transformed into a timeinvariant equivalent circuit
that makes analysis and design much easier. The model of the highorder
PWM is also developed. It is useful to analyze the LPWMcontrolled converter
and evaluate time delay caused by sampling effects.
5
The dc converters are used mostly in delicate and lowpower applica
tions, such as computers and microprocessors. The threephase PWM convert
ers are usually used in rugged, highpower applications, such as active
filtering [2], UPS [3], VAR compensation [4], power generation [5], motor
drives [6, 7], and multilevel converters [8]. The most popular threephase
PWM converters [9] are shown in Figure 14. The voltage conversion ratios
Buck inverter
Boost inverter
QTYYY
1
1
p
V
L
' x
Boost rectifier
Figure 14 The threephase inverters and rectifiers.
11
circuit. With the help of smallsignal equivalent circuit, the controltooutput
transfer function of the converter can be easily found graphically.
Although the boost inverter is used as an example to demonstrate the
whole procedure, the analysis and modeling techniques in this chapter can be
applied to any other threephase PWM converter. To simplify explanation, it is
assumed throughout the thesis that the components are ideal and the
switches are lossless and fourquadrant.
2.1 Derivation of StateSpace Equations of PWM Converters
2.1.1 SwitchingFunction Averaging Model of PWM Switch
To analyze the steadystate and dynamic performance of a PWM con
verter, which contains reactive components, the statespace equations must be
presented. There are many approaches to derive the statespace equations for
PWM converters, among which the statespace averaging technique [13, 14] is
the most popular. This approach requires the identification of the switched
networks and the derivation of the statespace equations for all switched net
works that is easy to do in dc converters because of the small number of
switched networks. However, a threephase converter usually has a large
number of switched topologies, and with the increase of phase numbers, the
number of switched topologies will increase rapidly. For a given PWM method,
the switched networks in one switching cycle can be different from those in
another cycle. Moreover, different PWM schemes generate different switched
6 IMPLEMENTATION AND EXPERIMENTAL RESULTS OF
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR 116
6.1 Analog Implementation of HighOrder LPWM 118
6.2 Experimental Results 135
6.3 Practical Issues in Experiment 145
7 SUMMARY AND CONCLUSION 151
REFERENCES 155
BIOGRAPHICAL SKETCH 162
IV
74
TA
Y J Qx{vg,vr)dt = Px{vg,vr) (4.23)
5 o
TsdM
Y J fM(V Vr^dt = PMVg vr) (424)
* 0
Each of these integration equations can be implemented by a firstorder
LPWM circuit with a resetable integrator and one comparator, as shown in
Figure 45. The operation waveform can be referred to Figure 4l(b).
Figure 45 The firstorder LPWM circuit to synthesize one
of the duty ratios.
When the clock signal is coming, the Q function is integrated, and the
output of the integrator is compared with the P function. During this time, the
PWM signal vp is high, turning on the switch in the converter. When the out
put of the integrator ramps up to the P function, the PWM signal vp becomes
low, turning off the switch. The resulting duty ratio of PWM signal vp is
56
where
Dm< 1 (3.17)
The dc offset in Equations (3.15) and (3.16) is to keep the duty ratios positive.
Equation (3.15) represents the sinusoidal modulation function for the top (sin
glepoletriplethrow) SPTT switches shown in Figure 31. Equation (3.16)
represents the sinusoidal modulation function for the bottom SPTT switches
shown in Figure 31. The amplitudes of the sinusoidal modulation function for
the same switch group must be the same to constitute balanced threephase
sinusoids. The amplitudes and phases of sinusoidal modulation function for
different switch groups could be different, as long as two switch groups are
topologically independent. Note that the duty ratios in Equation (3.16) have
the same amplitude and oppose phase from the duty ratios in Equation (3.15)
that results in the optimal effective duty ratios:
d
a
dn~d22
db
=
d\2~d22
d
c
d\?>~ d23
2D
m
sin(coO
2D
m
3
2D
m
sinoco?
sinoco/ +
2k
2k
(3.18)
that is related only to the continuous sinusoidal modulation techniques.
The amplitude Dm and the phase of duty ratio da in Equation (3.18) can
be obtained from Equation (3.12), which is given by
W = Dm = 5
3VmJ\+(aRCy
RI,
(3.19)
59
where the amplitude Dm and the phase of the duty ratio d1;L can be obtained
from Equation (3.22) as the following:
Kill = Dn
vmJi + ((oRcy
RI,
1.
Zda = tg (CORC)
The effective duty ratios are
d
a
dn ~d22
db
=
d\2~ d22
d
c
d\3~ d23
D
D
Â£>msin(cor)
m
sin
imf
m
sin
sin^coc +
271
Combination of Equations (3.11) and (3.29) yields
v = ^Vl+(*C)2
DLJm
(3.29)
(3.30)
(3.31)
(3.32)
In fact, the threephase boost inverter works like a dc boost converter.
When S12 is on, the inductor gets energy from the source. Wlien either Sn or
S13 is on, the inductor transfers the energy to the load. Since only one SPTT is
switched at high frequency and the other is switched at low frequency, the
SVM has less switching loss than continuous SPWM.
The duty ratios du d23 for the six segments are listed in Table 3.1, and
their waveforms over one period are shown in Figure 33. It is obvious that the
duty ratio functions in the SVM are piecewise sinusoidal and have sixstep
symmetry.
14
To avoid short circuit, no two throws turn on at the same time, and all
switching functions must add up to one at any instant to avoid open circuit. In
other words, one and only one of the switching functions of the SPMT switch is
one at any instant. This can be expressed as follows:
M
X1 *
X ^1 k 1
(2.1)
k = 1
It is obvious that there are only Ml independent switching functions of the
onepoleMthrow switch.
At any moment, only one throw is connected to the pole. Therefore, the
throw current equals pole current during its connection with the pole:
i k dlki p (2.2)
Over one switching cycle, the pole is connected to the throws one by one;
thus, the pole voltage v*p is just a linear combination of the products of throw
voltages and corresponding switching functions:
M
vP = X d*\kv*k (2.3)
k= 1
The SPMT switches are switched at a high frequency. The voltages or
currents connected with the pole and throws, either dc or sinusoidal ac, are
varying slowly, relative to the switching frequency. Therefore, over one switch
ing cycle, the terminal voltages and branch currents of the pole and throws
can be assumed as constant. The duty ratio d^ (without asterisk), which is
the average of the switching function d*]^ over the switching period Ts, is
90
Comparing Equations (4.60) and (4.61) with (4.44) and (4.45), it is not difficult
to find
12 = vy
(4.62)
22 =
(4.63)
>
II
(N
(4.64)
1
*
>
II
(4.65)
i = ^
(4.66)
* 3
1
<
* 3
1
(4.67)
Replacing the inputs of the highorder LPWM in Figure 411 with the
sixstep reference voltages in Equations (4.62) (4.66), the highorder LPWM
for the threephase boost inverter is then synthesized, as shown in Figure
415. It is worth noting that the inputs to the integrators are linear functions
of the reference and input voltages, and analog implementation of which
involves only adders/subtractors, as shown in Figure 415.
The threephase boost inverter, as shown in Figure 42, is simulated
with the proposed highorder LPWM circuit shown in Figure 415 in Saber.
The control voltage is va = Fmsin(ilf) with Vm = 262 V and Q = 2tc(60Hz). The
inputs of the LPWM circuit in Figure 415 are the sixstep reference signals vx
and Vy, which are generated from va, v^ and vc. The circuits that generate
sixstep reference signals are not shown here, but they can be easily built in
Saber by some analog switches and some comparators. The waveforms of vx
54
dva , v
~ (11 21 ^1L~~r
Multiplying va on both sides yields
(3.6)
dva
VaC~JÂ¡ = (dU~d2l)lLVaJ
(3.7)
The same procedure is applied to Equations (3.3) and (3.4) yields
dvb vb
= (d\2~d22 )iLvb~~^
(3.8)
dv. v
vcC^T = (d\3d23)iLvc~
(3.9)
Under steady state, the inductor current is assumed as dc; thus, dij/dt
in Equation (3.1) is zero. Substituting Equations (3.7) (3.9) into (3.1) yields
V2
VI ^
* L 2 R
(3.10)
V2
I = 1
L 2flV
(3.11)
It is obvious that Equation (3.10) is the conservation of power. The input
inductor current is dc, the value is determined by Equation (3.11). Once II is
obtained, duty ratios in Equations (3.7) (3.9) can be solved from
(II 21> = ~ =
(3.12)
(d\2~d22^ ~
dv v.
Cb + Â£
dt R
= du
(3.13)
88
reference voltages va, v^, and vc linearly when the boost inverter is controlled
by the LPWM to solve Equations (4.55) and (4.56). In other words, the inverter
would have lowdistortion sinusoidal waveforms at the output, and nonlinear
ity of the boost type inverter is eliminated.
If the steadystate equations for the six segments are listed, one can see
that they have the same forms as Equations (4.55) and (4.56). Thus, they can
be expressed as a general form as follows:
= dxvx + dyvy
(4.57)
(2 dx + dy)vy = (2 dy + dx)vx
(4.58)
dz 1 d x dy
(4.59)
where Vg is the dc input voltage of the inverter.
The coefficients vx and vy in Equations (4.57) and (4.58) are the refer
ence voltage signals to the LPWM. They are sixstepped piecewise sinusoidal
linetoline voltages, as shown in Figure 414. Within different sixstepped
segments, vx and vy takes different linetoline voltages, as shown in Table 4.1,
which are synthesized from continuous threephase reference signals va, vj,,
and vc. The outputs of the modulator are PWM signals with duty ratios dx, dy
and dz. For each segment of the SVM, dx, dy, and dz, are assigned to three
switches of the inverter based on Table 4.1. During the first segment, for
instance, dx = djj, dy = d13, dz = d12. The positions of dx and dy over one complete
period are shown in Table 4.1.
131
Tek 10.0MS/S 574 Acqs
Figure 618 The experimental waveforms of the comparator for
the PWM signal S2.
are shown in Figure 617 and Figure 618, respectively. The output waveforms
of the highorder LPWM are shown in Figure 619. All the waveforms agree
with the analysis.
The output signals of the highorder LPWM, S^, S2, and S3, are
assigned to the six switches of the inverter by the gatedrive logic shown in
Figure 620. The experimental waveforms of the switching signals for the six
switches Sj! S23 are shown in Figure 621. After being averaged by lowpass
filters, the duty ratios of PWM signals are obtained and shown in Figure 622.
Obviously, in spacevector modulation, the duty ratios of switching signals are
piecewise sinusoidal waveforms that agree with the discussion in Chapter 3.
104
line voltage regulation is improved because the output voltages are mainly
determined by control voltage and a constant gain set by the voltage divider. A
phase shift between the control and output voltages is caused by the load
resistance and filter capacitor.
5.1.3 SmallSignal Analysis
This section discusses the smallsignal property of the LPWMcon
trolled threephase boost inverter. Its smallsignal equivalent circuit is shown
in Figure 53 that is the combination of the smallsignal model of the modula
tor, as shown in Equation (5.15), and the ofb smallsignal equivalent circuit of
threephase boost inverter, as shown in Figure 218 of Chapter 2.
Let vg = 0, the controltooutput transfer function of the power stage is
shown in Equation (2.64) of Chapter 2 and given by
Figure 53 The smallsignal model of the LPWM controlled three
phase boost inverter in the ofb coordinates.
60
Figure 33 Duty ratios for spacevector modulation.
153
The synthesis procedures and analysis theories for the highorder
LPWM and balanced threephase PWM converters are demonstrated and sim
ulated through a threephase boost inverter in this thesis and checked with an
experiment. A prototype 1 kW, 24 kHz threephase boost inverter is used to
test the highorder LPWM implemented by analog circuits. The experimental
results agree with the simulation and the analysis. It is shown that the output
voltages of the threephase boost inverter can track the control voltages lin
early with good sinusoidal waveforms. The disturbance of dc input voltage can
be reduced in the output voltages, and line voltage regulation is improved.
The analog circuits of the highorder LPWM modulator is simple and can be
implemented easily.
In conclusion, the threephase PWM converter can be linearized by the
highorder LPWM modulator that uses integrators with the reset, and sam
ple/hold to compute commutation instants. The inputs to the integrators are
linear functions of control voltages. With the highorder LPWM control, the
output voltages can track the control signals linearly with good waveforms,
and the disturbance from the input voltage can be reduced. The control is sta
ble, and analog implementation is simple. Time delay caused by sampling
effects of the highorder LPWM is small because of high switching frequency,
and it can be neglected in the design. The method of synthesis and analysis in
this dissertation are general methods that can be applied to other threephase
topologies.
161
[60] Chongming Qiao, Keyue M. Smedley, A General ThreePhase PFC
Controller Part II. for Rectifiers with a SeriesConnected Dual
Boost Topology, Proceedings of the 1999 IEEE Industry Applica
tions Conference, vol. 4, Phoenix, 1999, pp. 25122519.
99
Vg by a voltage divider. The ratio of the voltage divider is A, then Vgr is given
by:
(5.4)
The ratio A is big enough so that could not saturate the operational circuit
in the analog highorder LPWM. Similarly, the amplitude of the control volt
ages, Vm, must be small enough, so that the control voltages would not satu
rate the operational circuit in the LPWM.
The modulation equations (5.1) and (5.2) are derived from the first seg
ment of spacevector modulation (SVM). In the first segment of the SVM, duty
ratios d21 = d23 = 0, d22 = 1. Thus, the duty ratios d11; d13 and d12 can be
replaced by the effective duty ratios da, dc and d^. Then, the modulation equa
tions (5.1) and (5.2) become
Vgr = daVab + dcVcb
(5.5)
(2 da + dc)vcb = (2dc + da)Vab
(5.6)
db ~ da~ dc
(5.7)
In fact, as shown in the above equations, only effective duty ratios play a role
of modulation.
For the highorder LPWM, Vgr, vab, and vcb in Equations (5.5) and (5.6)
are known. The task of the highorder LPWM is to solve da, dc and d^ from
Equations (5.5) (5.7). By applying a simple algebra to Equations (5.5) (5.7),
da, db, and dc can be expressed as the functions of v3) v^, and vc:
43
SFA statespace equations expedites the simulation. The equivalent circuit of
the threephase converters in the ofb coordinates is constructed graphically by
replacing sets of threephase components with appropriately connected ofb
components. With the help of the ofb equivalent circuit, the steadystate and
smallsignal analyses of the threephase converters can be worked out graphi
cally, which is proven to be easier than the equationoriented method.
135
gate and source, thus, three separate power supplies, Vcci, Vcc2, and Vcc3; are
required to drive them, as shown in Figure 623. MOSFETs M2i, M22, and
M23 have the same ground, only one power supply Vcc4 shown in Figure 623 is
required to drive them. The allowed forward current of optocoupler 6N137 is
from 6.3 mA to 15 mA, so the resistor in series with diode is designed to be
500 Q (5 V/10 mA). The collector resistor Rc is designed to be 2 Kil that gives
onstate current 6 mA (the maximum value is 13 mA).
6.2 Experimental Results
The highorder LPWM developed in this chapter is used to control a
threephase boost inverter, as shown in Figure 61 (a). The reference voltage is
given by
(6.7)
vfl = sin cor
The highorder modulator uses a portion of input voltage as its dc reference
voltage and the value is given by
(6.8)
According to Chapter 5, the output voltage is given by
voa = AVmJl + (coRC)2sin(cor 0)
(6.9)
where the phase is decided by
0 = tan 1 (co RC)
(6.10)
86
Figure 412 Waveforms of the highorder LPWM.
segments as shown in Figure 413. In each segment, one SPTT switch in Fig
ure 47 is permanently attached to one of the three capacitors as the other
sweeps through all three. The position of the stationary switch as well as the
sweeping ones are determined by a sixstepped sequence.
In the first segment, 060, < va and < vc. Let d22 = 1, and d21 =
d23 = 0, that is, S22 is on while S21 and S23 are off all the time during this seg
ment. The switches, S11; S12 and S13, are switched at high switching fre
quency. In steady state, the averaged statespace equations are
35
component and backward component are totally decoupled. Therefore, the
analysis of threephase converters with the ofb equivalent circuit is easy.
To construct the ofb equivalent circuit, we need to identify dc and ac
components in the abc coordinates. The ac components are replaced by their
graphical models in the ofb coordinates, and dc components remain in the ofb
equivalent circuit. As a result, the threephase boost inverter is divided into
five parts, as shown in Figure 212. Part one and part two are in the dc side of
the inverter, including the dc voltage source and the inductor. In steadystate
conditions, the inductor current is dc. Therefore, it is not necessary to trans
form the voltage source and inductor. Parts three, four and five are in the ac
side of the inverter and include timevariant switches, capacitors, and resis
tors.
Figure 212 Partitioning the threephase boost inverter.
65
Slope =
u(t)
^7
(4.2)
The average value of u(t) over one switching cycle is given by
d(tyr,
4 [ u(t)dt = u(t)d(t) (4.3)
T 0
which equals the comparator input v(t):
u(t)d(t) = v(t) (4.4)
In most of dc converters, the relationship between the output voltage
and the duty ratio can be expressed in the form of Equation (4.4). For example,
the average output voltage V0 of a dcdc boost converter is the function of duty
ratio and the input voltage:
V
O
1 D
(4.5)
Transformed into the form of Equation (4.4), then (4.5) becomes
VD=V0Vg (4.6)
If the boost converter is controlled by the firstorder LPWM shown in Figure
4l(a), and let v(t) = Vc Vg and u(t) = Vc, the duty ratio can be given as
V V
D = 4^ (4.7)
V C
Substitution of Equation (4.7) to (4.5) yields
= Vc (4.8)
Note that the average output voltage V0 of a nonlinear boost converter can
track the control voltage linearly, as indicated in Equation (4.8).
4
not listed here, because they can be derived from these basic topologies and
have one independent control variable like basic dc converters.
AYY\
V.jrH
Boostl_ ^
r
i
T
k I
n
BuckBoos
j
+
a
Buck
+
lt
V
T
A 
V V
M
H.
rrm.
Cuk_
/YYTL
Vgf H
SEPIC J +
:v,
11 (YYY\
:r 1 \
' r
r
.,
+
Dual SEPIC
Figure 13 Basic dcdc converter topologies.
Table 1.1 Voltage conversion ratios.
Converter Topology
Voltage Conversion Ratio, V/Vg
Buck
D
Boost
1/dD)
BuckBoost
D/CLD)
Cuk
D/(lD)
SEPIC
D(lD)
Dual SEPIC
D/(lD)
117
II
(a)
(b)
MOSFET IRF360
(400 V, 20 A, 0.2 Q)
H
snubber
8.6 Q
:680 pF
T Diode MUR3060
(600 V, 30 A, 1.7 V)
Figure 61 (a) The test system of a threephase boost inverter
controlled by the highorder LPWM; (b) ideal switch
and its implementation in the inverter.
134
To drive MOSFETs in the inverter shown in Figure 61, signals S;q 
S23 generated by gatedrive logic shown in Figure 620 need to be transferred
from 5 V to 12 V by MOSFET drivers. To drive MOSFETs in the inverter that
have floating gate and source, isolation circuits, such as optocouplers, are
required. The circuit that transfers signals S1;L S23 to MOSFETs and pro
vides isolations is shown in Figure 623. MOSFETs M^ M13 have floating
Figure 623 Optocouplers and MOSFET drivers.
122
ure 61(a). Sgl Sg6 are generated by the circuit shown in Figure 66. By
comparing threephase voltages va, v^, and vc with zero volt, the circuits gen
erate three digital signals Sa, S^, and Sc that are in phase with va, v^, and vc,
respectively, as shown in Figure 67. The sixstep signals Sgl Sg6 are
Figure 67 Signals S1; S2, and S3 generated from threephase
voltages va, v^, and vc.
Tek asa 25.0kS/s 751 Acqs
Sg2
Sg3
sg4
Sg5
Sg6
20 May 2000
23:06:46
Figure 68 The experimental waveforms of sixstep signals of
Sgl' Sg6
3 : J 1 ;
Ch2 3.00 V M 4.00 ms Ch1 f 200 mV
Ch3 5.00 V HiE 5.00 V
Ref4 5.00 V 4.00ms
66
Some dc converters have quadratic duty ratios in the controltooutput
relationship [11] [24]. They can still be linearized by the LPWM, as shown in
Figure 41, but adding one more integrator and gain block to it. This is
because
uD
2
(4.9)
4.2 Nonlinear Problem in ThreePhase Converters
The nonlinear relationship between the output and control voltages
exists in most threephase PWM converters that are controlled by conven
tional PWM modulations, such as sinusoidal PWM (SPWM) and spacevector
modulation (SVM). The highfrequency carrier signal in the conventional
SPWM and SVM has a constant slope. The duty ratios of switching signals
generated by these PWMs are proportional to the control voltages, which are
not able to cancel the nonlinear dutyratiotooutput relationship of the con
verters. As a result, output voltages are not able to track the control signals
linearly. In the balanced threephase converters, the output waveform is sinu
soidal, not affected by this nonlinear controltooutput relationship. The
amplitude, however, is affected by the nonlinear controltooutput relation
ship.
As an example, a balanced threephase boost inverter, as shown in Fig
ure 42, is used to demonstrate the nonlinear problem in threephase convert
ers. It is controlled by the conventional SPWM. The input voltage Vg is dc, the
32
Figure 210 (a) Threephase SPDT switches; (b) switch model in
abc coordinates; (c) switch model in ofb coordinates.
Threephase singlepoletriplethrow (SPTT) switches. The SPTT
switches shown in Figure 2ll(a) are commonly found in, e.g., the boost
inverter and buck rectifier [9], Their lowfrequency model in the abc coordi
nates is shown in Figure 211(b), where one choice for the duty ratios is
i D
3 + t cos(V
d,
la
d2 a
dlb
=
1 ( 2icA
3 3 \d 3 )
d2b
d,
1 c
d2c
1 m (a
 + cos 0 +
3 3 y d 3 )
1 D
1 m
3 3
D
cos(e^)
m fn 2n
rcos(e
D
1 m
 COS 0 J +
2k
d + T
(2.43)
The effective duty ratios are
64
Figure 41 (a) A firstorder linearizing PWM; (b) its operation
waveform.
The firstorder LPWM modulator usually consists of a resetable inte
grator and a comparator, as shown in Figure 4l(a). Its operation waveform is
illustrated in Figure 4l(b). When the clock signal comes, the output signal vp
of the comparator becomes high, turning on the switch in the converter. At the
same time, the integrator starts to integrate the input signal u(t). When the
integrator output reaches comparator input v(t), the output pulse drops to low
and turns the switch off, as shown in Figure 4l(b).
It is supposed that the clock is sufficiently fast so that the function u(t)
and v(t) can be assumed as constant over each switching cycle. Therefore, the
amplitude of the ramp generated by the integrator is
t,
V = 4 f ()* (') (4.1)
To
The slope of the carrier ramp varies with u(t), which is
30
_ d\Labe
vLabe ~ dt
(2.33)
application of Equations (2.22) (2.28) to (2.33) yields
d(Tir )
Lofb
dt
di,
_ TiL Lofb Lfb dT;
dt + dt lLofb
di,
dt \ dt ) Lofb
(2.34)
where
0 0
0 j(L
0
0
(2.35)
0 0 jcoL
The ofb inductor set is thus as shown in Figure 210(f). The ofb inductor is a
real dynamic inductor L in series with an imaginary static resistor jcoL.
Capacitors. The circuit models for the capacitors are the duals of those
for the inductors and are shown in Figure 210(g) 1(h).
Threephase singlepoledoublethrow (SPDT) switches. The SPDT
switches shown in Figure 210(a) are commonly found, for example, in the
buck inverter and boost rectifier [9], Their lowfrequency model in the abc
coordinates is shown in Figure 210(b), where one choice for the duty ratios is
d
d
d
a 1
b\
cl
i D
1 m /n \
 + COS(0)
1 { 2n
 + cos 0
2 2 yd 3
D
1 m
~ + TTCOS
2 2
(Vt
d a
a 2
l~dal
db2
=
l~dbl
dc2
}dcl
(2.36)
where
19
the same PWM converter, a different PWM strategy leads to different coeffi
cients in the SFA equations and state solutions.
The SFA statespace equations shown in Equations (2.10) and (2.12) 
(2.14) are derived in the stationary reference frame or abc coordinates, in
which all the state variables and coefficients in the equations are timevari
ant. Obviously, solving the timevariant statespace equations is very tedious
and difficult. Therefore, they are transformed into the ofb coordinates to
remove the time dependencies from the statespace matrices [15] by the abc
ofb transformation [17]. These coordinates consist of an 0sequence phasor, a
/orward(rotating) phasor, and a ackward(rotating) phasor. After the trans
formation, the SFA statespace equations become timeinvariant, and the
threephase boost inverter can be analyzed by solving the statespace equa
tions in the ofb coordinates. For a balanced threephase system, the equations
containing the 0sequence, forward, and backward phasors are completely
decoupled. The steadystate backward phasors are directly related to the volt
age and current phasors in the circuit. Unfortunately, the steadystate and
dynamic analysis of converters [15], based on ofb statespace equations, con
tain intensive algebraic calculation and matrix manipulation. In addition, the
equationoriented model of the converter is not intuitive to computer simula
tion.
In contrast, circuitoriented techniques [18, 19, 20, 21] are preferred for
handanalysis/calculation and computer simulation. Such circuitoriented or
graphical techniques not only produce the averaged equivalent circuit
83
integrators to solve these equations involves only linear inputs to the integra
tors. Thus, the resulting circuitry would not require multipliers/dividers to
synthesize the nonlinear inputs to the modulator circuits, making analog
implementation much easier.
In order to use analog circuits to synthesize Equations (4.44) and (4.45),
these equations are transformed into the integration forms:
TA 2
[ 12dt = kxand'\x 1 (4.46)
T* 0
Tsd"u
~~~ f #2] dt k>2 a22^\2 (4.47)
r' 0
The duty ratio djj can be obtained by comparing the integration of a2i with
k2 a22dn through an integrator and a comparator. The duty ratio d12
can be solved in the same way. However, to do so, 22^2"' and fln^ii on
the left side of the equations should be available. Note that
TAi
J audt~audu (4.48)
5 0
TA2
TrT f 22^ ~ a 22^12 (4.49)
T 0
Then these sampled terms can be implemented by the integrator with reset
and hold, as shown in Figure 410. The integrator starts to integrate a^ after
reset by the RESET signal. At the moment djjTg, the integration is stopped by
the HOLD signal. The output of the integrator is held at the value of a^du
68
the control voltages in Equations (4.10) and (4.11). According to the theory of
the balanced threephase inverter in Chapter 2 and the SPWM in Chapter 3,
the output voltages remain balanced sinusoidal waveforms, but their ampli
tude becomes inversely proportional to the amplitude of the control voltage:
Vm = ^Vl + (coi?C)2 (4.12)
Therefore, the output voltages are not able to track the control voltage linearly
in the boost converter. This nonlinear relationship is verified by the simula
tion results. The amplitudes of the output voltages for different control volt
ages for shown in Figure 43.
Figure 43 The amplitude of the output voltages versus the
amplitude of control voltages for the ideal case.
15
modulated at a frequency sufficiently slower than the switching frequency.
Therefore, the duty ratio is also assumed as constant over the switching cycle.
In the analysis and modeling of switchedmode converters, attention
usually is restricted to lowfrequency components of voltages and currents.
The highfrequency components (also called ripples) are designed to be small
and can be neglected due to the combination of fast switching and proper
placement of filter corner frequencies. Therefore, the exact value with the
asterisk in the previous switchingfunction equations can be, approximately,
replaced by their lowfrequency value for analysis and modeling of lowfre
quency components. This modeling technique is called switchingfunction
averaging herein. The duty ratios in Equation (2.1) are then replaced by their
averaged values:
M
Iu= 1
k = 1
(2.4)
The pole voltage and the throw currents in Equations (2.2) and (2.3) are
replaced by their averaged values:
m (2.5)
 1
\kvk
k = 1
h = dklp (26)
All values in Equations (2.4) (2.6) vary slowly relative to the switching
frequency; thus, they characterize the lowfrequency properties of the SPMT
switch shown in Figure 21. With these equations, the SPMT switches can be
treated as components in the way we treat other components. The derivation
159
[39] ShihLiang, YingYu Tzou, Sliding Mode Control of a ClosedLoop
Regulated PWM Inverter under Large Load Variations, IEEE
Power Electronics Specialists Conference Record, Seattle, 1993, pp.
616622.
[40] Nadira SabanovicBehlilovic, Asif Sabanovic, Tamotsu
Ninomiya, PWM in ThreePhase Switching Converter Slid
ing Mode Solution, IEEE Power Electronics Specialists Confer
ence Record, Taipei, 1994, pp. 560565.
[41] Vietson M. Nguyen, C. Q. Lee, Indirect Implementations of Slid
ingMode Control Law in BuckType Converters, Proceedings of
IEEE Applied Power Electronics Conference, San Jose, 1996, pp.
111115.
[42] A. Shonung, H. Stemmier, Static Frequency Changers with Sub
harmonic Control in Conjunction with Reversible VariableSpeed
A.C. Drives, Brown Boveri Review, vol. 51, no. 8/9, 1964, pp. 555
557.
[43] K. Taniguchi, H. Irie, Trapezoidal Modulation Signal for Three
Phase PWM Inverters, IEEE Transactions on Industry Electron
ics, vol. 33, 1986, pp. 193200.
[44] K.B. Bose, A HighPerformance Pulsewidth Modulator for an
InverterFed Drive System Using a Microcomputer, IEEE Trans
actions on Industry Applications, vol. 19, 1983, pp. 235243.
[45] K. Taniguchi, H. Irie, S. Kaku, Real Time Operation of Three
Phase P.W.M. Inverter, IEEE Industry Application Society Annual
Meet Record, Toronto, 1985, pp. 442447.
[46] Andrzej M. Trzynadlowski, NonSinusoidal Modulating Functions
for ThreePhase Inverters, IEEE Power Electronics Specialists
Conference Record, Kyoto, 1988, pp. 477484.
[47] K. Ngo, et al., A New Flyback DCtoAC Threephase Converter
with Sinusiodal Outputs, IEEE Power Electronics Specialists Con
ference Record, Albuquerque, 1983, pp. 377388.
[48] J. Zubek, Pulsewidth Modulated Inverter Motor Drive with
Improved Modulation, IEEE Transactions on Industry Applica
tions, vol. LA11, November/December 1975, pp. 695703.
[49] H. W. van, der Broeck, H. Skudelny, G. Stanke, Analysis and Real
ization of a Pulse Width Modulator Based on Voltage Space Vec
41
the imaginary part is given by
vg n l1 mz
2D2e(Â£>p D(s)
where
D{s)
(
+
RC +
V
q.2lc2r
2Dl +
L ) LC 2 LC2R 3
5 + ~5 + Ys
2 D2eR) D] 2 D]
co
z
2 zr
i+i
CO,
R
2 L
CO
p
RC
If the design allows
1
where the LC corner is located at
co
O
J2
e_
Jlc
(2.56)
(2.57)
(2.58)
(2.59)
(2.60)
(2.61)
it suffices to approximate the poles by
D(s)
1 s
Qu0 +
(2.62)
where
cooL
(2.63)
39
can be modeled as a voltagecontrol voltage source and a currentcontrol cur
rent source, as shown in Figure 217(a).
+
Rv
bw
+
R
Vf
Figure 216 The equivalent circuit of the threephase boost
inverter in the ofb coordinates.
Figure 217 (a) The largesignal model of the dc transformer; (b)
its smallsignal model.
Application of smallsignal perturbation to the capacitor voltage vpa,
the inductor current ic, and the control variable de in the circuit yields
de De + de
 7c + *c
v
9 v pa
= V +v
pa pa
(2.54)
track the control voltages linearly, and they have lowdistortion sinusoidal
waveforms.
In summary, the synthesis and analysis techniques are developed for
linearization of a threephase boost inverter in the dissertation. As general
methods, they can be applied to other threephase topologies, multiphase or
multilevel converters.
Vll
108
tin, ^13 and 12 from Equations (5.33) (5.35). From Equation (5.33), dnu can
be expressed as
,n ygr~d ll" Vab
13 ~
vcb
(5.36)
Substitution of Equation (5.36) into Equation (5.34) and applying simple alge
bra yields:
2dnuv2cb = 2dnu \vabvcbv2ab) + Vgr(2vabvcb)
(5.37)
,n 1 2
Adding 2du vcb to both sides of Equation (5.37), yields
,n ,n 1 ,nl(VabVcb V ab Vcb) (2Vab Vcb)
11^11 = 2 + Vgr 71
V
cb
2v
cb
(5.38)
Multiplying with Equation (5.38) and assuming
1 S
d\\ dn d, j s
T. =dt[d")
(5.39)
where dn is a continuous function of time, Equation (5.38) then becomes a
continuous differential equation:
dtJ N (vabvcbv2abv2cb) T, (2vabvcb)
iSd> o = = + v
2 T
Vcb^s
gr 2
2 vc*r,
(5.40)
From Equation (5.33), + 1 can be expressed as
.+1 T/
13
(5.41)
Substitution of dnn in Equation (5.34) into (5.41) yields
,n +12 1 yil 2 1 2 .
13 vcb = vab~2d\3(2vabvabvcb)
(5.42)
49
very good sinusoidal waveforms. In recent years, much effort has been made
toward digitization of the SPWM [4346]. Online computation of instants of
intersection of the triangular carrier and sinusoidal reference waveforms is
not possible because no closedform solution is available for intersection
instants. Therefore, the reference sinusoidal waveforms have been replaced by
trapezoidal [43], stepped [44], or triangular waveforms [45]. The carrierbased
SPWM technique has disadvantages, such as attenuation of the fundamental
component and large switching losses. Most of all, the slope of the highfre
quency carrier in the PWM is constant and the duty ratios are linear functions
of the reference signals. Therefore, the SPWM is unable to implement linear
ization of nonlinear PWM converters.
3.2.2 SpaceVector Modulation
Spacevector modulation [49] (SVM) can utilize most of the power
source and reduce switching losses, which makes it the most popular PWM
technique in threephase converters. The SVM technique generates PWM sig
nals by averaging the three switchingstate vectors to equal the reference vec
tor over each switching cycle. Since the SVM involves a significant amount of
computation to determine the commutation instants of the switches, it is usu
ally implemented by digital signal processor (DSP) or microprocessor [50]. The
clock speed of the DSP or microprocessors, however, could impede the progress
of PWM toward higher frequency. Analog implementation is an alternative to
DSP for highspeed SVM. As with sinusoidal pulsewidth modulation, the SVM
114
20
>
10
o
S4
o
>4
o
predicted
Dm = 0.333
0^0
Vm with a step distur Dance
predicted
Dm = 0.167
da = 0.29
da = 0.145
db = 0.0065
Si
db = 0.147
dc = 0.295
dc= 0.0005
Sc
T
T
t(ms)
16.62 16.64 16.66 16.68
16.7
i
16.72
Figure 57 The simulation results of the step response of the
highorder LPWM modulator.
Dm\QO_ = 0.338 (5.60)
During the switching cycle after 0 =0, the amplitude of control voltages jumps
to 20 V, resulting in the amplitude of duty ratios Dm = 0.167 according to
Equation (5.48). The measured effective duty ratios of S1; S2, and S3 are 0.145,
0.147, and 0.0005. Similarly, the amplitude of effective duty ratios is
0.169
(5.61)
100
d=^
3 V g
a 2 A 2 2 2
V + Vl + V VViVV V v,
a /? c a 7 a c c b
(5.8)
Yjl
b 2 A 2 2 2
V + Vl + V VVVV v Vl
a c a 7 a c c b
(5.9)
3^
2 A 2 2 2
V + V + V V Vl V V V Vl
a b c a b a c c b
(5.10)
For a balanced threephase system,
2 2 2 9 2
V + Vl + V V Vl V V V Vl = ~V
va b e a b avc c b ^ m
(5.11)
where Vm is the amplitude of the balanced threephase voltages. Substitution
of Equation (5.11) into (5.8) (5.10) yields the effective duty ratios:
d
a
d\\~d2\
db
n
d\2~d22
d
c
dl3~d23
3AV
IXjl
3AVt
2V
sin
3AV,
sin
sin(cor)
(5.12)
The effective duty ratios of the PWM signals in Equation (5.12) are balanced
threephase sinusoids. The amplitude of the sinusoids is inversely propor
tional to the amplitude of control voltages.
The model of the highorder LPWM, as shown in (5.12), is derived in the
abc coordinates. Its model in the ofb coordinates can be obtained by applying
abcofb transformation, as described in Chapter 2, to Equation (5.12). In the
derivation, assume that all the initial phases relative to the transformation
37
capacitors by open circuits in the ofb equivalent circuit shown in Figure 213,
the resulting steadystate equivalent circuit is shown in Figure 214.
Figure 214 The steadystate equivalent circuit of the three
phase boost inverter in the ofb coordinate.
Reflecting the resistors (real and complex) in the secondary of the
transformer to the primary, the circuit in Figure 214 becomes a simple circuit
shown in Figure 215. Two conjugate resistors in Figure 215 form a voltage
+
+
D 0V f
V,
ev f
g +
D eV bw"
:D
2 R
e 1 j(Â£>RC
2 R
e 1 + jcRC
Figure 215 A simple circuit to solve the steadystate output volt
age and inductor current.
divider; thus the backward voltage and inductor current can be obtained eas
ily. The backward phasor is given by
20
of a PWM converter expeditiously, but also result in a model that is
insightful and amenable to implementation in standard circuit simula
tors.
In the following section, the SFA statespace equations shown in Equa
tion (2.10) and (2.12) (2.14) are transformed into an equivalent circuit in the
abc coordinates using the PWM Switch Model described in refs 18 and 19.
This equivalent circuit is useful in fast simulation and prediction of various
waveforms in the converter even though it is a timevariant circuit. Following
this section, the thesis provides a technique that transforms a timevariant
threephase converter into a timeinvariant equivalent circuit in the ofb coor
dinates. With the help of the ofb equivalent circuit, the steadystate and
dynamic analysis of the threephase converter becomes much easier.
2.2 Equivalent Circuit in the ABC Coordinates
According to the PWMSwitchModel technique [18], the PWM switch
can be modeled as a dc transformer that is a standard component in the simu
lator (such as Saber). The turns ratio is the duty ratio of the switching signal
of the PWM switch. This technique is used in dc converters [18], but its idea
can be extended to threephase PWM converters or other PWM converters.
Therefore, the SFA statespace equations of the PWM converter derived in the
previous section can be transformed into an equivalent circuit using several dc
transformers. This equivalent circuit is constructed by appropriate connec
tions between the dc transformers and other components. The connections are
28
phases. Note that Xf and xare complex conjugates and constant (dc) under
steady state.
2.4 Equivalent Circuit in the QFB Coordinates
2.4.1 Models of ThreePhase Components in the OFB Coordinates
A threephase converter consists of resistors, inductors, capacitors,
sources, and switches. Their models in the ofb coordinates are obtained by
applying abcofb transformation and retaining Kirchhoffs voltage and current
laws to their connectivity, that is, after transformation, circuit topology is the
same as before. In the following analysis, R is the resistor matrix, L is the
inductor matrix, C is the capacitor matrix, and I is the 3x3 identity matrix:
R = IR L IL C = IC (2.30)
Voltage sources. For the set of abc voltage sources in Figure 210(a),
application of Equations (2.22) (2.28) yields the set of ofb voltage sources in
Figure 210(b). The ofb voltages/currents are found from the abc voltages/cur
rents by Equation (2.29).
Resistors. For the set of abc resistors shown in Figure 210(c),
VRabc R*Rabc
application of Equations (2.22) (2.28) to (2.31) yields
vRofb = RlRofb
The ofb resistor set is thus as shown in Figure 210(d).
Inductors. For the set of abc inductors in Figure 210(e),
(2.31)
(2.32)
CHAPTER 5
ANALYSIS OF HIGHORDER LINEARIZING
PULSEWIDTH MODULATOR
This chapter analyzes the highorder linearizing pulsewidth modulator
(LPWM) for a balanced threephase boost inverter. In the first section, the
sampling effects of the highorder LPWM are neglected for simplicity. The
duty ratios of the output PWM signals of the modulator are shown as balanced
threephase sinusoids, and their amplitudes are inversely proportional to the
amplitude of the control voltages. The output voltages of the inverter equal
the control voltages multiplied by a constant gain. The nonlinearity of the
threephase boost inverter is eliminated by the highorder LPWM. Mean
while, line voltage regulation is improved because the output voltages of the
inverter are mainly controlled by the control voltages.
The sampling effects of the highorder LPWM are discussed in Section
2. A pole is contributed by the sampling effects. The location of this pole is
determined by the sampling frequency. When the sampling frequency is high,
the bandwith of the modulator is wide. Simulation results show that the mod
ulator is able to follow change in the control voltage within one switching
cycle. Therefore, sampling effects of the modulator can be neglected in the
design.
97
71
mentation of the LPWM is discussed in this thesis, it is intended to parallel
the recent advances in analog firstorder LPWM techniques for dc or sin
glephase converters [11, 24].
Analog implementation of the LPWM could be done by the conventional
PWM and nonlinear modulation function in Equation (4.16). To synthesize the
control voltage given by Equation (4.16), multipliers/dividers or other sophis
ticated circuits must be used. As a result, the complexity of the resulting cir
cuitry makes them impractical.
To avoid complicated circuits such as multipliers/dividers, the integra
tor (with reset) are used by the firstorder LPWM to solve modulation Equa
tion (4.16) in dc or singlephase converters [11, 24], as shown in Section 4.1 of
this chapter. The firstorder LPWMs can be used for some threephase con
verters, as long as the modulation equation does not have nonlinear terms of
control signals [59, 60].
For most threephase converters, a modulation equation (4.16) usually
contains some nonlinear terms of control signals. The synthesis of the LPWM
with firstorder modulators, therefore, will involve multipliers/dividers. The
highorder LPWM technique developed in this thesis is able to eliminate the
nonlinear terms of control voltages in the modulation equation (4.16). The
resulting circuitry, called highorder LPWM, contains integrators with reset
and hold, and also comparators. The inputs to integrators are just linear func
tions of control voltages. Therefore, the highhigh LPWM is simple and easy to
use.
139
Table 6.3 Measured and theoretical output voltage and input current.
Variables
Measured Value
Theoretical Value
Voltage amplitude
115 V
121 V
Voltage phase
22.5
25
Input current
6.84 A
9.98 A
With the highorder LPWM control, the nonlinear threephase boost
inverter becomes a linear converter. The linearity of the converter is tested
and verified by the prototype circuit. To test the linearity of the inverter, input
voltage Vg is kept constant; the output voltage is measured for different con
trol voltages. The curve of output voltage versus control voltage is shown in
Figure 627. It is tested under the condition of Vg =50 and A =40.
Figure 627 The linearity of the threephase boost inverter con
trolled by the highorder LPWM.
102
5.1.2 SteadyState Analysis
In this section, the output voltages of the threephase boost inverter
that is controlled by the highorder LPWM will be derived. From Equation
(3.32) of Chapter 3, the amplitude of the output voltages of SVM threephase
boost inverter is given by
Vom = ?h7i+(t0iiC)2 (5.18)
J Um
Dm in Equation (5.18) is provided by the highorder LPWM that is given by:
2V.
= 8
3AV.
(5.19)
Then, the amplitude Vom of the output voltages becomes
Vom = AVmJl+WC)
(5.20)
Of course, this result can be obtained by combining the steadystate ofb
equivalent circuit of the threephase boost inverter, as shown in Figure 214 of
Chapter 2, and the ofb model of the modulator, as shown in Equation (5.14).
The resulting ofb equivalent circuit is shown in Figure 52. The resistors in
Figure 52 are reflected to the primary side of the transformers. The voltage
V3W is then solved by the voltage divider as following:
v = Wfg{l~iaRC) <5'21)
Substitution of De in Equation (5.14) into Equation (5.21) yields
vbw = ^AVJljQRC)
(5.22)
156
Power Electronics Specialists Conference Record, Maggiore, 1996,
pp. 10131018.
[9] K. D. T. Ngo, Topology and Analysis in PWM Inversions, Rectifica
tion, and Cycloconversion, Ph.D. Thesis, California Institute of
Technology, Pasadena, May 1984.
[10] K. M. Smedley and S. Cuk, OneCycle Control of Switching Con
verters, Proceedings of IEEE Applied Power Electronics Confer
ence, Dallas, 1991, pp. 888896.
[11] Barry Arbetter and Dragan Maksimovic, FeedForward Pulse
Width Modulators for Switching Power Converters, IEEE Power
Electronics Specialists Conference Record, Atlanta, 1995, pp. 601
607.
[12] L. Calderone, L. Pinola, V. Varoli, Optimal Feedforward Compen
sation for PWM DC/DC Converters with Linear and Quadratic
Conversion Ratio, IEEE Transactions on Power Electronics, vol. 7,
no. 2, April 1992, pp. 349355.
[13] R. D. Middlebrook and S. M. Ck., A General Unified Approach to
Modeling Switching Converter Power Stages, IEEE Power Elec
tronics Specialist Conference Proceedings, Cleveland, 1976, pp. 18
34.
[14] Seth R. Sanders, George C.Verghese, Generalized Averaging
Method for Power Conversion Circuits, IEEE Transactions on
Power Electronics, vol. 6, no. 2, April 1991, pp. 251259.
[15] K. D. T. Ngo, Low Frequency Characterization of PWM Con
verter, IEEE Transactions on Power Electronics, vol. PE1, no. 4,
October 1986, pp. 223230.
[16] L. P. Huelsman, Basic Circuit Theory with Digital Computations,
Englewood Cliffs, NJ: PrenticeHall, Inc., 1972.
[17] D. D. White and H. H. Woodson, Electromechanical Energy Conver
sion, New York, John Wiley and Sons, 1959.
[18] V. Vorprian, Simplified Analysis of PWM Converters Using
Model of PWM Switch Part I: Continuous Conduction Mode, IEEE
Transactions on Aerospace and Electronic Systems, vol. 26, no. 3,
May 1990, pp. 490496.
8
a simple analog highorder linearizing PWM prototype circuit without
multipliers/dividers.
a circuitoriented analysis technique for balanced threephase convert
ers.
model and analysis of highorder linearizing PWM modulator.
simulation and experimental verification.
This dissertation is organized as follows. Chapter 2 characterizes the
lowfrequency property of the PWM switch and reviews the switchingfunction
averaging (SFA) technique. The derivation of the SFA statespace equations of
a threephase converter is presented. Components of balanced threephase
converters are modeled in the ofb coordinates, by which the timevariant
threephase converter can be graphically transformed into a timeinvariant
equivalent circuit for steadystate and dynamic analyses.
Chapter 3 reviews PWM techniques for dc and threephase converters,
in which largesignal linearization is emphasized. Two popular PWM tech
niques, sinusoidal PWM (SPWM) and spacevector modulation (SVM), are dis
cussed in details in this chapter.
Chapter 4 identifies the nonlinear problem in threephase converters.
Two largesignal linearization techniques for threephase PWM converters
are proposed in this chapter. One technique uses several firstorder linearizing
PWM circuits to synthesize duty ratios for the switches in the converter indi
vidually. It involves multipliers/dividers to compute the inputs to the integra
tors. The other technique employs the proposed highorder LPWM circuit to
27
0^(0 = J^C0(t)7t
where co is the instantaneous frequency;
T = 4
1
73
1
A
;(eTf)
1 e e
;(0r + y) '(er + y
1 e e
(2.25)
(2.26)
where
1 _
73
1
70
1
1
r 7rf)
J*T ~^T 23
A sr+f
rt
&T(t) = J^co(x)<7t())^
Note that T1 = (T*)t (the conjugate transpose matrix of T),
(2.27)
(2.28)
ofb
r 
0
X
O
Itsi
!
X
1

Xf
xbw
73 ;'(VM
_ 2
(2.29)
where x0 is the zerosequence component, Xf is the forward (rotating) phasor,
and xÂ¡,w is the backward (rotating) phasor. Both <)x and ())p are the initial
125
Figure 610 The experimental waveforms of sixstep reference
voltages of v^ and v2.
Figure 611 The highorder LPWM modulator used in the simu
lation.
52
been made to solve the variable frequency problem, they increase the complex
ity of the control circuit.
3.2.7 Direct Amplitude Control
The direct amplitude control [58] can make fundamental amplitude of
the output voltage directly follow the reference voltage. Using Fourier analy
sis, the algorithm is to equalize the subamplitude of the output voltage with
the subamplitude of the reference voltage for a complete fundamental cycle.
This technique involves a significant computation; thus, it usually is imple
mented by DSP or microprocessor.
Among the above PWM techniques, the SPWM and SVM are the most
popular in various threephase converters. However, due to the constantslope
carrier, both PWM methods can produce a nonlinear relationship between the
control and output voltages. This results in the output voltage failing to track
the reference voltage linearly. Nevertheless, it can be shown that both SPWM
and SVM can be developed into the linearizing PWM (LPWM) through the
proposed largesignal linearization technique in this thesis. In the following
sections, the conventional SPWM and SVM are discussed and synthesized, so
that the proposed linearizing PWM can be better appreciated.
A threephase boost inverter, shown in Figure 31, is used as an exam
ple to demonstrate how to synthesize the conventional SPWM and SVM.
The statespace equations of the inverter were given by Equations (2.15) 
(2.18) in Chapter 2 and repeated here:
6
Vm/Vg are listed in Table 1.2, where Vm is the amplitude of the output volt
ages; Vg is the amplitude of the input voltages. The conversion ratios in Table
1.2 are derived from the balanced threephase converters, and the input volt
age and current are assumed in phase in the rectifiers. In the table, Dm is the
amplitude of the sinusoidal control signal. D is the duty ratio of the dc switch
in the flyback topology. It should be noted that these conversion ratios are
derived by assuming that the impedance of input/output reactive components
are small at input/output frequency and can be neglected.
Table 1.2 Voltage conversion ratios Vm/Vg.
Converter Topology
Inverter
Rectifier
Buck
Dm/2
Dm
Boost
1/Dm
2m
Flyback
D/Dm
D/Dm
Although PWM converters are the most popular in various power con
versions, they have an inherent problem: nonlinearity. It keeps the output
voltage from tracking the control signal, gives rise to waveform distortion, and
degrades the performance of the converter. The reason that generates the non
linearity can be found by investigating the voltage conversion ratios of PWM
converters in Table 1.1 for dc converters, in Table 1.2 for threephase convert
ers, and duty ratios shown in Equation (1.1). The conventional PWM with a
constant slope carrier produces a linear relationship between the duty ratio
and the control signal as shown in Equation (1.1). When the duty ratio is used
22
Figure 24 The de transformer with the duty ratio of d.
Figure 25 The equivalent circuit of the threephase boost
inverter in the abc coordinates.
that in Equation (2.15). The capacitor currents in the equivalent circuit also
are found to be the same as those in Equations (2.16) (2.18). Therefore, the
equivalent circuit exactly represents the lowfrequency properties of the
threephase boost inverter. Because there are no real switches in the equiva
lent circuit, the simulation of this circuit is expedited and memory space of the
computer is also greatly saved. The simulation results of the equivalent cir
cuit are the lowfrequency components of the voltages and currents in the
148
Commonmode current are shown in Figure 640(a) that flow in both
signal path and its return path in the same direction, they generate common
mode noise and contaminate reference signals in the experiment. PWM sig
nals generated from these contaminated reference signals are distorted, and
would not operate the inverter correctly. The technique used to reduce com
_ < N differentialmode
signal paths / \currents
>i
its return.
4*
1CMi^
1 > i
commonmode
i currents
system ground
t<
(a)
(b)
reference Q @ 24 KHz
input voltage
of the inverter
rm?
2 mH
signals
TAA/
i/YY\
control
voltage TvA^r
irm.
1 mH
test system
(control and
inverter)
(c)
Figure 640 (a) commonmode currents; (b) commonmode filter; (c)
test system showing three commonmode filters.
monmode noise in this experiment is to add commonmode filters to the refer
ence signals. The commonmode filter is constructed just by wrapping signal
and its return wires on the same magnetic core, as shown in Figure 640(b).
130
vol
V02
vo3
V04
20 May 2000
23:29:08
Figure 616 The experimental waveforms of the integrators of
the prototype highorder LPWM.
TekgHjJii 10.0MS/S 79 Acqs
Figure 617 The experimental waveforms of the comparator for
the PWM signal Sj.
23
inverter that are sufficient for us to predict various waveforms and design the
inverter.
To appreciate how fast and accurate the equivalent circuit is, the circuit
in Figure 25 is simulated in Saber. The simulation results are compared with
the realtime simulation of the threephase boost inverter. Supposing that the
PWM method applied to the threephase boost inverter is continuous SPWM,
one choice for duty ratios is
i D
1 m .
 + sin(0)
dll
d2\
dl2
=
1 Dm (a 271^
3 3 yd 3 )
d22
d\3
d2S_
1 m 2jr^
 + sin 0 +
3 3 yds)
i D
1 m /Q .
rTsm(Od)
1 Dn 2n
3 3 V d 3
1 Dm (a 2k
3 3 {d 3
(2.19)
The effective duty ratios are
d
a
dn~d22
db
=
d\2~d22
d
c
d\S~d2S
2D
m
2D
m
3
2D
m
sin
sin(coi)
(T
l
sin m +
2k
(2.20)
The simulated inverter has the following parameters: Dm = 0.9, Vg = 200 V, Q =
2k(100 Hz), R = 10 Ohm, C = 100 (iF. The realtime simulation result is shown in
Figure 26, and the time for 40 ms simulation is 30 seconds. The results of the
simulation with the equivalent circuit is shown in Figure 27, and the time for
40 ms simulation is only 0.4 seconds.
33
d
a
d 1 d r\
1 a 2a
db
=
d\b~d2b
d
d 1 d~
c
l c 2c
2D
m
3 cos(e)
2Dm ( 2n
C0Vd~T
2D
m 2k
,cos(9+t
(2.44)
The pole voltages and the throw currents can be expressed as
v =
ps
d did
a b c
ta
'tb
tc
d abcV tabc
(2.45)
i
d
ta
a
ltb
*tabc
db
/
d
tc
c
i d i
ps abc ps
(2.46)
The voltage reference node of the proceeding equations is assumed to be the
common node of the threephase voltages. Application of Equations (2.22) 
(2.28) to (2.45) and (2.46) yields
* T
Vps ~ 0fb^ Vtofb
ltofb d ofb^ps
(2.47)
^ rn
where (d q^) is the conjugate transpose matrix of dofb;
ofb
q D\nd ^T^ ~j($d ~ ^j)
J3 J3
(2.48)
The ofb model for the threephase SPTT switches is thus as shown in Figure
2ll(c). Before leaving this section, it is worth noting that, unlike the dq
133
Tek IHiIiV 50.0kS/s l7Acqs
[ T ]
Sn
Sis
Sl2
^21
$22
^23
21 May 2000
00:04:41
Figure 621 Experimental waveforms of PWM signals for the
six switches in the inverter.
R1>
R3> l
... /. uim
UUH
JPPPPPPL
aim
Ml.
1
HlUfc I
JWPfl A
WWi
mu
HUH
4,.m m
HIUH
JIIIIIIL
Ch2 5.00 VM2.00ms Ch1 J 2.48V
5.00 V Ch4 5.00 V
Ref4 5.00 V 2.00ms
Tek anmi so.oks/s
72 Acqs
T
]
Ref4 5.00 V 2.00ms
dli
di3
di2
21
23
22
21 May 2000
11:38:49
Figure 622 Experimental waveforms of duty ratios of PWM sig
nals for the six switches in the inverter.
CHAPTER 4
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR
This chapter investigates the feasibility of largesignal linearization of
threephase PWM converters by analog linearizing pulsewidth modulator
(LPWM). The study shows that threephase PWM converters have nonlinear
relationships between the control and output voltages when they are con
trolled by the conventional analog SPWM or SVM modulators. Some sophisti
cated analog circuits may employ analog multipliers/dividers to compute the
switching instants for threephase converters to implement linearization.
However, the complexity of the resulting circuitry makes them impractical.
The firstorder linearizing PWM circuit uses integrators to compute commuta
tion instants to linearize controltooutput relationship in dc or singlephase
converters. They can also be used to control threephase converters, but, as
indicated in this chapter, the inputs to the integrators are nonlinear function
of control voltages, resulting in use of analog multipliers/dividers.
A highorder linearizing PWM modulator is developed in this chapter.
It is able to make output voltages of threephase PWM converters track con
trol signals linearly even in the nonlinear topologies. Instead of multipliers/
dividers, the highorder linearizing PWM uses only integrators with the reset,
and sample/hold to compute the switching instants for the switches. The
62
57
Z.da = tan l((RC)
(3.20)
where II is determined by Equation (3.11).
Combining Equations (3.11) and (3.19), one can find the amplitude of the out
put voltage is
(3.21)
m
3.4 Synthesis of SpaceVector Modulation
The balanced threephase voltages va, vb, and vc are shown in Figure 3
2. In spacevector modulation (SVM), the phase voltages are divided into six
segments, and each segment occupies 60. In each segment, one SPTT switch
in Figure 31 is permanently attached to one of the three capacitors as the
other sweeps through all three. The position of the stationary switch, as well
as the sweeping ones, are determined by sixstep sequence.
In the first segment, 060, vb < va and vb < vc. Let d22 = 1, and d2i =
d23 = 0, i. e., S22 is on, S21 and S23 are off all the time during this segment.
The switches, S11; S12, and S13, are switched at high switching frequency. The
corresponding duty ratios, d11; d13, and d12, are determined by Equations
(3.12) (3.14), respectively. Taking advantage of d22 = 1 and d21 = d23 = 0, then
Equations (3.12) (3.14) become
(3.22)
36
Figure 213 The equivalent circuit of the threephase boost
inverter in the ofb coordinates.
The ac components are transformed to their ofb models in the ofb equiv
alent circuit of the inverter. The resulting timeinvariant equivalent circuit in
ofb coordinates is shown in Figure 213. Since zerosequence is zero in the bal
anced threephase converter, the zero sequence circuit is excluded from Figure
213. The transformer turns ratios df and dt>w in the ofb circuit are time
invariant, and they have the same value when (j)d = <])T = 0 in Equation (2.48)
D
so that Djr = D^w
which is represented by De in Figure 213:
D
D, =
orD
bw
m
J3
(2.49)
2.5 Graphical SteadyState Analysis
To analyze the threephase boost inverter in the ofb coordinates under
steadystate condition, replace all the inductors by short circuits and all the
136
Since the gain of the integrator K is not equal to the switching frequency due
to the component errors, Equation (6.9) is modified by
voa = AVmjJl + ((RC)2sin((te) (6.11)
J S
where fs is the switching frequency. Therefore, the output voltages of the
threephase boost track the control voltages linearly. The amplitude is deter
mined by the amplitude of the control voltage and other parameters such as
the ratio of the inputvoltage divider, load resistance, filter capacitance, and
ratio between switching frequency and the gain of the integrator. However, all
these parameters are independent of the operating condition of the inverter.
There is a phase shift between the output and control voltages that is caused
by the output capacitor.
The output voltages of the threephase inverter measured at 600 W are
shown in Figure 624; the output voltages measured at 1000 W are shown in
Figure 625. Both measurement results show that the highorder LPWM mod
ulator is able to generate lowdistortion sinusoidal waveforms in the three
phase boost inverter.
Figure 626 shows inductor current, input voltage, control voltage, and
output voltage. A phase shift between the control and output voltages exists
due to the load resistor and filter capacitor. The value can be found:
0 = vl'l68 360 = 25 (6.12)
16.67 ms
This value is quite close to its theoretical values decided by Equation (6.10):
7
to control nonlinear converters that have a nonlinear relationship between the
duty ratio and the output voltage, as shown in Table 1.1 and Table 1.2, the
output voltage is proven to be a nonlinear function of the control voltage.
The nonlinear problem of PWM converters has been solved mainly by
the smallsignal linearization technique of negative feedback control.
Recently, largesignal PWM linearization techniques were proposed in [10]
and [11]. As an alternative linearization technique, the largesignal PWM lin
earization features an openloop, steadystate linear controltooutput rela
tionship, regardless of operating conditions, leading to simple and stable
control circuit design. Moreover, this technique has better line voltage regula
tion not only for the linear converters, but also for the nonlinear converters
that are difficult for the feedforward control [12],
The largesignal PWM linearization techniques in [10, 11] can success
fully solve the nonlinear problem for dcdc converters and singlephase invert
ers, in which the PWM controller deals only with a single control variable.
However, threephase converters or multiphase converters have more than
one control variable. Therefore, the firstorder PWM linearization is limited in
threephase converters or multiphase converters. Nevertheless, the idea of
the largesignal linearization is a useful concept that could be extended to the
threephase converters, thus motivating the present research and leading to
the following objectives of the thesis:
a general way to synthesize the highorder linearizing PWM for bal
anced threephase converters.
42
From Equation (2.62), the poles of the threephase boost inverter consist of a
real pole and complex poles. Like the dc boost converter, the bandwidth is
affected by the duty ratio De.
The controltooutput transfer function of the inverter can be found
from Equations (2.55) and (2.56), which is given by
(2.64)
Letting de = 0, the inputtobackward phasor transfer function of the inverter
can be found in the circuit in Figure 218, which is given by
(2.65)
The audiosusceptibility, that is, the inputtooutput transfer function, is
solved from Equation (2.65)
(2.66)
The transfer functions graphically derived from the smallsignal equivalent
circuit in Figure 218 completely agree with those derived by the equationori
ented method [9]. However, the graphical derivation is much simpler than the
manipulation of statespace equations.
In conclusion, among existing modeling techniques, the switchingfunc
tion averaging is the easiest technique to model threephase converters. The
equivalent circuit of the converter in the abc coordinates constructed from the
81
and P and Q functions, and it synthesizes the LPWM directly from the
reduced SFA equations. The resulting LPWM circuit is called highorder
LPWM because it uses more than one integrators to get one duty ratio. The
analog highorder LPWM modulator is developed for a general PWM con
verter in this section. It employs only integrators (with reset and hold) to com
pute the commutation instants of the switches. The inputs to the integrators
and comparators are linear functions of the control and input voltages. The
synthesis procedure of the highorder LPWM is demonstrated through a
threephase boost inverter. The modulator, together with the inverter, is simu
lated in Saber. The result shows that the output voltages can track the control
voltages linearly, and the highorder LPWM modulator is simple and easy to
use.
The synthesis of the highorder LPWM is based on the steadystate SFA
equations of the PWM converter, which are just linear functions of state vari
ables and duty ratios of the switches, as described in Chapter 2. In the
steadystate condition, the derivative terms in statespace equations are zero.
As an example, the steadystate SFA equations of a PWM converter with two
independent duty ratios are given by
1 jd 1 j "4* Â£Z)2^12 ^"1 (4.40)
a2\d\\ + a22^\2 = ^2 (4.41)
where coefficients a22, ki and k2 are related to control and input voltages.
For the LPWM modulator, they are reference and input voltages.
67
threephase output voltages va, v^, and vc are purely sinusoidal. The control
voltages vcnti_a and vcnti_b into the SPWM are sinusoidal waveforms with the
amplitude Dm:
vc*tla = 5 + Tpsin(G>t) (4.10)
Vclt =  + 3!sin0120') (4.11)
The dc offset in the control voltages is needed to guarantee the duty ratio pos
itive.
The control voltages are compared with the constantslope carrier in
the SPWM. The duty ratios of the resulting PWM signals are proportional to
Figure 42 The threephase boost inverter controlled by the
conventional SPWM modulator.
158
[29] H. Jin, G. Joos, M. Pande, P. D. Ziogas, Feedforward Techniques
Using Voltage Integral DutyCycle Control, IEEE Power Electron
ics Specialists Conference Record, Toledo, 1992, pp. 370377.
[30] Manish Pande, Hua Kin, Geza Joos, Modulated Integral Con
trol Technique for Compensating Switch Delays and Nonideal
DC Bus in ThreePhase Voltage Source Inverters, IEEE
IECON93 Proceedings, Hawaii, 1993, pp. 12221227.
[31] Regan Zane and Dragan Maksimovic, Nonlinearcarrier Con
trol for HighPowerFactor Rectifiers Based on Flyback, Cuk or
SEPIC Converters, Proceedings of IEEE Applied Power Electron
ics Conference, San Jose, 1996, pp. 814820.
[32] Dragan Maksimovic, Yungtaek Jang, and Robert W. Erickson,
NonlinearCarrier Control for HighPowerFactor Boost Rectifi
ers, IEEE Transactions on Power Electronics, vol. 11, no. 4, July
1996, pp. 501510.
[33] Regan Zane and Dragan Maksimovic, Modeling of HighPower
Factor Rectifiers Based on Switching Converters with Nonlinear
Carrier Control, IEEE Transactions on Power Electronics, vol. 7,
no. 2, April 1992, pp. 349355.
[34] Esam H. Ismail, Robert W. Erickson, Application of onecycle con
trol to three phase high quality resonant rectifier, IEEE Power
Electronics Specialists Conference Record, Atlanta, 1995, pp. 1183
1190.
[35] Joel P. Gegner, C. Q. Lee, Linear Peak Current Mode Control: A
Simple Active Power Factor Correction Control Technique, IEEE
Power Electronics Specialists Conference Record, Maggiore, 1996,
pp. 196202.
[36] R. Redi and N. O. Sodal, NearOptimum Dynamic Regulation of
DCDC Converters Using Feedforward of Output Current and
Input Voltage With CurrentMode Control, IEEE Transactions on
Power Electronics, vol. 1, no. 3, July 1986, pp. 181192.
[37] Dennis Gyma, A Novel Control Method to Minimize Distortion
in AC Inverters, Proceedings of IEEE Applied Power Electronics
Conference, Orlando, 1994, pp. 941946.
[38] YanFei Liu, and Paresh C. Sen, A Novel Method to Achieve Zero
Voltage Regulation in Buck Converter, IEEE Transactions on
Power Electronics, vol. 10, no. 3, May 1995, pp. 292301.
157
[19] K. D. T. Ngo, Simplified Analysis of PWM Converters Using Alter
nate Forms of the PWM Switch Models, IEEE Transactions on
Aerospace and Electronic Systems, vol. 35, no. 4, October 1999, pp.
12831292.
[20] E. van Dijk, J. Ben Klaassens, PWMSwitch Modeling of DCDC
Converters, IEEE Transactions on Power Electronics, vol. PE10,
no. 6, November 1995, pp. 659665.
[21] C. T. Rim, et al., Transformers as Equivalent Circuits for
Switches: General Proofs and DQ TransformationBased Analy
ses, IEEE Transactions on Industry Applications, vol. 26, no. 4,
July/August 1990, pp. 777785.
[22] Huang Xu, Jun Chen, K.D.T. Ngo, Graphical DC Analysis of
ThreePhase PWM Converters Using a Complex Transformation,
IEEE ISCAS 2000 Proceedings, vol. Ill, Geneva, 2000, pp. 243246.
[23] Keyue M. Smedley, Integrators in Pulsewidth Modulation, IEEE
Power Electronics Specialists Conference Record, Maggiore, 1996,
pp. 773781.
[24] Zheren Lai, Keyue Ma Smedley, A General Constantfrequency
Pulsewidth Modulator and Its Applications, IEEE Transactions
on Circuits and SystemsI: Fundamental Theory and Applications,
vol. 45, no. 4, April 1998, pp. 386396.
[25] Zheren Lai, Keyue Ma Smedley, A Family of ContinuousConduc
tionMode PowerFactorCorrection Controllers Based on the Gen
eral Pulsewidth Modulator, IEEE Transactions on Power
Electronics, vol. 13, no. 3, May 1998, pp. 501510.
[26] K. Mark Smith, Zheren Lai, Keyue M. Smedley, A New PWM Con
troller with One Cycle Response, IEEE Power Electronics Special
ists Conference Record, St. Louis, 1997, pp. 970976.
[27] Zheren Lai, Keyue M. Smedley, and Yunhong Ma, Time Quantity
OneCycle Control for Power Factor Correctors, Proceedings of
IEEE Applied Power Electronics Conference, San Jose, 1996, pp.
821827.
[28] Zheren Lai, Keyue M. Smedley, A Low Distortion Switching Audio
Power Amplifier, IEEE Power Electronics Specialists Conference
Record, Atlanta, 1995, pp. 174180.
126
+5 V
0.1.uF
Reset
Hold
v
o
Figure 613 The operation waveforms of the integrator with sam
ple and hold.
tor becomes zero. After reset, the integrator starts to integrate the input volt
age Vj. The output voltage v0 is
v
O
0
(6.4)
HIGHORDER LINEARIZING PULSEWIDTH MODULATOR
FOR THREEPHASE POWER CONVERTERS
BY
JUN CHEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000
147
A better layout is shown in Figure 638(b), in which signal path is directly con
nected to the source of power MOSFET.
+12 V
mosfetJ
driver >o
bad layout
MOSFET
parasitic
MOSFET
driver
+12 V good layout
MOSFET
parasitic
Figure 638 (a) power and signal share ground; (b) power and signal
grounds are separated.
The test system consists of four circuit boards. There are signal connec
tions among these boards. Each signal path should leave one circuit board and
arrive another board together with its return path, as shown in Figure 639.
signal pat
component
component
ground connections
between two boards
Figure 639 Signal connections between two circuit boards.
In general, the bottom line for layout is to keep the layout small and neat.
Each current path and its return should be clearly identifiable and run paral
lel, in close proximity.
80
Figure 49 The simulation results of output and control voltages
of the threephase boost inverter controlled by
firstorder LPWM circuits. va = Fmsin(Q.t) with Vm =
262 V and Q = 27t(60Hz).
4.5 Linearization by HighOrder LPWM
The main problem for firstorder LPWM circuits to linearize
threephase converters is that multipliers/dividers may be employed to syn
thesize the inputs to the integrators. The problem is solved by a technique
presented in this section. With the help of this technique, the SFA equations of
a threephase converter are reduced into a set of SFA equations that have only
one unknown duty ratio in each of them, and they have coefficients of linear
functions of the control voltages. Different from the firstorder LPWM imple
mentation, this technique does not need to find the expressions of duty ratios,
118
The analog highorder LPWM, the reference circuit, and the gatedrive
logic in Figure 6l(a) are presented in the first section of this chapter. The cor
responding waveforms measured from the prototype circuits are also included.
The second section presents experimental results and discusses how the high
order LPWM improves the performance of a nonlinear threephase boost
inverter. Some practical issues in the experiment are discussed in the last sec
tion.
6.1 Analog Implementation of HighOrder LPWM
The highorder LPWM synthesizes the PWM signals S1; S2, and S3
from the sixstep reference voltages, Vj and v2, and input voltage Vg, as shown
in Figure 6l(a). The circuit that generates and v2 consists of three parts.
The first part is to generate threephase reference voltages from a single
phase voltage. The second part is to generate digital sixstep signals Sgl Sg6.
The third part is to produce the sixstep reference voltages Vi and v2 using
analog multiplexers. The circuit in Figure 62 takes a singlephase voltage va
Figure 62 The circuit to generate balanced threephase volt
ages.
69
4.3 LargeSignal Linearization of PWM Converters
A typical PWM converter controlled by the LPWM is shown in Figure
44. The PWM converter could have single or multiple input/output variables.
The variables at the input/output side could be dc or ac. For example, in
threephase inverters, the input is dc voltage and the output are threephase
voltages, in which the output voltages are controlled variables. In threephase
rectifiers, the inputs are threephase voltages and the output is dc voltage,
where both output dc voltage and the input currents are controlled variables.
In order to simplify the explanation of largesignal linearization technique
presented in this thesis, we consider only the output voltage vQ as the con
trolled variable. The objective is to make the output voltage vQ track the con
trol voltage vc linearly through the LPWM.
vg(ac, dc)
ig (ac, dc)
Clock
POWER STAGE
d(ac, dc)
HIGHORDER
LPWM
vG(ac, dc)
iG (ac, dc)
vc (ac, dc)
ic (ac, dc)
Figure 44 The converter controlled by the highorder LPWM.
9
solve SFA equations. The inputs to the highorder LPWM circuit are linear
functions of the control and input voltages. Therefore, no multipliers/dividers
are required in the circuit, making analog implementation simple.
Chapter 5 focuses on the analysis of the LPWMcontrolled converters.
The largesignal and smallsignal models of the LPWM are derived in this
chapter. The time delay caused by sampling effects in the highorder LPWM is
also investigated.
Chapter 6 concentrates on implementation and experimentation of the
proposed analog highorder linearizing PWM. The experimental circuits and
results are presented in this chapter.
Chapter 7 consists of the summary and conclusion of this dissertation.
13
The operation of the throw k in Figure 21 is specified and modulated
by the switching function d*]^, as shown in Figure 22, where the asterisk *
denotes the instantaneous switching function. The function is one when the
throw is closed and zero when the throw is open. A switching function defined
this way can always be assigned to any throw in the converter without prior
knowledge of modulation strategy or sequence of switched topologies. There
fore, this switch model allows derivation of statespace equations of a PWM
converter without specifying modulation strategy. After the statespace equa
tions are derived, they can be used for any PWM strategy to do a specific anal
ysis [9],
In Figure 21, v^v*]^ are throw voltages, i*ii*M are throw currents,
i*p is the pole current, and v*p is the pole voltage. The asterisk denotes the
exact value. In switching functions shown in Figure 22 d11d1jy[ stand for duty
ratios of switching functions. Ts is the switching period.
1
Figure 22 Switching functions of SPMT switches.
34
transformation, the ofb transformation results in decoupled zerosequence,
forward, and backward components subcircuits.
SPTT switch
+
*ps 
dla/
31
'ps
die/
d2a V d2b V d2c
ta
o vf
. hb ta
+ vtb
+
V
ps
1,
tc
o vtC
(a)
(c)
i\i
p +
. ta
db:l
. bb
M
w +
. tb
dc:l
^tc
5 ii
p +
vtc
(b)
Figure 211 (a) Threephase SPTT switches; (b) switch model in
the abc coordinates; (c) switch model in the ofb coordi
nates.
2.4.2 Derivation of Equivalent Circuit in the OFB Coordinates
The equivalent circuit for a balanced threephase PWM converter can
be constructed graphically in the ofb coordinates just by replacing each set of
threephase switches by appropriately connected ofb transformers, and each
set of threephase components by the corresponding ofb component models.
The resulting ofb equivalent circuit is timeinvariant, in which the forward
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to my research committee
chairman, Dr. Khai D. T. Ngo, for welcoming me to the power electronics group
at the University of Florida. He provided constant support and encourage
ment for my study and research. I learned not only the knowledge of power
electronics, but also a work attitude that has greatly reshaped my career. I
also wish to thank Dr. Dennis P. Carroll, Dr. Alexander Domijan, Dr. Vladimir
A. Rakov, and Dr. Loc VuQuoc for their participation on my research commit
tee.
I am very grateful to American Research Corporation for its financial
support and projects, and also Texas Instruments for a TI fellowship.
My special thanks go to my colleagues, Jun Xu and Paiboon Nakmah
achal, for their helpful discussions and suggestions in my project and disserta
tion.
There is no word for me to describe my gratitude to my wife, Yin Xie,
who spent her time taking care of our family when I was working at the lab
day and night. Without her help and patience, I would not know how to finish
my research and dissertation.
n
BIOGRAPHICAL SKETCH
Jun Chen was born in Hubei Province, P. R. China, on April 8, 1966. He
received his B.S. degree and M.S. degree in electrical engineering from
Chendu University of Science and Technology, Chendu, P. R. China, in 1986
and in 1989, respectively. Between 1989 and 1994, he joined the Automation
and Electrification Institute in Southwest Jiaotong University, Chendu, P. R.
China, as a research engineer, where he was in charge of development of tele
control equipment for power systems. In May 1994, he was admitted to the
electrical engineering department of Hong Kong Polytechnic University as a
Ph.D. student, where he conducted research in power electronics. Since 1996,
he has been studying in the Power Management (PMag) Group of the electri
cal and computer engineering department at the University of Florida,
Gainesville, Florida. His research is focused on power conversion and manage
ment.
162
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53
diL
L = Vg(dnd2l)va(dnd22)vb{dud13)\
(3.1)
dvn 2v vhvr
(3.2)
dv, 2 vh~vnvr
c = (*,/)<*4^'
(3.3)
dv
C = (du d23)iL
2vc~vavb
3 R
(3.4)
In the above equations, va, v^, and vc are balanced threephase voltages. Their
frequency and amplitude are known from the specifications:
va Vmsin(cor) vb = V^siihcof120) vc = Vmsin(coi + 120) (3.5)
Duty ratios in the statespace equations d^ d23 are unknown, which will be
synthesized from Equations (3.1) (3.4).
For a balanced threephase system, Equation (3.2) can be expressed as
101
matrix T and duty ratio vector d are zero. The resulting timeinvariant ofb
model of the highorder LPWM is
Dofb=[DoDf DbJ =
V
8
V
~\T
g
J3AVm J3AV
m J
(5.13)
The zerosequence component of the effective duty ratios is zero; the forward
and backwardsequence components become the same:
De Df Dbw 
V
73 AV
(5.14)
The smallsignal model of the highorder LPWM can be obtained by
perturbation of (5.14) around a dc operating point specified by the input volt
age Vg, the control voltage Vm, as well as the effective duty ratio De. The
smallsignal component of the duty ratio de becomes a linear combination of
the smallsignal perturbations in Vg and Vm:
de KJm + Kgvg (5.15)
The gain Km and Kf are found by differentiation of (5.15) with respect to Vg
and Vm:
= T = _J__ _Rl
V,n 73 AV2m Vm
K = h = 1 1
* *7 73 AVm
(5.16)
(5.17)
105
\ om
G(s) = =
id:
h
(
A
v
s
(0.
1 +
CO
z 1
D(5)
(5.26)
Combined with transfer function Km of the modulator, the controltooutput
transfer function Gc of the LPWMcontrolled threephase boost inverter is
given by
v v d
G.f \ om om ue ^, v v
c(s) = = = G(s)Kn
Vm
de vm
(
h
A
co
z\
D(s)
(5.27)
Therefore, at the low frequency, the smallsignal controltooutput gain is a
constant A, independent of operating conditions, which can simplify feedback
loop design of the LPWMcontrolled power converter. It is to note that the
LPWM does not have any effect on corner frequencies of the transfer function.
These corner frequencies remains the same as those in the conventional
PWMcontrolled inverter.
Let de = 0, the inputtooutput transfer function of the power stage is
shown in Equation (2.66) of Chapter 2 and given by
v i
H(s) =
vg 2 De
1+1
A
UJ
1
l + J
D(s)
(5.28)
From Equations (5.26) and (5.28), one can find that
gw=4:(i;CK)
(5.29)
From Figure 53, the inputtooutput transfer function of the LPWMcon
trolled threephase boost inverter can be expressed by
89
Figure 414 Sixstepped reference voltage signals to the LPWM.
Table 4.1 Sixstep reference voltages and duty ratios.
vx
vy
dx
dy
dz
a 5
0
0
Seg.I
vab
vcb
dn
dn
d12
d22
d23
d21
Seg. II
vac
vab
d23
d22
d21
dn
d12
d13
Seg. Ill
vbc
vac
d12
dn
d13
d23
d21
d22
Seg. IV
vba
vbc
d21
d23
d22
di2
dl3
dll
Seg. V
vca
vba
d13
d12
dll
d21
d22
d23
Seg. VI
vcb
vca
d22
d21
d23
d13
dn
d12
In order to use the proposed highorder LPWM, it is necessary to trans
form the preceding equations into the following forms:
dnv = V d" 1 v
y y g x yx
(4.60)
, n 1 ,n 1 1 ,n.
dxvy = 2^dx Vx> + 2 V2vjcV
(4.61)
TABLE OF CONTENTS
pages
ACKNOWLEDGMENTS ii
ABSTRACT v
CHAPTERS
1 INTRODUCTION 1
2 MODELING AND ANALYSIS OF THREEPHASE CONVERTERS 10
2.1 Derivation of StateSpace Equations of PWM Converters 11
2.2 Equivalent Circuit in the ABC Coordinates 20
2.3 ABCOFB Transformation 26
2.4 Equivalent Circuit in the OFB Coordinates 28
2.5 Graphical SteadyState Analysis 36
2.6 Graphical SmallSignal Analysis 38
3 REVIEW OF PULSEWIDTH MODULATION 44
3.1 Pulsewidth Modulation for DC Converters 44
3.2 Pulsewidth Modulation for ThreePhase Converters 48
3.3 Synthesis of Continuous Sinusoidal Pulsewidth Modulation 55
3.4 Synthesis of SpaceVector Modulation 57
4 HIGHORDER LINEARIZING PULSEWIDTH MODULATOR 62
4.1 FirstOrder Linearizing Pulsewidth Modulator 63
4.2 Nonlinear Problem in ThreePhase Converters 66
4.3 LargeSignal Linearization of PWM Converters 69
4.4 Linearization by FirstOrder LPWM 72
4.5 Linearization by HighOrder LPWM 80
5 ANALYSIS OF HIGHORDER LINEARIZING
PULSEWIDTH MODULATOR 97
5.1 Analysis of HighOrder Linearizing PWM 98
5.2 Sampling Effects in HighOrder LPWM 107
iii
63
inputs to the integrators are just linear functions of the control and state vari
ables.
In the first section of this chapter, the firstorder LPWM is reviewed,
which is helpful to understand the concept of largesignal linearization and
analog implementation of the LPWM. The nonlinear problem, caused by the
conventional PWM modulator in threephase PWM converters, is identified in
the second section. A general way to linearize PWM converters is discussed in
the third section. Implementation of the LPWM modulator by firstorder
LPWM circuits for a threephase inverter is given in the fourth section. The
fifth section presents the highorder LPWM that linearizes threephase con
verters with simple analog circuits. The techniques to synthesize a highorder
LPWM and eliminate multipliers/dividers in the LPWM circuit of threephase
converters is discussed. An analog implementation of the highorder LPWM
for a threephase converter is derived and simulated.
4.1 FirstOrder Linearizing Pulsewidth Modulator
The carrier signal in the conventional PWM modulator has the fixed
frequency and constant slope. The duty ratio of switching signals generated by
the conventional PWM is directly proportional to the control signal. The car
rier signal in the linearizing PWM (LPWM) has the fixed frequency, but vary
ing slope. The duty ratio generated from the LPWM is a nonlinear function of
the control signal and input voltage.

