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SMALL SIGNAL MODELING OF DC-DC POWER

CONVERTERS BASED ON SEPARATION OF

VARIABLES

BY

NG POH KEONG

(B.S.E.E, University of Kentucky, USA)

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

A THESIS SUBMITTED

FOR THE DEGREE OF MASTER OF ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2003

ACKNOWLEDGEMENT I would like to thank my family and parents for giving me the moral support and

encouragement to complete my thesis. Many thanks to my fiancee, Joy Tay, who encouraged me

to give my best.

I am very thankful to my thesis main advisor, Dr. Oruganti and co-advisor Dr. Y. C

Liang. Words are inadequate to express my appreciation and gratitude to Dr. Oruganti for his

guidance, patience, and expertise to my thesis and academic life.

I am also very thankful to all my friends in the Center for Power Electronics (1999~2001)

in the National University of Singapore, for providing me a lively student life throughout my

academic work in the research laboratory.

SUMMARY

Small Signal Modeling of DC-DC Converters Based on Separation of Variables

Most DC-DC power converters have a negative feedback system to regulate the output

voltage. The performance of this regulator depends on the design of the feedback control loop,

which is in turn based on the small signal AC behavior of the converter. Finding this small signal

behavior would be the first step that a design engineer has to take. Although the literature

material on this topic is abundant, they are usually far too complex for practical use. Even some

practical methods deemed suitable for design engineers can be quite mathematically cumbersome

especially if there is a new or unfamiliar DC-DC power converter. This discourages design

engineers from using them. In view of these problems, a more general, simpler approach for

modeling DC-DC converters is developed in this thesis based on separation of variables method.

The separation of variables method (SVM) is developed intuitively based on the

assumption that a converter circuit can be separated into two sub-circuits. One sub-circuit consists

of variables that respond instantaneously fast under small signal perturbations, which is named as

fast dynamic group (FDG), while the other sub-circuit variables react slowly to these

perturbations, which comprises (relatively large) dynamical components. Then these dynamical

components are replaced with constant sources, which result in a new circuit called fast dynamic

equivalent circuit (FDEC). Thus, the analysis has reduced to finding small signal characteristic of

the fast sub-circuit whose model parameters or FDM (fast dynamic model) are constants. Then a

small signal equivalent circuit is constructed by connecting the FDM to the large dynamical

components that were once removed. Thus, the frequency responses of the converter can be

i

obtained either through circuit simulation or analytical approaches. The separation of variables

method is applied to hard-switched converters (buck, boost, and buck-boost converters), soft-

switched (zero-current-switch quasi-resonant buck) converters, and resonant converters (series

resonant converter with diode conduction angle control). Model predictions are verified by

comparing with those obtained from ‘brute-force’ SPICE simulation of the actual circuit or with

existing results from other methods.

The proposed method, so far, is not suitable for converters with dynamic FDM

coefficients. To account for this effect, two modifications to the separation of variables method

are proposed. The first modification, named as separation of variables method type 1 (SVM-1),

employs system identification method to model the transient response of the fast dynamic model

coefficients. The poles and zeros are extracted and then multiplied appropriately to the FDM (fast

dynamic model) constant coefficients, as correction factors. This results in a new (more accurate)

equivalent circuit, which can be analyzed through circuit simulation or mathematically. The

second modification or separation of variables method type 2 (SVM-2) employs sampled data

approach to model the FDM dynamics. It is demonstrated that the SVM-2 is easier than other

sampled-data contemporaries. This technique results in another (more accurate) circuit model,

which is also suitable for circuit simulation or mathematical analysis.

ii

TABLE OF CONTENTS

Summary i Table of Contents iii Nomenclatures vi List of Abbreviations/Acronyms viii List of Figures ix List of Tables xii Chapter 1: Introduction 1

1.1: Background 1

1.2: Literature Review 3

1.3: The Present Work: Motivations and Aims 7

1.4: Project Results in Summary 8

1.5: Organization of Thesis 10

Chapter 2: Development of Separation of Variables Method 11

2.1: Introduction 11

2.2: Fast Dynamic Model (FDM) Development 12

2.3: Small Signal Circuit Model 15

2.4: Separation of Variables Method (SVM) Algorithm 16

2.5: Summary 19

Chapter 3: Application Examples to Hard-Switching Converters 20 3.1: Introduction 20

3.2: Fast Dynamic Model (FDM) for Continuous 21

Conduction Mode (CCM) Cases

iii

3.2.1: Buck Converter Under CCM: Finding FDM 22

by Circuit Perturbation

3.2.2: Boost Converter: Finding FDM by Analytical 32

Method

3.2.3: Buck-Boost Converter: Finding FDM by 39

Simulation Method

3.3: Fast Dynamic Model for Discontinuous Conduction 46

Mode Cases

3.3.1: Buck Converter (DCM): Finding FDM by 47

Analytical Approach

3.3.2: Boost Converter: 55

3.3.3: Buck-Boost Converter 55

3.3.4: Note on FDMs for DCM Cases 56

3.4: Small Signal Circuit Model 56

3.5: Obtaining Small Signal Responses 58

3.5.1: Analytical Method 58

3.5.2: Simulation Method 63

Chapter 4: Application Examples to Soft-Switching Converters 69 4.1: Introduction 69

4.2: Brief Review of Zero-Current-Switch 70

Quasi-Resonant-Current (ZCS-QRC) Buck Converter.

4.3 Fast Dynamic Equivalent Circuit (FDEC) of 71

Zero-Current-Switch Quasi-Resonant-Converter (ZCS-QRC)

Buck Converter in Half Wave Mode

4.4 Finding FDM by Analytical Method 73

4.4.1 Finding FDM by Simulation Method 78

4.5 Small Signal Equivalent Circuit Model 84

4.6 Simulation Results 85

4.7 Summary 88

iv

Chapter 5: Extension of SVM Technique to Resonant Converter 89 5.1: Introduction 89

5.2: Basic Operation of Series Resonant Converter (SRC) 91

5.2.1: Brief Review of State Plane Analysis 95

5.3: DC Analysis 98

5.4: Limitation on Small Signal Analysis by SVM Approach 101

5.5: SVM-1 Approach: Dynamical FDM 109

5.6: SVM-2 Approach: Resonant Tank Fast Dynamic 116

Model (RFDM)

5.6.1: RFDM and FDM Coefficients 120

5.6.2: Practical Method to Compute RFDM Coefficients 127

5.6.3: Odd Symmetry Problem of RFDM 130

5.6.4: Small Signal Analysis by SVM-2 Approach 132

5.7: Verification and Comparison of SVM Techniques for 137

SRC-DCA

5.7.1: Pspice Simulation for SRC-DCA 137

5.7.2: Simulation Results 139

5.8: Summary 143

Chapter 6: Conclusions and Future Work 145 6.1: Summary of Work 145

6.2: Basic Separation of Variables Method (SVM) 145

6.3: Extension of SVM Technique to Resonant Converters 148

6.4: Overall Summary of SVM Methods 150

6.5: Future Work 154

References 155

Appendix A 158

v

NOMENCLATURES

Note that these symbols may be accompanied by superscripts or subscripts in some

chapters, which are not indicated here. Other rarely used symbols in following chapters

are not included here.

Symbols

∆X - Perturbation of ‘X’ where X is an arbitrary variable

α - Diode Conduction Angle [28]

β - Switch Conduction Angle [28]

C, Co - Relatively Large Value Capacitor Component

Cr - Resonant Capacitor Component

D - Duty Ratio Control

Da,Db,D1,D2 - Diode Component

fr - LC Resonant Frequency

fs - Switching Frequency

iC - Instantaneous Current of the Capacitor C, Co

iCr - Instantaneous Current of the Resonant Capacitor Cr

id - Instantaneous Current of the Voltage Source Vd

iL - Instantaneous Current of the Inductor L, Lo

iLr - Instantaneous Current of the Resonant Inductor Lr

iLr(W) = iLr(tW) - Resonant Inductor Lr Current at Instant ‘W’

io - Instantaneous Current of the Resistor R, Ro

irect - Instantaneous Rectifier Current

ix - Instantaneous Current of the Voltage Source Vx

IC - Average Current of the Capacitor C

ICr - Average Current of the Resonant Capacitor Cr

Id - Average Current of the Voltage Source Vd

IL - Average Current of the Inductor L (Can be a Component as well)

ILr - Average Current of the Resonant Inductor Lr

vi

Io - Average Current of the Load Resistor Ro

Irect - Average Rectifier Current

Ix - Average Current of the Voltage Source Vx

L, Lo - Relatively Large Value Inductor Component

Lr - Resonant Inductor Component Component

Q, Q1, Q2 - Switch Component

Ro - Load Resistor Component

R - Switch Radius [28]

t - time

tα - Diode Conduction Time [28]

tβ - Switch Conduction Time [28]

tM - Particular Instant at ‘M’; M = arbitrary symbol

Ts - Switching Period

vC - Instantaneous Voltage of the Capacitor C, Co

vCr - Instantaneous Voltage of the Resonant Capacitor Cr

vL - Instantaneous Voltage of the Inductor L, Lo

vLr - Instantaneous Voltage of the Resonant Inductor Lr

vLr(W) =vLr(tW) - Resonant Capacitor Cr Voltage at Instant ‘W’

vx - Instantaneous Voltage

VC - Average Voltage of the Capacitor C

Vd - Input Voltage Source Component

VL - Average Voltage of the Inductor L

Vx - Constant Voltage Source (Can be a Component as well)

Zr - Characteristic Impedance of Resonant Tank

vii

LIST OF ABBREVIATIONS/ACRONYMS

CCM - Continuous Conduction Mode

DCA - Diode Conduction Angle

DCM - Discontinuous Conduction Mode

FDEC - Fast Dynamic Equivalent Circuit

FDG - Fast Dynamic Group

FDM - Fast Dynamic Model

IAC - Injected Absorbed Current [14]

PRC - Parallel Resonant Converter

RFDM - Resonant Tank Fast Dynamic Model

SRC - Series Resonant Converter

SSA - State Space Averaging [1]

SVM - Separation of Variables Method

SVM-1 - Separation of Variables Method Type 1

SVM-2 - Separation of Variables Method Type 2

viii

LIST OF FIGURES

Fig. 2.1: A Boost Converter 14 Fig. 2.2: Fast Dynamic Equivalent Circuit (FDEC) for the Boost Converter 14 Fig. 2.3: Small Signal Circuit Model 16 Fig. 2.4 Separation of Variables Method Algorithm 18 Fig. 3.1: Buck Converter in CCM 23 Fig. 3.2: FDEC for Buck 24 Fig. 3.3: Buck Converter under CCM: Perturbing Input Voltage Converter 24 Fig. 3.4: Buck Converter under CCM: 25

Inductor Voltage for Input Voltage Perturbation. Fig. 3.5: Buck Converter under CCM: Perturbing Output Voltage 27 Fig. 3.6: Buck Converter under CCM: Perturbing Inductor Current 28 Fig. 3.7: Buck Converter under CCM: Perturbing Duty Ratio 30 Fig. 3.8: Boost Converter in CCM 33 Fig. 3.9: FDEC for Boost Converter 33 Fig. 3.10: Waveforms of Fast Variables 34 Fig. 3.11: Buck-Boost Converter 40 Fig. 3.12: FDEC for Buck-Boost Converter 40 Fig. 3.13: Buck-Boost Converter: Perturbation of Vd 42 Fig. 3.14: Buck-Boost Converter: Fast Variables Response to Vd Perturbation 43 Fig. 3.15: FDEC for Buck Converter in DCM 48 Fig. 3.16: Buck Converter under DCM: Inductor Waveforms 49 Fig. 3.17: Averaged Small Signal Circuit Model of the Converter under CCM 57

ix

Fig. 3.18: Averaged Small Signal Circuit Model of the Converter under DCM 57 Fig. 3.19: Buck Converter Small Signal Model Using FDM – Pspice Diagram 64 Fig. 3.20: Small Signal Frequency Response for Buck Converter under CCM 65 Fig. 3.21: Small Signal Frequency Response for Boost Converter 67 Fig. 3.22: Small Signal Frequency Response for Buck-Boost Converter 67

under CCM Fig. 4.1: ZCS QRC Buck Converter (Half-Wave) 70 Fig. 4.2: Resonant Inductor Current (iLr) and Resonant Capacitor Voltage (vCr) 70

Behavior of the ZCS-QRC Buck Converter in Half-Wave Mode

Fig. 4.3: FDEC of the ZCS QRC Buck Converter (Half-Wave) 72 Fig. 4.4: FDEC Pspice Model for ZCS-QRC-Buck (HW) Having Two 79

Different Input Voltages to Emulate a Step Change Fig. 4.5: The Perturbed Response of the Average Id Current from the 79

Pspice Model in Fig. 4.4 Fig. 4.6: The Perturbed Response of the Average Vx Voltage from the 80

Pspice Model in Fig. 4.4 Fig. 4.7: Two Steady State Pspice Models to Emulate Output Current Step 81

Change from 0.63A to 0.6363A Fig. 4.8: Behavior of Id Current under Steady State and Perturbed Conditions 82 Fig. 4.9: Behavior of Vx Voltage under Steady State and Perturbed Conditions 82 Fig. 4.10: Small Signal Circuit Model of ZCS QRC Buck HW 84 Fig. 4.11: Control-to-Output Frequency Response Curves For the 85

ZCS-QRC-Buck Half Wave Fig. 4.12: Input-to-Output Frequency Response Curves For 86

ZCS-QRC-Buck Half-Wave Fig. 4.13: Small Signal AC Model of ZCS-QRC-Buck Half-Wave Based 87

on [21]

x

Fig. 5.1: Series Resonant Converter in Half-Bridge Switch Arrangement 92 Fig. 5.2: Resonant Inductor Current of SRC 93 Fig. 5.3: Resonant Capacitor Voltage of SRC 94 Fig. 5.4: State-Plane Trajectory of SRC under CCM in Below Resonant[28] 96 Fig. 5.5: SRC-DCA: Switching Frequency versus Switch Radius 101 Fig. 5.6: FDEC for SRC-DCA 102 Fig. 5.7: FDG for SRC-DCA 103 Fig. 5.8: Small Signal Circuit Model of SRC-DCA Using SVM 106 Fig. 5.9: Input-to-Output Frequency Responses of SRC-DCA via SVM 107 Fig. 5.10: Control-to-Output Frequency Responses of SRC-DCA via SVM 108 Fig. 5.11: Sampled Inductor Current Dynamics Against Number of 110

Switching Cycle, due to: (a) Perturbed Vd, (b) Perturbed Vo, and (c) Perturbed α

Fig. 5.12: First Order System’s Step Response Curve 101 Fig. 5.13: Small Signal Circuit Model of the SRC-DCA with Corrected FDM 113

Coefficients Fig. 5.14: Input-to-Output Frequency Responses of SRC-DCA via SVM and 114

SVM-1 Fig. 5.15: Control-to-Output Frequency Responses of SRC-DCA via SVM and 115

SVM-1 Fig. 5.16: Resonant Tank Behavior with (a)Discrete Time Inductor Current 119

(b) Discrete Time Capacitor Voltage

Fig. 5.17: Interconnection of FDM and RFDM with Injected Variables and 120 External Circuit

xi

Fig. 5.18: Response of the resonant inductor current after one-half period under 122 (a) Perturbed resonant inductor current, ∆iLr(k) (b) Perturbed resonant capacitor voltage, ∆vCr(k) (c) perturbed input voltage, ∆Vd(k) (d) perturbed output voltage, ∆Vo(k) and (e)perturbed control angle,∆α(k)

Fig. 5.19: Arbitrary Periodic Waveform (a) Odd Symmetry (b) Even Symmetry 131 Fig. 5.20: Small Signal Equivalent Circuit Model for SRC-DCA using 135

FDM and RFDM Fig. 5.21: PSPICE Realization of the Model in Fig. 5.20 136 Fig. 5.22: PSpice Realization of SRC with DCA Control 138 Fig. 5.23: Input-to-Output Frequency Responses of the SRC-DCA 140

using SVM, SVM-1, and SVM-2 Method Fig. 5.24: Control-to-Output Frequency Response of the SRC-DCA 141

using SVM, SVM-1, and SVM-2 Methods Fig. 5.25: Input-to-Output Frequency Response of the SRC-DCA 142

using SVM-2 with Different Sampling Interval Fig. 5.26: Control-to-Output Frequency Response of the SRC-DCA 143

using SVM-2 with Different Sampling Interval

xii

LIST OF TABLES

Table 3.1: Design Parameters of the Buck-Boost Converter 41

Table 3.2: Calculation of Coefficients 45

Table 3.3: Design Parameters of the Buck Converter 64

Table 3.4: Design Parameters of the Boost Converter 66

Table 4.1: Design Parameters for ZCS-QRC-Buck Half-Wave 75

Table 4.2: Perturbation of Input Variables 75

Table 5.1: Design Parameters of the Series Resonant Converter 98

With Diode Conduction Angle Control (SRC–DCA)

Table 5.2: Calculation of Perturbed Slow Variables 104

Table 5.3: Perturbation of Variables by Numerical Computation 125

Table 5.4: Comparison of Different SVM Techniques for SRC – DCA 142

Table 6.1: Comparison of Different Small Signal Analysis Methods 152

xiii

Chapter 1 Introduction

1

Chapter 1

Introduction 1.1 Background

Small signal analysis of a power converter is concerned with modeling of the

averaged, linearized, small signal AC behavior around a steady state operating point of

the system. The information obtained from this investigation can later be used to predict

the system transfer functions that are useful in the feedback loop design.

Literature on small signal analysis, also known as AC analysis, for power

converters is very well established. There is a never-ending list of techniques and

methods. To this day, literature on small signal modeling are still growing. Some are

compact and effective such as the state-space averaging (SSA) technique by Middlebrook

and Cuk [1]. Others are more mathematically intensive such as the work by Caliskan and

et al [2], Tymerski [3], and Sanders and et al [4] that utilize describing function theory to

establish the transfer function. Besides, they can be conceptually abstract too. This

sometimes hinders practicing engineers from applying these modeling techniques to their

power supply design. This is particularly true when frequency responses of new converter

topologies are desired but not available in literature yet. Because power supply designers

have very short time-to-market product design phase due to competitive business, these

engineers will not have time to digest the complicated techniques like [2], [3], [4]. The

use of a straightforward method like system identification, which treats a converter as a


2

black box can be mathematically abstract such as [5], [6]. Thus, the method is suitable,

perhaps, only for researchers to explore.

Even a well-established ‘classical’ canonical small signal tool such as the state-

space-averaging (SSA) has shortcomings. For example, the complexity of the SSA [1]

technique increases when the order of the converter analyzed is increased. It may be

noted that the order of the converter is proportional to the number of passive elements,

namely inductor, L, and capacitor, C. Sometimes, when enhanced filtering effect is

required, the aforementioned large passive components are added such as C-L-C

network. Also, in resonant converter and other soft-switched converters, additional

inductor and capacitor may be used in order to achieve soft-switching. This in turn

increases the order. Another shortcoming of the state-space-averaging technique is that it

is not applicable to resonant type of converters due to the presence of fast varying state

variables in the converter.

The primary motivation for this thesis is to present a small signal model that is

suitable for practicing engineers. The technique is based on an intuitive approach in

which the converter is separated into a slow part and a fast part. The slow part of the

converter consists of components that react very slowly to perturbations. The fast part of

the converter, on the contrary, consists of components that react almost instantaneously to

perturbations. The fast part of the converter is analyzed first and its averaged, linear

response function is determined. In carrying out this analysis, the slow variables

associated with the slow part are treated as DC sources. The linearized response function


3

of the fast part is then coupled to the slow part of the converter to obtain the overall small

signal model of the converter.

Since the technique involves division of the converter circuit into a slow part and

a fast part, it is given the name Separation of Variables Method (SVM). This technique

was first developed by Dr. Ramesh Oruganti [8]. In the present research work, the

method is developed and presented in a more in-depth manner. More importantly, the

SVM has been applied to a greater range of power converter including converters in

discontinuous-conduction-mode (DCM), quasi-resonant converters and resonant

converters. In particular, extension of SVM technique to resonant converters has led to

the generalization of the method to account for resonant tank dynamics also.

1.2 Literature Survey

The state-space-averaging (SSA) method proposed by [1] is a widely known

small signal analysis tool, which employs averaging technique. This model is expressed

by a canonical state matrix that predicts the low frequency response behavior of the

converter. Although this technique is simple to use and provides good insight into the

poles and zeros behavior of the converter circuit, it has several limitations. First, as

mentioned earlier, the state-space-averaging complexity increases as the order of the

converter is increased. This is due to the fact that the order of the canonical matrix is

proportional to the order of the system. Second, the method can only be applied to slowly


4

varying state variables. Therefore, SSA is not applicable to converters with fast varying

state variables such as quasi-resonant converters or resonant converters. Third, the end

result of the analysis is not a small signal circuit model. A circuit model is deemed

beneficial because a circuit simulation approach can be used to obtain desired transfer

functions [9]. This is also very practical as it does not only save time but avoids

mathematical complexities that usually arise during small signal analysis. Hence it is

relative easier to use by the average power electronics engineer on a given power

converter.

The state-space-averaging is extended to quasi-resonant-converters in [7], known

as the ‘extension of state-space-averaging’ method or ESSA. Although one of the

limitations of state-space-averaging is resolved through this technique, other remaining

limitations are still unsolved. The implementation of this method, for instance, is still

very complicated due to intensive mathematics. This is undesirable for practicing

engineers. Furthermore, the ESSA method cannot be applied to resonant converters.

Finally, the analysis does not result in an equivalent circuit form.

The pulse-width-modulated (PWM) switch modeling [10] technique whose

analysis results in an equivalent circuit model may seem to offer a practical modeling tool

to practicing engineers. In this method, a linearized, small signal, circuit model (or large

signal model) replaces the nonlinear switch and diode component at their three terminal

points in the hard-switched type or quasi-resonant type converters. Once a small signal

linear circuit model replaces these components, frequency responses of the converter can


5

be obtained through circuit simulation of the circuit model. An analytical approach to

obtain transfer functions through circuit analysis is also possible. Although the

application of this method is quite straightforward and simple, a disadvantage is that the

equivalent small signal model must be derived analytically through averaging

waveforms, which is quite tedious. Another drawback of this method is that it cannot be

applied to resonant type converters.

There are various modeling techniques [18], however, capable of performing

small signal analysis of resonant converters along with other converters such as [2] - [6],

[12]. But most of these techniques are mathematically intensive. [13] offers a quick way

for engineers to determine resonant converters’ transfer functions by the formulas

presented. However, the analysis is not general as the formulas are developed solely for

series and parallel resonant converters. A more general approach to deal with resonant

converters is presented in [11]. The method employs both continuous and discrete

approaches to develop a small signal circuit model. This method would appear to have

similar concept to our proposed technique but, nevertheless, there are differences. The

underlying concept for the method [11] is to treat the state variables in the filter tank as

continuous quantities while the state variables in the resonant tank as discrete quantities.

Here, the authors [11] claimed that the explicit form of equivalent circuit model for the

resonant tank is too complicated for practical use; therefore a two-port lumped parameter

equivalent circuit model is proposed, which absorbs the resonant tank variables. In effect,

these resonant state variables do not appear in the resultant circuit model. However, to

arrive to this lumped parameter network, one must work through rigorous mathematical


6

analysis. This is, of course, not favored by practicing engineers. This lumped parameter

approach is found to be unnecessary as there is a better way to implement the model

parameters, without engaging cumbersome mathematics. This is discussed in Chapter 5

under Section 5.6.4.

Until now, a practical (which involves less mathematics), simple, and general

small signal modeling tool may be offered by the injected absorbed current method [14].

In this method, the slowly varying state variable (capacitor voltage) is extracted from the

system. This results in an order reduction of one, which simplifies the analysis. The

method then probes the behavior of the converter’s input and output currents. By

analyzing these variables’ small signal behavior, the analysis yields an equivalent small

signal circuit model, which can be used for circuit simulation. In addition, the method can

be applied to a broad class of converters such as hard-switched converters [14], [15],

quasi-resonant type converters [16], series-resonant-converter (SRC) [14], and parallel-

resonant-converters (PRC) [17]. Still, the injected absorbed current method has some

limitations. First, analysis of high order system converters may not be simplified much by

just reducing the system order by one. Second, the resonant converters analyzed with this

injected absorbed current method, so far, did not include the inner feedback control of the

converter. Since the inner-feedback control is inherent to resonant converters [27], it must

be included in the model analysis otherwise the small signal modeling of the converter is

incomplete.


7

1.3 The Present Work: Motivations and Aims.

While the injected-absorbed-current (IAC) method has many merits, there are two

problems, which were pointed out earlier. Order reduction by one may offer some relief

for low order converter’s control analysis but this would not have substantially simplify

high order converters control analysis. Then the effect of resonant tank dynamic due to

inner feedback control must be addressed otherwise the small signal modeling is

incomplete. In view of these, the separation-of-variables (SVM) technique is developed

and presented in this project report, which has features deemed practical and advantages

over the injected absorbed current method. The followings are the advantages of the

proposed SVM.

1. Conceptually simple and easy to use. The proposed method separates

the slow and fast variables of a converter. This allows order reduction

of several magnitudes in the analysis.

2. Applicable to resonant converters with the inner-feedback dynamic,

besides being applicable to quasi-resonant converters and hard-

switched converters. Also, SVM is applicable to both continuous-

conduction-mode (CCM) and discontinuous-conduction-mode (DCM).


8

3. Model parameters can be determined either analytically or through

circuit simulation. The latter is particularly important as this allows a

practicing engineer to use this technique without much knowledge on

mathematical modeling methods.

1.4 Project Results in Summary

In this thesis, the following results are achieved and reported:

1. The proposed separation-of-variables-method (SVM) is developed based on

intuitive understanding of the converter dynamics. The basic method,

identified as SVM method, simplifies the analysis to analyze the fast part

(named Fast Dynamic Group or FDG). This results in the Fast Dynamic

Model (FDM), which models the behavior of the circuit. The small signal

model of the FDG to perturbations is then constructed by merely inserting in

the slower responding components and variables to FDM.

2. The method is then applied to a buck type, a boost type, and a buck-boost type

DC-DC power converter. Both analytical and simulation based approaches to

determine the model parameters are discussed. Also, the separation of

variables method is applied to both continuous conduction mode (CCM) and

discontinuous conduction mode (DCM) operation of the converters. The linear


9

small signal models obtained from the separation of variables method are

verified by comparing their frequency responses with those obtained from

applying classical state-space averaging approach.

3. The separation of variables method is applied to quasi-resonant-type

converters. Both analytical and circuit simulation-based implementation of the

technique are presented. It is shown here that the latter method can allow an

engineer to determine the frequency response of a quasi-resonant converter

without an in-depth knowledge of the converter operation.

4. The separation of variables method is finally applied to resonant converter,

with a series resonant converter (SRC) under diode conduction control (DCA)

operating below resonant frequency serving as an example. Due to the

presence of resonant tank components, the fast dynamic group of the

converter does not respond infinitely fast (as assumed in the separation of

variables method) to perturbations. Two modifications (separation of variables

method type 1 or SVM-1 and separation of variables method type 2 or SVM-

2) to SVM method, which aim to account for the dynamics of the resonant

tank are proposed and evaluated. Separation of variables method type-1

(SVM-1) is simpler while slightly less accurate than separation of variables-

method type-2 (SVM-2), which is more complex. A simulation-based

technique is adopted in place of experimental verification to verify the

frequency response results provided by the application of various SVM


10

methods. The intention here is to show the proposed analysis results in

accurate low frequency model of the converter. It is not to show that such

theoretical low frequency model corresponds to practical converters. Such

justification has been provided by many researchers earlier. In this technique,

a Pspice []19], [20] model of the resonant converter under diode-conduction-

angle (DCA) control is developed. The control angle is then perturbed by a

small amount at various frequencies and the corresponding component in the

output voltage was extracted.

1.5 Organization of Thesis

This thesis consists of 6 chapters. Chapter 1 examines the motivation and aim of

the research project including a brief literature survey. Chapter 2 looks at the general idea

and concept of the SVM technique. In Chapter 3, application examples for hard-switched

converters are shown and illustrated with different approaches, all under the same

separation of variables method principle. Furthermore, converters operating in

discontinuous conduction mode condition are also investigated. Chapter 4 illustrates the

separation of variables method concept to soft-switching converters, namely quasi-

resonant converters. In Chapter 5, separation of variables method technique is extended

to resonant converters by considering the resonant tank dynamics. Two possible

modifications to the separation-of-variables-method techniques are presented and

compared. Finally Chapter 6 concludes the thesis.

Chapter 2 Development of Separation of Variables

11

Chapter 2

Development of Separation of Variables Method

2.1 Introduction

The proposed Separation of Variables Method (SVM) offers a simple approach

to obtain the low frequency small signal model of a power converter. Most power

converters can be viewed as having two distinct sub-circuits. One reacting slowly and the

other rapidly to external perturbations. Typically, components that react slowly are the

large passive components like filter inductors and filter capacitors. On the other hand,

components such as switches, diodes, resistors, ideal transformers, etc respond very fast

with very little dynamics. Typically, in most power converters, these components may be

viewed as reacting in an ‘infinitely’ fast manner.

Recognizing that there are differences in terms of response speeds, the SVM

divides the power circuit into two main sub-groups, the slow and the fast sub-circuit, and

separates them. The components in the slow sub-circuit are then replaced by appropriate

constant voltage/current sources leaving only the fast part of the sub-circuit to be

analyzed for small signal modeling purpose. The fast part of the power converter

hereafter shall be referred as the Fast Dynamic Group, FDG.


12

One of the main advantages of the proposed method is the simplicity of the

technique and procedures to obtain small signal transfer functions. The method is mainly

procedural without complex mathematics. Also, only the fast dynamic part of the

converter, which contains little or no dynamics, is analyzed/simulated for the small signal

model development, which simplifies the method considerably. Therefore, the SVM

should be attractive for practicing engineers who are not inclined towards complex

mathematical modeling techniques. The method can be used to derive a circuit model for

the small signal behavior directly. The circuit model can be analyzed further to derive

transfer functions.

The objective of this chapter is to introduce the concept and methodology of the

proposed SVM. The organization of this chapter is as follows: Section 2.2 explains the

development of the Fast Dynamic Model (FDM) of the converter. Section 2.3 then show

how the small signal circuit model is obtained. Section 2.4 outlines the SVM algorithm

based on earlier discussion to obtain small signal model. Section 2.5 summarizes the

developments in this chapter.

2.2 Fast Dynamic Model (FDM) Development

Strictly speaking, the SVM separates the slow and fast variables instead of the

slow and fast sub-circuits. These two groups of variables are assumed to have very

different frequency responses. As such, from the perspective of slow variables, the fast


13

variables have ‘infinitely’ fast response behavior. Likewise, from the perspective of the

fast variables, the slow variables have ‘infinitely’ slow response behavior.

Therefore the components associated with the slow variables can be replaced by

constant voltage/current sources where values are equal to their averaged values because

within a switching period, any slow variable behaves like a constant. Consider a boost

converter, for example (See Fig. 2.1), in which the inductor current and the output

voltage variables are considered as slow variables and may be replaced by constant

sources as described earlier. The resulting circuit has been named as Fast Dynamics

Equivalent Circuit (FDEC) as shown in Fig. 2.2.

Notice that the average inductor current in Fig. 2.1 varies according to the voltage

VL applied to inductor. Likewise, the voltage across the output voltage capacitor varies

according to the current IC through it (or current Ix through C and R). Thus the output

variables of the FDEC are treated as Ix, VL, and Id. Here, Id is also included in case the

small signal impedance (∆Vd/∆Id) of the converter needed to be determined. By

perturbing the slow variables (IL, Vx, Vd, and D) the output response of the FDEC is

determined by analyzing the FDEC mathematically or through simulation. Note that here

Vd and D are included in the slow variables as response to Vd and D changes are usually

required of the small signal model.


14

+Vd-

QD

id

+

Vo

-

+

vC

-

iL

+ vL -

iC

IoDiL

C R

Fig. 2.1: A Boost Converter

+Vd-

QD

id+ vL - Di

FDG

IL

Vx+-

Ix

Fig. 2.2: Fast Dynamic Equivalent Circuit (FDEC) for the Boost Converter

The ratios between the FDEC perturbed output variables and the perturbed DC

sources are then obtained. These are ∆VL/∆D, ∆Ix/∆D, ∆Id/∆D, ∆VL/∆D, etc. The matrix

formed by these ratios is called the Fast Dynamic Model (FDM), which models the small

signal response of the FDEC. The FDM for the boost converter will be in the form given

by the following equation:


15

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

DIVV

aaaaaaaaaaaa

VII

L

x

d

L

x

d

34333231

24232221

14131211

(2.1)

Here a11 to a34 are the model coefficients determined by small signal perturbation

analysis of the FDEC. Note that FDM is relatively easy to determine because the

complexity and dynamics of the original converter is reduced considerably in FDEC

through the non-considerations of slowly varying variables.

2.3 Small Signal Circuit Model

The FDM provides the linearized, averaged, small signal model of the fast part

(FDEC) of the converter. The outputs of the FDM are either fast variables (such as VL)

that affect slow variables (such as IL) or output variables such as Id needed for transfer

function determination. The inputs of the FDM are either converter inputs (such as Vd, D)

or slow variables such as Vx or IL.

By simply combining the FDM and the slow variables/components, the small

signal circuit model of the converter may be easily obtained (Fig. 2.3). The interaction

between the circuit slow variables and fast variables are automatically taken care of in

this model.


16

+

-

+ -

+∆VL

-

∆Id

a11∆Vd a12∆Vc a13∆ IL a14∆D

a21∆Vd a22∆Vc a23∆ IL a24∆D

+ - + - + -a31∆Vd

a32∆Vc a33∆ IL a34∆D∆IL

+∆VC

-

∆ IC

∆Vd

∆D

Fig. 2.3: Small Signal Circuit Model

Note that with the development of the linear circuit in Fig. 2.3, the modeling is

complete. The circuit in Fig. 2.3 may be simulated in SPICE [19] in order to obtain the

small signal frequency response of the converter. Alternatively, the circuit in Fig. 2.3 can

be analyzed further to derive the mathematical transfer function of the converter.

2.4 Separation of Variables Method (SVM) Algorithm

Fig. 2.4 shows the steps involved in determining the small signal model of a given

converter. In STEP 1, the power converter’s operating point is determined. This may be

done either through analysis or through circuit simulation. The fast and slow components

are also identified. The steady state values of the slow variables are then determined. In


17

STEP 2, slow and fast components associated with slow and fast variables are separated

into two sub-circuits. Then the FDG is identified within this step. STEP 3 involves a

simple procedure. All identified slow dynamical components (the slow sub-circuit

element) are replaced by constant sources. This new equivalent circuit with constant

sources is called FDEC, as mentioned before. Perturbing the constant sources in STEP 4

allows responses of fast variable to be measured for computing FDM. The FDM found in

step 4 is used in STEP 5 to obtain the complete small signal FDM circuit model. This is

done by reconnecting the FDM to its original slow components as shown in Fig. 2.3. The

linearized and averaged small signal circuit model of STEP 5 may be simulated directly

to obtain the desired small signal frequency plots. Alternatively, the circuit may be

analyzed further to determine mathematical expression for the converter transfer

function.

An important point to note is that in STEP 1, STEP 4 or STEP 6, either

mathematical analysis or circuit simulation can be used as desired. This allows a

practicing engineer to determine the small signal model of the converter completely

through simulation only without any recourse to complex mathematics.


18

Power Converter Operating atSteady State Condition

Separate Slow and FastComponents

Replace Slow DynamicalComponents with Constant

Sources to Obtain FDEC 1

Obtain FDM 2 by PerturbingConstant Sources of the Slow

Variables

Obtain SVM 3 Small SignalCircuit Model

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

1 FDEC = Fast Dynamics Equivalent Circuit

2 FDM = Fast Dynamics Model

3 SVM = Separation of Variables Method

Obtain Transfer Function(s)

STEP 6

Fig. 2.4 Separation of Variables Method Algorithm


19

2.5 Summary

This chapter has outlined a basic SVM procedure to develop an averaged small

signal model of a power converter based on separation of the circuit into a slow and a fast

dynamical part. This basic method is intuitive in its approach and simple to use compared

to other conventional methods. In the next two chapters (Chapter 3 and 4), the basic SVM

technique is applied to several converters and operating conditions to establish the

validity of the proposed method. In Chapter 5, the method is extended to resonant

converters, where the fast dynamics variables are not fully fast.

Chapter 3 Application Examples to Hard-Switched Converters

20

Chapter 3

Application Examples to Hard-Switched Converters

3.1 Introduction

The objective of this chapter is to show the applicability of the proposed SVM to

modeling some of the most common hard-switched converters, which are the buck, boost,

and buck-boost converters in both continuous conduction mode (CCM) and

discontinuous conduction mode (DCM) operations. These converters are the basic hard-

switched topologies that can be extended to various isolated power supply topologies

such as flyback, forward, and push-pull.

A second objective of this chapter is to show that the SVM procedure can be

applied either through simulation or analytically. Such flexibility allows a converter’s fast

dynamic model (FDM) and transfer functions to be obtained either way depending on the

engineer’s preference.

Section 3.2 develops the FDM introduced in Chapter 2 for buck, boost, and buck-

boost converters when operating under continuous conduction mode (CCM). The model

parameters are obtained analytically in the case of buck and boost converters. In the case

of the buck-boost converter, the model parameters are obtained through OrCad Pspice


21

[20] simulation to illustrate the feasibility of obtaining the coefficients through

simulation. Section 3.3 develops the fast dynamic model for the same converter under

discontinuous conduction mode operation. Section 3.4 first develops the linearized,

averaged small signal circuit model for the converters under discussion. In Section 3.5,

the small signal performances of the converters are then obtained first through

mathematical analysis and then through simulation of the small signal circuit mode.

These results are compared with those obtained from classical state space averaging

techniques [1]. Finally, Section 3.6 summarizes the chapter.

3.2 Fast Dynamic Model (FDM) in Continuous Conduction Mode

(CCM) Cases

In the next several sub-sections, the development of the FDM in a few hard-

switched converters are explored. Different approaches for obtaining the fast dynamic

model coefficients are considered and presented. These illustrate the versatility of the

proposed separation of variables method. In sub-section 3.2.1, a buck converter under

continuous conduction mode operation is considered. The technique used in this sub-

section relies on intuition and knowledge of the converter operation. Such approach is

likely to be easy for experienced designers to use. On the other hand, an analytical

approach to obtain the fast dynamic model is adopted in sub-section 3.2.2 for a boost

converter operating under continuous conduction mode. In sub-section 3.2.3, a circuit

simulation approach is presented to compute a buck-boost converter’s fast dynamic


22

model parameters. The last mentioned approach neither requires much intuition with

regard to circuit operation nor does it require mathematical skills. This approach tends to

‘mechanize’ the development of the small signal model and may be favored by practicing

engineers.

Once a converter’s FDM (fast dynamic model) is obtained, it can be used for

further analysis to obtain the small signal circuit model of the converter and to obtain

small signal responses, as will be discussed in Section 3.5.

3.2.1 Buck Converter under CCM: Finding FDM by Circuit

Perturbation

This section illustrates how a buck converter’s FDM can be obtained using basic

knowledge of the converter operation. Fig. 3.1 depicts a buck converter circuit, which is

assumed to be operating in steady state under CCM (continuous conduction mode)

condition. The inductor current iL and capacitor voltage vC in Fig. 3.1 are the

instantaneous values. The slow variables for this converter are the inductor current iL,

capacitor voltage vC, input voltage Vd, and duty ratio D. For the low frequency

(averaged) model, these slow variables can be approximated by their averaged values

taken over a switching period. The components L and C that are associated with the slow

variables can be replaced by their averaged values. In other words, L is replaced by a

constant current source and C is replaced by a constant voltage source. Here, the output


23

load Ro is lumped together with the capacitor C as a single voltage source, Vx. The

capacitor C is assumed to be very large so that its voltage is ripple free and constant.

Furthermore, as can be seen, the voltage across the load Ro and the capacitor C are the

same. This justifies the output circuit Ro and C be lumped together and replaced by a DC

voltage source. Although Ro can be separated from C and be included in the fast dynamic

part of the converter, it offers no advantage as it does not simplify the analysis.

The resultant fast dynamic equivalent circuit (FDEC) for the buck converter under

continuous conduction mode (CCM) is shown in Fig. 3.2. Here, after averaged over a

cycle, IL, Vx, Vd, and D are the slow variable inputs to the FDEC while the fast variable

responses of the FDEC are Id, VL, and Vx. Hence, in order to determine the fast dynamic

model (FDM) of the converter, the averaged, small signal ratios of the output and input

variables of the fast dynamic equivalent circuit must be obtained. This is obtained by the

perturbing a small amount one slow variable at a time, while holding all other slow

variables constant, and obtaining the resultant perturbations (averaged over a switching

cycle) of the fast variables in the FDEC.

+

Vd

-

Q

D

id

+

Vo

-

+

vC

-

iL

+ vL -

iC

Io

Da

L

C Ro

Fig. 3.1: Buck Converter in CCM


24

+

Vd

-

Q

D

Id

Vx

IL

+ VL -

IxDa+-

FDG

Fig. 3.2: FDEC for Buck Converter

Input Voltage Perturbation

+Vd-

Q

D

Id i

Vx

IL

+ VLi -

IxiDa

+-

+-∆Vd

Fig. 3.3: Buck Converter under CCM: Perturbing Input Voltage

Here, the input voltage, Vd, is perturbed by a small DC value ∆Vd. The other slow

variables Vx, IL, and D are kept constant. The perturbation in Vd causes all fast variables

to change by a small amount from their steady state values. This is indicated in Fig. 3.3


25

by the addition of a superscript ‘i’ sign to the variable name. Since there is no dynamical

component in the FDG (fast dynamic group), all fast variables respond instantaneously.

Of course, it must be borne in mind that the fast variable outputs of FDEC (fast dynamic

equivalent circuit) are averaged variables. By inspecting Fig. 3.3, it can be seen that the

averaged input current Id depends on the averaged inductor current IL and the duty ratio

D. Because these IL and D did not change during the ∆Vd perturbation, the input current

Id remains the same. By understanding the circuit perturbation, the above observation is

written as Eq. (3.1). By similar reasoning, the inductor current IL is found to be the same

as Ix. Since IL was not perturbed, Ix will not change as well. This observation is written as

Eq. (3.2).

did VI ∆×=∆ 0 (3.1)

dix VI ∆×=∆ 0 (3.2)

DT

T

Vd-Vx

-Vx

0

(Vd+∆ Vd)-VxvL(t)

t

Fig. 3.4: Buck Converter under CCM:

Inductor Voltage for Input Voltage Perturbation.


26

Next, the change in average inductor voltage is obtained for Vd perturbation (See

Fig. 3.4). The waveform in that figure shows clearly that when the input voltage is

perturbed, the averaged inductor voltage is also affected. The relationship between the

input and output changes can be derived from a consideration of the inductor’s volt-

second balance. The averaged steady state inductor voltage is given by,

( ) ( )T

TDVDTVVV xxdL

−−−=

1 (3.3)

Perturbing Vd by ∆Vd results in,

( ) ( )DVDVVVV xxddi

L −−−∆+= 1 (3.4)

Defining iLL

iL VVV ∆+= ,

( ) ( ) dxxdi

LL VDDVDVVVV ∆+−−−=∆+ 1 (3.5)

Thus,

di

L VDV ∆=∆ (3.6)

Eq. (3.6) gives the response ∆VLi of the FDEC (fast dynamic equivalent circuit) to a

small signal perturbation in Vd. Note that the relationship is a constant since the FDEC

has no dynamical components.


27

Output Voltage Perturbation

+Vd-

Q

D

Idii

Vx

IL

+ VLii -

Ixii

Da+-

+- ∆Vx

Fig. 3.5: Buck Converter under CCM: Perturbing Output Voltage

Figure 3.5 shows the fast dynamic equivalent circuit of the buck converter circuit

with its voltage source Vx being perturbed by ∆Vx. From the circuit, one can easily

deduce that this perturbation will not change Ix and Id. The only change occurs in the

inductor voltage.

Thus,

xiid VI ∆×=∆ 0 (3.7)

xiix VI ∆×=∆ 0 (3.8)


28

To determine the perturbed average inductor voltage, Vx + ∆Vx is substituted for Vx in

Eq. (3.3),

( ) ( )( )DVVDVVVVV xxxxdiiLL −∆+−∆−−=∆+ 1 (3.9)

Retaining only the first order perturbation terms of Eq. (3.9),

xii

L VV ∆×−=∆ 1 (3.10)

Here, the superscript ‘ii’ refers to the effect due to perturbation of Vx.

Inductor Current Perturbation

+Vd-

Q

D

Idiii

Vx

IL

IxiiiDa

+-

∆ IL

+ VLiii -

Fig. 3.6: Buck Converter under CCM: Perturbing Inductor Current


29

The additional parallel current source ∆IL in Fig. 3.6 creates a small change to IL.

This results in changes to Id and Ix variables. In order to obtain the perturbation

coefficients, consider the DC relationship between input current and inductor current.

Ld DII = (3.11)

Perturbing IL in Eq. (3.11),

( )LLiiidd

iiid IIDIII ∆+=∆+= (3.12)

Retaining only the first order perturbation terms,

Liiid IDI ∆=∆ (3.13)

Since IL=Ix, the perturbed Ix is,

LiiiL II ∆×=∆ 1 (3.14)

From Fig. 3.4, the inductor current can be seen to have no influence on the inductor

voltage. Thus,

Liii

L IV ∆×=∆ 0 (3.15)

Here, the superscript ‘iii’ refers to the effect due to perturbation of IL.


30

Duty-Ratio Perturbation

+Vd-

Q

∆ D

Idi

Vx

IL

+ VLi -

IxiDa

+-

Fig. 3.7: Buck Converter under CCM: Perturbing Duty Ratio

Figure 3.7 shows the FDEC (fast dynamic equivalent circuit) with duty ratio D

being perturbed by ∆D. Eq. (3.11) indicates that the input current is affected by the

perturbation in D. By substituting D+∆D term into this equation,

DII Livd ∆=∆ (3.16)

Since Ix = IL, which is a constant source, perturbing D has no effect on Ix. Thus,

DI ivx ∆×=∆ 0 (3.17)

Next, Fig. 3.4 shows that the averaged inductor voltage depends on D. Substituting D +

∆D into Eq. (3.3),

( )( ) ( )DDVDDVVVV xxdiv

LL ∆−−−∆+−=∆+ 1 (3.18)


31

Retaining only the first order perturbation terms,

DVV div

L ∆=∆ (3.19)

Here, the superscript ‘iv’ refers to the effect due to perturbation of D.

Fast Dynamic Model (FDM) Coefficients

Using the superposition principle, Eqs. (3.1), (3.7) , (3.13), and (3.16) can be added

together,

DIIDVVIIIII

LLxd

ivd

iiid

iid

idd

∆+∆+∆×+∆×=∆+∆+∆+∆=∆

00 (3.20)

Similarly, from Eqs. (3.2), (3.8), (3.14), and (3.17),

DIVVIIIII

Lxd

ivx

iiix

iix

ixx

∆×+∆×+∆×+∆×=∆+∆+∆+∆=∆

0100 (3.21)

Likewise, from Eqs. (3.6), (3.10), (3.15), and (3.19),

Dd

VL

Ix

Vd

VD

ivL

ViiiL

ViiL

ViL

VL

V

∆+∆×+∆×−∆=

∆+∆+∆+∆=∆

01 (3.22)


32

Combining Eqs. (3.20) - (3.22), the fast dynamic model (FDM) can be written as,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

DIVV

VD

ID

VII

L

x

d

d

L

L

x

d

010100

00 (3.23)

Eq. (3.23) describes the dynamic behavior of the fast sub-circuit (FDEC) of the

buck converter. This model has four slow variable inputs, ∆Vd, ∆Vx, ∆IL and ∆D, and

three fast variable outputs, ∆Id, ∆Ix and ∆VL. Except for the constant input ∆D, each

output has a complementary input and the input-output pair relates to an external

component. Thus, the ∆Id response is paired up with ∆Vd and together relate to the input

small signal voltage source. Similarly, ∆Ix and ∆Vx are coupled to the lumped Ro and C

output circuit, and ∆VL and ∆IL are coupled to the inductor L.

3.2.2 Boost Converter: Finding FDM by Analytical Method

In the previous example, the FDM (fast dynamic model) coefficients are derived directly

from circuit largely based on intuition. In this example, the FDM parameters are derived

analytically.


33

+Vd-

QD

id

+

Vo

-

+

vC

-

iL

+ vL -

iC

IoDaL

C R

Fig. 3.8: Boost Converter in CCM

A boost converter, which is assumed to be operating in steady state under CCM is

shown in Fig. 3.8. The inductor L and capacitor C are slow components because their

associated current and voltage variables are relatively slow. The low frequency

approximation is again made here so that all slow variables behave as constants during a

switching period. Other slow variables are the input voltage Vd and duty ratio D.

Replacing the slow components by sources as before, we obtain the FDEC (fast dynamic

equivalent circuit) for the boost converter (Fig. 3.9). Here, the value Vd, Vx, IL and D

represent the average steady state values of these variables.

+Vd-

QD

id+ vL - Da

FDG

IL

Vx+-

Ix

Fig. 3.9: FDEC for Boost Converter


34

In this analytical approach, all DC relationships between the input and the output

variables are determined first. The output variables are Id, VL and Ix. Of these, Id equals

IL. By inspecting the ‘real-time’ waveforms (See Fig. 3.11) of the fast variables namely ix

( = id ) and vL, one can determine the DC relationship between the output variables Ix, VL,

and the input variables Vd, IL, Vx, and D.

DT

T

IL

ix(t)

0

0

vL(t) Vd

-(Vx-Vd)

t

t

Fig. 3.10: Waveforms of Fast Variables


35

Averaging the instantaneous current ix in Fig. 3.10 over a switching period,

[ ]( )

( ) L

L

TDTL

T

DTL

T

xx

IDT

DTTI

IT

dtIT

dttiT

I

−=

−=

=

=

=

∫

∫

1

1

1

)(1

0

(3.24)

Again from Fig. 3.10, the average inductor voltage VL may be obtained as follows,

( )

( )( )

( )( )dxd

dxd

T

DTdx

DT

d

T

LL

VVDDV

DTTVVDTVT

dtVVdtVT

dttvT

V

−−−=

−−−=

⎭⎬⎫

⎩⎨⎧

−−=

=

∫∫

∫

1

1

1

)(1

0

0

(3.25)

Rewriting the above results,

Ld II = (3.26)

( ) Lx IDI −= 1 (3.27)

( )dxdL VVDDVV −−−= )1( (3.28)


36

In general, the output variables are dependent on the input variables as follows.

( )DIVVfI Lxdd ,,,1= (3.29)

( )DIVVfI Lxdx ,,,2= (3.30)

( )DIVVfV LxdL ,,,3= (3.31)

Small signal input current, ∆Id

From Eq. (3.29), a Taylor series expansion of 1st order is performed.

DDf

IIf

VVf

VVf

I LL

xx

dd

d ∆∂∂

+∆∂∂

+∆∂∂

+∆∂∂

=∆ 1111 (3.32)

Since f1 = IL,

DDI

III

VVI

VVI

I LL

L

Lx

x

Ld

d

Ld ∆

∂∂

+∆∂∂

+∆∂∂

+∆∂∂

=∆ (3.33)

Thus,

DIVVI Lxdd ∆×+∆×+∆×+∆×=∆ 0100 (3.34)


37

Small signal lumped output circuit current, ∆Ix

From Eq. (3.30), in a similar manner as before and thus,

DDf

IIf

VVf

VVf

I LL

xx

dd

x ∆∂∂

+∆∂∂

+∆∂∂

+∆∂∂

=∆ 2222 (3.35)

From Eqs. (3.30) and (3.27),

0

)1(2

=∂−∂

=∂∂

d

L

d VID

Vf

(3.36)

0

)1(2

=∂−∂

=∂∂

x

L

x VID

Vf

(3.37)

DI

IDIf

L

L

L

−=∂−∂

=∂∂

1

)1(2

(3.38)

L

L

ID

IDDf

−=∂−∂

=∂∂ )1(2

(3.39)


38

Hence, from Eqs. (3.35) to (3.39),

DIVVI Lxdx ∆×−∆×+∆×+∆×=∆ 1000 (3.40)

Small signal inductor voltage, ∆VL

Likewise, from Eq. (3.31),

DDf

IIf

VVf

VVf

V LL

xx

dd

L ∆∂∂

+∆∂∂

+∆∂∂

+∆∂∂

=∆ 3333 (3.41)

From Eqs. (3.31) and (3.28),

( )

1

3

=∂

+−∂=

∂∂

d

xxd

d VDVVV

Vf

(3.42)

( )

( )DD

VDVVV

Vf

x

xxd

x

−−=+−=

∂+−∂

=∂∂

11

3

(3.43)

( )

0

3

=∂

+−∂=

∂∂

L

xxd

L IDVVV

If

(3.44)

( )

x

xxd

VD

DVVVDf

=∂

+−∂=

∂∂ 3

(3.45)


39

Thus, from Eqs. (3.41) to (3.45),

( ) DVIVDVV xLxdL ∆+∆×+∆−−∆×=∆ 011 (3.46)

We can now combine Eqs. (3.34), (3.40), and (3.46) to form the FDM for the boost

converter under CCM.

( )( ) ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

DIVV

VDID

VII

L

x

d

x

L

L

x

d

011100

0100 (3.47)

3.2.3 Buck-Boost Converter: Finding FDM by Simulation Method

In section 3.2.1, the buck converter’s FDM (fast dynamic model) is derived using

intuition regarding the converter operation. Section 3.2.2 took an analytical approach to

obtain the FDM of a boost converter. In this section, the fast dynamic model of a buck-

boost converter (see Fig. 3.11) will be obtained through circuit simulation. The converter

is assumed to be operating under CCM (continuous conduction mode). One of the

strengths of the SVM (separation of variables method) technique lies in that the method

allows an engineer to obtain the model parameters by circuit simulation only, thus

avoiding the need for mathematical analysis. In other words, practicing engineers without


40

detailed theoretical knowledge of the converter operation can still find the desired

transfer function by simulating the actual circuit design. Another advantage, this

approach allows us to incorporate the effects of parasitic elements into the small signal

model with relative ease. The following explains how this is done. Since this method is

based on actual simulation of the FDEC (fast dynamic equivalent circuit) of the power

converter, the specifications of the converter must be known. For this purpose, the

following design specifications are assumed (See Table 3.1).

+Vd-

D

id

-

Vo

+

-

vC

+

iL

+

vL

-

iC

IoDa

L

C R

Q

Fig. 3.11: Buck-Boost Converter

+Vd-

D

id

Vx

iL +vL-

Ix

DaQ

FDG

-+


41

Fig. 3.12: FDEC for Buck-Boost Converter

Table 3.1: Design Parameters of the Buck-Boost Converter

The fast dynamic equivalent circuit for the buck-boost converter is obtained as

before and is shown in Fig. 3.12. The input variables are identified as Vd, Vx, IL and D

whereas the output variables are Id, Ix and VL. The form of fast dynamic model is shown

as in Eq. (3.48).

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

DIVV

aaaaaaaaaaaa

VII

L

x

d

L

x

d

34333231

24232221

14131211

(3.48)

The coefficients a11, a12, etc can be found by taking the ratio between an input variable

and an output variable.

VariableInputinonPerturbatiVariableOutputinonPerturbatiamn = (3.49)

Where amn is the coefficient corresponding to mth row and nth column.

Input Voltage: Vd = 220V Output Voltage: Vo = 150V Filter Inductor: L = 20mH Filter Capacitor: C = 2µF Load Resistor: Ro = 500Ω Switching Frequency: fs = 30kHz Duty Ratio: D = Vo/(Vd+Vo) ≅0.4054 Output Current: Io=Vo/Ro =0.3A Inductor Current: IL=Io/(1-D) ≅ 1.3513A


42

Thus, for example x

d

VI

a∆∆

=12 , L

d

II

a∆∆

=13 , etc.

Using OrCad PSpice [20] simulation, each slow variable is perturbed a small

amount one at a time, between one and ten percent of its steady state value. As a general

rule of thumb, the perturbation magnitude should be sufficiently small so that circuit

responds linearly. In this simulation, the average variables are directly probed in the

circuit. The command to average a cyclic waveform over a period in the OrCad PSpice is

‘AVGX( V , Ts )’, where V is the probed waveform of interest and Ts is the averaging

period, which in this case is the switching period. Fig. 3.13 shows a method to simulate

the FDEC response to step change in input voltage Vd. Similar approach may be taken

with respect to other slow variables. In Fig. 3.13, Vd is perturbed from 220V to 242V by

switching ‘Vd’ off and ‘Vd1’ on at time 0.999ms. The simulation results are shown in

Fig. 3.14.

SW2

0.999ms1

2

IL

1.3513

0

Di

SW1

0.999ms1

2

Vd1242

R1

1k

+

Q

1m

Vx

150

Vd220

Vpwm

PW = 0.8108usPER = 2us

V1 = 0V2 = 10


43

Fig. 3.13: Buck-Boost Converter: Perturbation of Vd

Fig. 3.14: Buck-Boost Converter:

Fast Variables Response to Vd Perturbation

There are three waveforms displayed in Fig. 3.14. The instantaneous inductor

voltage vL is shown by the waveform with ‘ ’ symbol and oscillating with a period of

2µs. This vL waveform is scaled down by 25 times to show the waveform more clearly in

the figure. The waveform with symbol ‘∇’ indicates the averaged inductor voltage VL,

averaged over 2µs period. The averaged voltage VL rises from 3.0377V during steady

state to 12.938V peak just after perturbation and then settles down at 12.168V. The

transient peak is ignored and only the new (quasi) steady state value 12.168V is

measured. Ideally at the steady state, the inductor voltage has zero averaged voltage due

to volt-second balance principle. However there exist a 3.0377V offset for the steady

state average inductor voltage. This is caused by the voltage source ‘Vpwm’ (See Fig.

VL vL Id

Ix


44

3.13) and decreasing its amplitude reduces such unwanted effect. Alternatively, the DC

offset can be removed by normalizing the offset voltage to 0V. Thus, the perturbed

response of the averaged inductor voltage is (12.168 – 3.0377 = 9.1303)Vavg. Unlike the

averaged inductor voltage, the averaged input current Id shown with ‘∆’ symbol did not

respond to perturbation. The averaged value before and after the perturbation are

561.035mA and 561.050mA respectively. The difference is only 0.015mA as compared

to the current value of 561mA. Thus the coefficient corresponding to this is nearly zero.

The averaged diode current (Ix) with ‘o’ symbol changes from 796.301mA to 787.324mA

giving a difference of –8.977mA.

Therefore,

011 ≈∆∆

=d

d

VIa (3.50)

63

21 10046.40822

10977.8 −−

×−=×−

=∆∆

=d

x

VIa (3.51)

4150.0221303.9

31 ==∆∆

=d

L

VVa (3.52)

The procedure above is repeated for each of the perturbations of Vx (150V

165V), IL (1.3513A 1.4864A) and D (0.4054 0.4469). The changes in the output

variables to input variable perturbations after the initial transient period are shown in

Table 3.2.


45

Table 3.2: Calculation of Coefficients

Consolidating results obtained in Table 3.2 into FDM matrix,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

×−−××=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

−

−−

DIVV

VII

L

x

d

L

x

d

4.371102805450.04150.03627.15824.01098810408

3529.14150.000

6

66 (3.53)

(A) For ∆Vx = 15V Output Variables Before After Change FDM Coefficients (Averaged)

Id 561.038mA 561.047mA 0.009mA a12 = 600 × 10-9 ≈ 0 Ix 796.303mA 781.477mA -14.826mA a22 = 988.4×10-6

VL 3.0354V -5.7392V -8.7746V a32 = -0.5450

(B) For ∆IL = 0.1351A Output Variables Before After Change FDM Coefficients (Averaged)

Id 561.035mA 617.103mA 56.068mA a13 = 0.4150 Ix 796.305mA 874.986mA 78.681mA a23 = 0.5824

VL 3.0354V 3.0396 V 0.0042V a33 = 280×10-6

(C) For ∆D = 0.04054

Output Variables Before After Change FDM Coefficients (Averaged)

Id 561.035mA 615.883mA 54.848mA a14 = 1.3529 Ix 796.305mA 741.060 mA -55.245mA a24 = -1.3627

VL 3.0433V 18.100 V 15.0567 a34 = 371.4


46

For comparison purpose, the FDM (fast dynamic model) obtained for the buck-boost

converter under CCM obtained through analytical method is given below. The theoretical

derivation is not shown here.

( )( ) ( ) ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

DIVV

VVDDID

ID

VII

L

x

d

xd

L

L

L

x

d

01100

00 (3.54)

Substituting the parameters listed in Table 3.1 into Eq. (3.54), we obtain the FDM for the

converter under consideration.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

DIVV

VII

L

x

d

L

x

d

37005946.04054.03513.15946.000

3513.14054.000 (3.55)

It can be seen that the FDM parameters in Eq. (3.53) are very close to the theoretical

values in Eq. (3.55).

3.3 Fast Dynamic Model for Discontinuous Conduction Mode Cases

The inductor current (or more accurately, the magnetic flux) determines if a

pulse-width-modulated (PWM) converter is operating in CCM (continuous conduction

mode) or DCM (discontinuous conduction mode). In CCM operation, the inductor

current is continuous without becoming zero at any instant. In other words, there is

always stored energy in the inductor throughout the switching cycle. When the converter

is operated under DCM condition, there is duration in a switching period when the


47

inductor energy depleted entirely. Therefore, under DCM, at the beginning of a switching

cycle, the inductor energy or current will always begin from zero initial condition. Thus

the inductor core flux is ‘reset’ in every cycle when the inductor current goes to zero.

Therefore if the inductor current is perturbed a small amount in a cycle, the perturbation

will not be able to propagate to the next cycle. Therefore, there is no dynamics related to

inductor in DCM operation and the current settles to a new steady state instantaneously

within one switching period. As a result, the inductor current is now to be treated as a fast

variable. Therefore unlike previous CCM examples, the inductor current in DCM cannot

be separated from the FDG. Thus performing SVM in DCM will be slightly different

from CCM cases. The inductor component will be a part of FDG now.

3.3.1 Buck Converter (DCM): Finding FDM by Analytical Method

The buck converter (Fig. 3.1) has three operating intervals under DCM,

corresponding to Q ‘on’, Da conducting and both Q and Da ‘off’. The buck converter in

DCM (discontinuous conduction mode) maybe considered to have averaged slow

variables, Vd, D, and Vc. As before, the output circuit Ro and C are lumped together as a

single voltage source Vx. Thus, instead of Vc as the slow variable, Vx is considered. The

two output variables of the FDG, on the other hand, are the Id and Ix. Here, the inductor is

treated as a fast component and not a slow component. Separating the fast and slow

components, and replacing the slow components by constant sources will give the FDEC

(fast dynamic equivalent circuit) for the Buck converter in DCM shown in Fig. 3.15.


48

+

Vd

-

Q

D

Id

Vx

IL

+ VL -

IxDi+-

FDG

Fig. 3.15: FDEC for Buck Converter in DCM

From Fig. 3.15, the averaged large signal equations for the FDG model can be defined as,

( )DVVfI xdd ,,1= (3.56)

( )DVVfI xdx ,,2= (3.57)

Here, the output variables Id and Ix of the FDEC (fast dynamic equivalent circuit)

depends on the input variables Vd, Vx, and D. Also, note that IL = Ix.


49

Vd-VxDT

T

dxT

Vd

ILP

IL

iL(t)

0

0

vL(t)

Mod

e 1

Mod

e 3

Mod

e 2

m 1 m 2

t

t

Fig. 3.16: Buck Converter under DCM: Inductor Waveforms

Eqs. (3.56) and (3.57) describe the fast characteristics of the buck converter in

DCM (discontinuous conduction mode) and like before they are functions of input

variables. To find their explicit form, the associated waveforms are plotted over a

switching period, as in Fig. 3.16, and investigated. A typical DCM operation for non-

isolated hard-switching converters comprises of three operating intervals as depicted by

Interval 1, Interval 2 and Interval 3 in the figure. The first two intervals are similar to

those in the CCM case. The extra interval that occurs in DCM is Interval 3 in which the

inductor stored energy is zero and hence there is no current flowing through the inductor

and the voltage across the component is zero. From this figure, it is obvious that a small


50

perturbation in the inductor current during one switching cycle will not propagate to the

following switching cycle due to the fact the inductor current goes to zero in Interval 3.

Therefore, the initial condition for the inductor current in the next Interval 1 will always

be zero. This observation indicates that the inductor current settles rapidly in response to

a perturbation, indicating that the inductor current has become a fast variable in DCM

case.

In order to obtain the fast dynamic model (FDM) through small signal

perturbation, the averaged signal relationship in Eqs. (3.56) and (3.57) must be obtained.

A steady state analysis of the converter yields these relationships.

Applying volt-seconds balance to inductor voltage in Fig. 3.16, the net area over a cycle

must be zero under steady state operation.

( ) TdVDTVV xxxd =− (3.58)

Thus, the free wheeling interval is

( )x

xdx V

DVVd

−= (3.59)

The inductor peak current, Ip in Fig. 3.16 can be easily obtained from the inductor

voltage-current relationship during Interval 1.

( )DTILVVtv P

xdL =−=1Interval

(3.60)


51

From this,

( )L

DTVVI xdP −= (3.61)

The magnitude of slopes m1 and m2 are

LVV

DTIm xdP −

==1 (3.62)

LV

TdIm x

x

P ==2 (3.63)

The averaged input current, Id, is given by

∫

∫

∫

×=

=

=

DT

DT

L

T

dd

dttmT

dttiT

dttiT

I

01

0

0

1

)(1

)(1

(3.64)

Thus,

DT

od

tmT

I ⎥⎦

⎤⎢⎣

⎡=

21 2

1 (3.65)

Substituting for m1 from Eq. (3.62),

( )L

TDVVI xdd

2

21

−= (3.66)


52

This is the equation for the averaged input current Id in the form given in Eq. (3.56).

Here, Id is the output variable of the FDEC while Vd, Vx, and D are the slow (input)

variables.

Next, the expression for averaged lumped output circuit current, Ix, will be found.

⎥⎦

⎤⎢⎣

⎡+=

=

∫ ∫

∫DT T

DT

T

Lx

tdtmtdtmT

dttiT

I

021

0

1

)(1

(3.67)

Integrating,

22

22

21 TdmTDm

I xx += (3.68)

Substituting for m1, m2, and dx from Eqs. (3.62), (3.63), and (3.59).

TDVV

LVV

Ix

dxdx

2

21

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ −= (3.69)

Eqs. (3.66) and (3.69) are the explicit forms of Eq. (3.56) and (3.57).

Thus, let

( )LTDVVfI xdd

2

1 21

−== (3.70)

and let

LTDV

VVfI d

x

dx

22

2 21

⎟⎟⎠

⎞⎜⎜⎝

⎛−== (3.71)


53

Next, the small signal FDM coefficients will be obtained through perturbation of the slow

variables in Eqs. (3.70) and (3.71).

Small signal input current, ∆Id

From Eq. (3.70), the first order Taylor series expansion is given by

DDf

VVf

VVf

I xx

dd

d ∆∂∂

+∆∂∂

+∆∂∂

=∆ 111 , (3.72)

where f1 is given as in Eq. (3.70).

Determining the partial derivatives,

( ) DL

DTVVVLTDV

LTDI xdxdd ∆−+∆−∆=∆

22

22

(3.73)

The FDM (fast dynamic model) coefficients (a11 = ∆Id/∆Vd, a12 = ∆Id/∆Vx, a13 = ∆Id/∆D)

can be determined from Eq. (3.73).

Small signal lumped output circuit current, ∆Ix

From Eq. (3.71), the first order Taylor series expansion is given by

DDf

VVf

VVf

I xx

dd

x ∆∂∂

+∆∂∂

+∆∂∂

=∆ 222 , (3.74)

where f2 is as given in Eq. (3.71).


54

Again, determining the partial derivatives,

DL

DTVVV

VLTD

VV

VLTD

VVI

dx

d

xx

d

dx

dx

∆⎟⎟⎠

⎞⎜⎜⎝

⎛−+

∆⎟⎟⎠

⎞⎜⎜⎝

⎛−

∆⎟⎟⎠

⎞⎜⎜⎝

⎛−=∆

2

22

2

21

21

(3.75)

Here, again the FDM coefficients (a21 = ∆Ix/∆Vd, a22 = ∆Ix/∆Vx, a23 = ∆Ix/∆D) can be

determined from Eq. (3.75).

FDM of the Buck Converter in DCM

Eqs. (3.74) and (3.75) can be combined to form the fast dynamic model (FDM) of the

buck converter under DCM condition.

( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−=⎥

⎦

⎤⎢⎣

⎡∆∆

DVV

VVV

LTD

VV

LTD

VV

LDTVV

LTD

LTD

II

x

d

dx

d

x

d

x

d

xd

x

d2222

22

21

21

22 (3.76)

In Eq. (3.76), ∆Vd, ∆Vx, and ∆D are linearized, averaged perturbation around

steady state of the slow variables Vd, Vx, and D. ∆Id and ∆Ix are the averaged response of

the output (fast) variables Id and Ix again around the steady state operating point of the


55

converter. Notice again the absence of any dynamic term in the FDM indicating the ‘fast’

nature of this component group.

3.3.2 Boost Converter

By following the same approach above, the FDM (fast dynamic model) of a Boost

converter in DCM (discontinuous conduction mode) can be found.

( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=⎥⎦

⎤⎢⎣

⎡∆∆

DVV

LDT

VVV

LTD

VVV

LTD

VVVVV

LDTV

LTD

II

x

d

dx

d

dx

d

dx

dxd

d

x

d2222

2

2

2

212

21

02

(3.77)

3.3.3 Buck-Boost Converter

Once again following a similar approach, the FDM for the buck-boost converter can be

found.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎥⎦

⎤⎢⎣

⎡∆∆

DVV

LDT

VV

LTD

VV

LVTDV

LDTV

LTD

II

x

d

x

d

x

d

x

d

d

x

d2222

2

21

02

(3.78)


56

3.3.4 Note on FDMs for DCM Cases

The FDMs for buck, boost, and buck-boost converters have been derived for the

DCM cases. Unlike the CCM cases, the FDM coefficients in DCM cases possess the L

parameter. However, the appearance of the inductance value in the matrix does not

indicate any dynamics in the FDM. As pointed out earlier, the inductor current settles

rapidly within a switching period.

Also, in DCM cases, the FDMs are reduced to those derived using injected-

absorbed-current (IAC) method [14], [15]. Nevertheless the proposed SVM is

conceptually different from the IAC method. In the injected-absorbed-current method, the

input and output currents are averaged as a means of obtaining the small signal behavior

of the power converter. The proposed SVM, in contrast, is based on separating a circuit

into slow and fast sub-systems to develop a small signal model.

3.4 Small Signal Circuit Model

Once the FDM (fast dynamic model) of the converter is obtained as described in

Section 3.3, the averaged small signal circuit model of the converter can be obtained in a

straightforward manner. This is done by connecting the FDM to slow components as

shown in Fig. 3.17.


57

FDM

+-

+∆Vd

-

+∆Vo

-

∆VL

∆Ιx∆Ιd

∆ΙL

C R

L

+∆D

-

+∆Vx

- ∆ΙC ∆Ιo

P1

N1

Fig. 3.17: Averaged Small Signal Circuit Model

of the Converter under CCM

FDM

+∆Vd

-

+∆Vo

-∆ Ιx∆ Ιd

C R

+∆D

-

+∆Vx

- ∆ΙC ∆Ιo

N1

Fig. 3.18: Averaged Small Signal Circuit Model

of the Converter under DCM


58

The averaged small signal circuit of Fig. 3.17 is applicable to buck, boost and

buck-boost converters under CCM operation. The FDMs in the three cases are given by

Eqs. (3.23), (3.47), and (3.54) respectively. Fig. 3.18 shows the small signal model of the

converter under DCM operation. Notice that the inductor is not in the DCM equivalent

circuit as explained earlier.

3.5 Obtaining Small Signal Responses

As explained earlier, either an analytical or simulation based approach can be

adopted in order to obtain the small signal responses from the small signal circuit model.

These will be illustrated in Sections 3.5.1 and 3.5.2 respectively for the CCM cases only.

The small signal responses in the DCM cases are not included in the thesis. They can be

obtained in a similar manner.

3.5.1 Analytical Method

The small signal transfer functions to be determined are the control-to-output

(∆Vo(s)/∆D(s)) and the input-to-output (∆Vo(s)/∆Vd(s)) transfer functions. This can be

obtained by a straightforward frequency response analysis of the circuit Fig. 3.17.

Referring to the figure, the interaction between the slow components and the FDM is

established first. It may be noted that ∆Vx = ∆Vo. Applying KCL (Kirchoff’s Current

Law) at node N1 results in


59

RV

VsCIII oooCx

∆+∆=∆+∆=∆ (3.79)

The small signal averaged inductor voltage can be found by applying KVL (Kirchoff’s

Voltage Law) around loop P1 in Fig. 3.17.

LL IsLV ∆=∆ (3.80)

Given below is the general FDM equation applicable for buck, boost, and buck-boost

converters (This equation is the same as Eq. 3.48).

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆∆∆

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

DIVV

aaaaaaaaaaaa

VII

L

x

d

L

x

d

34333231

24232221

14131211

(3.48)

In Eq. 3.48, only the last two row equations corresponding to ∆Ix and ∆VL are

needed to obtain the desired transfer function. The expression for ∆Id is not developed as

the desired transfer function can be obtained without it. The first row equation in Eq. 3.48

will be needed if, for example, the input impedance (∆Vo(s)/∆Id(s)) is to be determined.

From Eqs. (3.48) and (3.80),

DaIaVaVaIsL LodL ∆+∆+∆+∆=∆ 34333231 (3.81)

( )DaVaVaasL

I odL ∆+∆+∆−

=∆ 34323133

1 (3.82)


60

From Eqs. (3.48), (3.79), and (3.82),

( )

Da

DaVaVaasL

a

Va

VaRVVsC

od

o

do

o

∆+

⎥⎦

⎤⎢⎣

⎡∆+∆+∆

−+

∆+

∆=∆

+∆

24

34323133

23

22

21

1 (3.83)

Thus,

DasL

aaa

VasL

aaaVasL

aaaR

sC do

∆⎟⎟⎠

⎞⎜⎜⎝

⎛−

++

∆⎟⎟⎠

⎞⎜⎜⎝

⎛−

+=∆⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−+

33

342324

33

312321

33

322322

1

(3.84)

Eq. (3.84) relates the output response to both Vd and D perturbations.

For Duty-ratio to Output Voltage transfer function, the ∆Vd is set to null in Eq. (3.84).

( )

3223332233

33222

34233324

aaaaR

aCaLa

RLsLCs

aaasLaDVo

−+−⎟⎠⎞

⎜⎝⎛ −−+

+−=

∆∆

(3.85)

For input to output transfer function, the ∆D is set to null in Eq. (3.84).

( )

3223332233

33222

31233321

aaaaR

aCaLa

RLsLCs

aaasLaVV

d

o

−+−⎟⎠⎞

⎜⎝⎛ −−+

+−=

∆∆

(3.86)

These are general transfer functions, applicable to all the three converters under CCM.


61

Transfer Function of Buck Converter under CCM

It was shown earlier in Eq. (3.23) that a buck converter’s FDM is

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

d

L

VD

ID

aaaaaaaaaaaa

010100

00

34333231

24232221

14131211

Substituting the FDM (fast dynamic model) parameters into Eq. (3.85), we obtain the

duty ratio-to-output transfer function.

12 +⎟⎠⎞

⎜⎝⎛+

=∆∆

RLsLCs

VDV do (3.87)

And substituting the same parameters into Eq. (3.86), we can obtain the input-to-output

transfer function.

12 +⎟⎠⎞

⎜⎝⎛+

=∆∆

RLsLCs

DVV

d

o (3.88)

These transfer functions are identical to those derived by state-space-averaging

(SSA) [1]. This validates the SVM technique for hard-switched converters under CCM.

No further verification is considered necessary for the SVM technique when applied to

this class of converters.


62

Transfer Function of Boost Converter under CCM

The boost converter’s transfer functions can be obtained in a similar manner.

The duty ratio-to-output transfer function,

( )

( ) ( ) ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+−

+−

−−

−=

∆∆

111

11

122

2

2

RDLs

DLCs

RDsL

DV

DV oo (3.89)

The input-to-output transfer function,

( )( ) ( ) ⎟

⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+−

−=

∆∆

111

11

1

222

DRLs

DLCsDV

V

d

o (3.90)

Once again, these transfer functions are identical to those derived by SSA [1].

Transfer Function of Buck-Boost Converter under CCM

Likewise, the buck-boost’s transfer functions can be obtained.

The duty ratio-to-output transfer function,

( )( )

( ) ( ) ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+−

+−

−−

−=

∆∆

111

11

122

2

2

RDLs

DLCs

RDLDs

DDV

DV oo (3.91)


63

And input-to-output transfer function,

( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+−

−=

∆∆

111

11

222

RDLs

DLCsD

DVV

d

o (3.92)

Again, these transfer functions obtained above are identical to that obtained using

SSA method [1]. As mentioned earlier, the SVM technique also produces identical results

to SSA when the three hard-switched converters are under DCM. There, however, are not

provided here.

3.5.2 Simulation Method

This section illustrates a Pspice simulation-based approach to obtain desired small

signal responses of the converter. These will be carried out for the three converter cases

under CCM. The DCM cases can be verified in a similar manner.

Table 3.3 contains relevant design parameters for the application example of Buck

converter under CCM. The design parameters listed in Table 3.3 are substituted into the

buck’s FDM in Eq. 3.23 to construct the FDM small signal circuit, as shown in Fig. 3.19

in the Pspice format.


64

Table 3.3: Design Parameters of the Buck Converter

+-

a14

G25

Vds0

+-

a22

G0

+-

+-

a21

G0

+-

a11

G0

R

0.2

RD1

1000G

-+

+-

a31

E0.625

0

C

2000uF

+-

a12

G0

-+

+-

a32

E-1

-+

+-

a34

E8

+-

a23

G1

-+

+-

a33

E0

+-

a24

G0

D10

+-

a13

G0.625

R1

0.001L

5uH

Fig. 3.19: Buck Converter Small Signal Model Using FDM – Pspice Diagram

Input Voltage: Vd = 8V Output Voltage: Vo = 5V Filter Inductor: L = 5µH Filter Capacitor: C = 2000µF Load Resistor: Ro = 0.2Ω Switching Frequency: fs = 200kHz Duty Ratio: D = Vd/Vo = 0.625 Inductor Current: IL=Vo/Ro = 25A


65

The dependent source blocks with labels a11, a12, a13, a14, a21, a22, a23, a24,

a31, a32, a33, and a34 in Fig. 3.19 correspond directly to a11, a12, a13, a14, a21, a22, a23, a24,

a31, a32, a33, and a34 of the FDM coefficients. The dependent current source ∆Id is

connected to the external small signal voltage source ∆Vd, shown as Vds. Similarly, ∆Ix

is connected to the lumped output Ro and C components. The inductor, L, has a small

series resistance, R1, to avoid convergence problem in Pspice simulation and is

connected to ∆VL. ∆D shown as D in the program is an independent duty-ratio injection

signal. The large resistor RD1 eliminates ‘floating’ problem in Pspice simulation. The

gain-phase plots from both SSA (State-Space-Averaging) and PSpice simulations are

plotted in Matlab for comparison and these are shown in Fig. 3.20.

(a) Gain Plot

(b) Phase Plot

Solid line – SSA ‘ * ’ line – SV Method


Fig. 3.20: Small Signal Frequency Response for Buck Converter under CCM


66

It is pointed out in the previous section that SSA method and SVM analytical

method yield identical results. From Fig. 3.20, it can be seen that both SSA and SVM

simulation result in the same response curves.

The same procedure is applied for the boost converter under CCM to find desired

small signal frequency response curves. The design parameters for this converter are

listed in Table 3.4. Since buck, boost, and buck-boost share the same number of rows and

columns of their FDM matrices, the PSpice schematic of Fig. 3.19 can be reprogrammed

for boost or buck-boost converter by merely changing the matrix coefficients.

Table 3.4: Design Parameters of the Boost Converter

The gain-phase comparison plots for Boost converter under CCM are given below.

Input Voltage: Vd = 22V Output Voltage: Vo = 40V Filter Inductor: L = 60µH Filter Capacitor: C = 100µF Load Resistor: Ro = 1.333Ω Switching Frequency: fs = 500kHz Duty Ratio: D = 1-Vd/Vo=0.45 Output Current: Io=Vo/Ro ≅ 30A Inductor Current: IL=Io/(1-D) ≅ 54.55A


67

(a)Gain Plot

(b)Phase Plot

Fig. 3.21: Small Signal Frequency Response for Boost Converter

Again, it can be noted that the SVM approach gives identical results to that provided by

classical state space averaging (SSA) approach.

The design parameters of the buck-boost converter under CCM can be found in Table

3.1. The corresponding frequency response plots are given in Fig. 3.22.




68

(a) Gain Plot

(b) Phase Plot

Fig. 3.22: Small Signal Frequency Response for

Buck-Boost Converter under CCM

Once again, it may be noted that the SVM method gives identical results when compared

to the SSA method.

3.6 Summary

This chapter demonstrated the use and validity of SVM in obtaining small signal

behavior of hard-switched converter circuit both under CCM and DCM conditions. The

proposed SVM allows the creation of an FDM (Fast Dynamic Model) that can be used to

form a small signal circuit, which can then be used to yield small signal responses either

analytically or by simulation. It is also shown for the case of buck, boost, and buck-boost

converters under CCM conditions that SVM produces transfer functions that are identical

to those provided by the SSA (State-Space-Averaging) method. Unlike the latter, the

SVM is simpler and less mathematical, which is one of the main attractions of the

proposed method.

Chapter 4 Application Examples to Quasi-Resonant-Converters

69

Chapter 4

Application Examples to Soft-Switched Converters

4.1 Introduction

The aim of this chapter is to show the applicability of the proposed SVM to quasi-

resonant type (soft-switched) of converters. The operating mechanism of these soft-

switched converters are different from their hard-switched counterparts. Due to this

variation, these converters can deliver higher output wattage per volume. However,

detailed operation of soft-switched converters is not discussed here as they can be found

in many literatures such as [21], [22]. Although there are many variations of soft-

switched converters, only zero-current-switch quasi-resonant-converter buck in half-wave

mode is presented in this report.

This chapter is divided as follows. Section 4.2 reviews briefly on the operation of

the quasi-resonant converter. The creation of the fast dynamic equivalent circuit (FDEC)

for the converter is shown in Section 4.3. Section 4.4 presents an analytical derivation for

obtaining the fast dynamic equivalent circuit (FDEC) model while Section 4.4.1 explores

a simulation-based approach to achieve the same goal. A small signal equivalent circuit is

developed in Section 4.5 and the frequency responses are obtained through circuit

simulation in Section 4.6. Section 4.7 summarizes the chapter.


70

4.2 Brief Review of Zero-Current-Switch

Quasi-Resonant-Converter (ZCS-QRC) Buck Converter

Fig. 4.1 depicts a circuit construction of a Zero-Current-Switch Quasi-Resonant-

Converter (ZCS-QRC) Buck converter for half-wave mode operation. If the diode Da is

removed from the circuit, the converter would operate in a full-wave mode.

+-

+

VO

-

CO

LOILr

+ VLr -

Lr

Id

Vd

Da

Db

+ VO -

ILO

+

VCr

-

ICr IO

Q

Cr

Fig. 4.1: ZCS QRC Buck Converter (Half-Wave)

Fig. 4.2: Resonant Inductor Current (iLr) and Resonant Capacitor Voltage (vCr) Behavior of the ZCS-QRC Buck Converter in Half-Wave Mode

iLr vCr

Io


71

The operation for the ZCS-QRC buck converter consists of four operating

intervals as depicted by T1, T2, T3, and T4 in Fig. 4.2 (Pspice simulated plots). From the

same figure, it can be seen that the inductor current iLr becomes zero at intervals T3 and

T4 whereas the capacitor voltage becomes zero at intervals T1 and T4. This operates

similarly to hard-switched converters under DCM (discontinuous conduction mode),

which was discussed in Section 3.3. Therefore, perturbed resonant inductor current iLr

and perturbed resonant capacitor voltage vCr of the ZCS-QRC buck converter will not

propagate to the following switching cycle. As a consequence, these resonant tank state

variables are treated as fast variables. This results in the converter’s FDM whose

coefficients are also constant as in hard-switched converter cases.

4.3 Fast Dynamic Equivalent Circuit (FDEC) of Zero-Current-Switch

Quasi-Resonant-Converter (ZCS-QRC) Buck Converter in Half

Wave Mode

At steady state operating condition with low ripple approximation, the equivalent

circuit seen from Cr to the output circuit (see Fig. 4.1) is considered as a constant current

source. Hence the output circuit Lo, Co, and Ro are replaced by this constant current

source. However, the resonant tank components Lr and Cr, due to their fast varying state

variables, cannot be replaced by constant sources. Thus, these resonant tank components

are then considered as fast dynamic group (FDG) along with other nonlinear components

Da, Db and Q, which respond almost instantly to perturbations. This results in a fast


72

dynamic equivalent circuit (FDEC) for the ZCS-QRC-Buck in half wave mode as shown

in Fig. 4.3.

+-

+

VO

-

ILr

+ VLr -

LrId

Vd

Da

Db

+

VCr

-

ICr

IO

Q

fs

FDG

Fig. 4.3: FDEC of the ZCS QRC Buck Converter (Half-Wave)

With the converter’s FDEC circuit, the slow and fast variables of this QRC circuit

can be easily identified. As before, all the variables are approximated by their averaged

values taken over a switching period. From Fig. 4.3, the average slow variables are Vd, Io,

and fs while the average fast variables are Id and Vx. Note that the output voltage Vo is the

same as Vx, however the latter symbol is adopted here. In this case, the control signal is

the switching frequency fs and not the duty-ratio D. By perturbing each slow variableby a

small amount at a time while keeping other slow variables constants, the resultant

perturbations of the fast variables are used to compute the converter’s FDM coefficients.

This is shown in the following section.


73

4.4 Finding FDM by Analytical Method

The converter’s fast dynamic model (FDM) can be obtained by inspecting its fast

dynamic equivalent circuit (FDEC) shown in Fig. 4.3. Here, the output variables were

found to be ILr (Note that ILr = Id) and Vx while the input variables were found to be Vd,

Io, and fs. By considering the variables’ small signal counterparts (of first order terms

only), the FDM equation is,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡∆∆

s

o

d

x

d

fIV

aaaaaa

VI

232221

131211 (4.1)

Where a11, a12, a13, a21, a22, and a23 are constants.

To obtain the FDM coefficients analytically, the following large signal equations

must be found explicitly. Note that by expanding Eqs. (4.2) and (4.3) in terms of Taylor

series and considering only the first order terms, this will result in the form of Eq. (4.1).

However, since the focus of this section is to show the applicability of the separation of

variables method (SVM) to soft-switched converters, the explicit form of Eqs. (4.2) and

(4.3) will be adopted from literatures such as [22].

( )sodd fIVfI ,,1= (4.2)

( )sodx fIVfV ,,1= (4.3)


74

From [22], Eqs. (4.2) and (4.3) are expressed explicitly as (respectively),

od II µ= (4.4)

dx VV µ= (4.5)

Where,

⎟⎟⎠

⎞⎜⎜⎝

⎛−++++= − 111sin

2 21

αααπα

ωµ

r

sf (4.6)

d

ro

VZI

=α (4.7)

r

rr C

LZ = (4.8)

rrr CL

1=ω (4.9)

Here, the FDM coefficients will be obtained numerically. In this case, a Matlab

program is written to compute Eqs. (4.4) and (4.5). The procedure for finding these FDM

coefficients is to compute the converter’s resonant state variables ILr and Vx at their

steady state by using the design parameters listed in Table 4.1. The steady state values are

shown in Eqs. (4.10) and (4.11).


75

Table 4.1: Design Parameters for ZCS-QRC-Buck Half-Wave

The steady state values for the fast variables Id and Vx are,

AId 4264.0= (4.10)

VVx 7678.6= (4.11)

Next, using Eqs. (4.4) and (4.5), all input variables are perturbed separately from

their steady states and then the resultant perturbed fast variables are measured. Then by

taking the ratio between an input variable and an output variable as in Eq. (3.49) in

Chapter 3, the QRC converter’s FDM coefficients are found. For this computation, the

input variable perturbation values used are given in Table 4.2.

Table 4.2: Perturbation of Input Variables

Input Voltage: Vd = 10V Output Voltage: Vo = 6.7V Resonant Inductor: Lr = 1µH Resonant Capacitor: Cr = 25nF Load Current: Io = 0.63A Switching Frequency: fs = 500kHz Filter Inductor: Lo = 100µH Filter Capacitor: Co = 10µF

Input Voltage Perturbation: ∆Vd = 0.1V Load Current Perturbation: ∆Io = 0.0063A Switching Frequency Perturbation: ∆fs = 5kHz


76

From numerical computation, under Vd perturbation, the perturbed output variables are

4287.0=∆+ dd II (4.12)

8724.6=∆+ xx VV (4.13)

Therefore,

023.0101.10

4264.04287.011 =

−−

=∆∆

=d

d

VI

a (4.14)

046.1101.10

7678.68724.621 =

−−

=∆∆

=d

x

VV

a (4.15)

Similarly, under Io perturbation, the perturbed output variables are

4283.0=∆+ dd II (4.16)

7318.6=∆+ xx VV (4.17)

Thus,

3016.063.06363.0

4264.04283.012 =

−−

=∆∆

=o

d

II

a (4.18)

7143.563.06363.0

7678.67318.622 −=

−−

=∆∆

=o

x

IV

a (4.19)


77

The perturbed output variables due to fs perturbation are

4306.0=∆+ dd II (4.20)

8355.6=∆+ xx VV (4.21)

Therefore,

613 1084.0

5005054264.04306.0 −×=

−−

=∆∆

=kHzkHzf

Ia

s

d (4.22)

623 1054.13

5005057678.68355.6 −×=

−−

=∆∆

=kHzkHzf

Va

s

x (4.23)

Substituting Eqs. 4.14, 4.15, 4.18, 4.19, 4.22, and 4.23 into Eq. 4.1,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×−

×=⎥

⎦

⎤⎢⎣

⎡∆∆

−

−

s

o

d

x

Lr

fIV

VI

6

6

1054.137143.5046.11084.03016.0023.0

(4.24)

Eq. (4.24) characterizes the fast part of the ZCS-QRC buck converter in half-wave

mode. The approach taken in this section demonstrated the mathematical technique for

obtaining model parameters of the soft-switched converter. In the next section, a

simulation-based approach to find the converter’s fast dynamic model (FDM) or model

parameters is presented.


78

4.4.1 Finding FDM by Simulation Method

In the previous section, the procedure to find FDM coefficients is achieved by

perturbing DC equations (Eqs. 4.2 and 4.3) around their steady state values. But these DC

equations must be analytically derived beforehand. For practicing engineers, this

approach may not be the best way as time-to-market factor is very crucial for the success

of power supply business. Therefore, an engineer who does not fully understand the

operation of a soft-switched converter would have some difficulties deriving the DC

equations. Here, the separation of variables (SVM) technique offers an easier way to

obtain the model parameters of the converter, which in turn are used for construction of

small signal equivalent circuit for the soft-switched converter.

An OrCad Pspice [20] version for the ZCS-QRC Buck half-wave FDEC circuit is

developed (See Fig. 4.4). Shown in the figure, the input voltage Vd is perturbed by 0.1V.

This is done by switching the voltage source Vd = 10V to Vd1 = 10.1V at time 2ms. The

perturbed response of averaged fast variables Id and Vx are then measured (Take note that

the averaging of waveforms in Pspice is explained in Section 3.2.3) as shown in Figs. 4.5

and 4.6.


79

R1

1k

Vpwm

PW = 0.8usPER = 2us

V1 = 0V2 = 10

Vd10

Io0.63Cr

25n

Vd110.1

V

Da

U1

2ms1

2

Lr

1u +

Q

1m

Db

U2

2ms1

2

0

Fig. 4.4: FDEC Pspice Model for ZCS-QRC-Buck (HW)

Having Two Different Input Voltages to Emulate a Step Change

Fig. 4.5: The Perturbed Response of the Average Id Current

from the Pspice Model in Fig. 4.4


80

Fig. 4.6: The Perturbed Response of the Average Vx Voltage

from the Pspice Model in Fig. 4.4

Note that the transient behaviors of the perturbed fast variables are ignored. This

is because as assumed by the separation of variables, the fast part of a converter circuit

would respond instantaneously to external perturbations.

Thus,

02285.01.0

426095.0428380.011 =

−=

∆∆

=d

d

VIa (4.25)

036.11.0

9582.50618.621 =

−=

∆∆

=d

x

VVa (4.26)


81

Alternatively, two steady state conditions can also be simulated since the transient

interval is ignored anyway. This is shown in Fig. 4.7 for output current Io perturbation.

Here, the quasi-resonant converter at the ‘top’ of the Pspice circuit in Fig. 4.7 is

simulated under steady state conditions whereas the quasi-resonant converter at the

‘bottom’ of the same figure is simulated under perturbed Io. This perturbed current is

indicated by component ‘Ioz’ in the Pspice configuration, which has a value of 0.6363A

or ∆Io = 0.0063A. The output variable responses Id and Vx are shown in Fig. 4.8 and 4.9

Vd10

Io0.63

R1z

1k

Vdz10

0

Vpwmz

PW = 0.8usPER = 2us

V1 = 0V2 = 10

Lr

1u

Dbz

Da

Daz

Cr

25n

Lrz

1u +

Qz

1m

+

Q

1m

Ioz0.6363

Crz

25n

Db

R1

1k

Vpwm

PW = 0.8usPER = 2us

V1 = 0V2 = 10

Fig. 4.7: Two Steady State Pspice Models to Emulate

Output Current Step Change from 0.63A to 0.6363A


82

Fig. 4.8: Behavior of Id Current under Steady State and Perturbed Conditions

Fig. 4.9: Behavior of Vx Voltage under Steady State and Perturbed Conditions

Perturbed State

Steady State

Steady State

Perturbed State


83

Therefore the a12 and a22 coefficients can be calculated as follows (Data is obtained from

Fig. 4.8 and 4.9).

3167.00063.0

426515.0428510.012 =

−=

∆∆

=o

Lr

II

a (4.27)

476.50063.0

9767.59422.522 −=

−=

∆∆

=o

Cr

IV

a (4.28)

The derivations for coefficients a13 and a23 are not shown (but their actual values

are included in the FDM, eq. (4.29)) as they are derived in similar manners.

Consolidating all the coefficients, the FDM for the converter is then given below.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×−

×=⎥

⎦

⎤⎢⎣

⎡∆∆

−

−

s

o

d

x

d

fIV

VI

6

6

1054.13476.5036.11084.03167.002285.0

(4.29)

The use of simulation-based approach in this section has demonstrated the

straightforwardness of finding the FDM coefficients as opposed to the analytical

approach. Moreover, a main advantage of this technique is that a design engineer does

not have to fully understand the operation of the QRC converter in order to use the

separation of variables method. As can be seen, the model parameters for the fast

dynamic model (FDM) were done entirely under circuit simulation, thus avoiding

mathematically difficulties. With this FDM obtained, a small signal equivalent circuit

model is to be constructed next and this is shown in the following section.


84

4.5 Small Signal Equivalent Circuit Model

In this section, a small signal equivalent circuit model is constructed from the fast

dynamic model (FDM). This is done, as before, by coupling appropriate external

components (which were replaced earlier by constant sources) to the input and output

variables of the FDM, as shown in Fig. 4.10. A Pspice version of the circuit can also be

developed based on this configuration and with coefficients in Eq. (4.29). And then the

frequency responses of the converter can be obtained from this Pspice circuit simulation.

Note that the resonant tank components are absence from the model.

FDM

+∆Vd

-

+∆Vo-

∆Ιo∆Ιd CoRo

+∆fs

-

+∆Vx

-

Lo

Fig. 4.10: Small Signal Circuit Model of ZCS QRC Buck HW


85

4.6 Simulation Results

This section demonstrates how small signal frequency responses of the quasi-

resonant soft-switched converter are obtained through Pspice simulation. Using the

circuit construction in Fig. 4.10, the circuit is simulated under Pspice version and the

frequency responses of the converter are shown for both control-to-output and input-to-

output frequency responses in Fig. 4.11 and 4.12.

101 102 103 104 105 106-80

-60

-40

-20

0GAIN/dB

101 102 103 104 105 106-200

-150

-100

-50

0PHASE/deg

Fig. 4.11: Control-to-Output Frequency Response Curves

For the ZCS-QRC-Buck Half Wave

Hz

Hz


86

101 102 103 104 105 10660

80

100

120

140GAIN/dB

101 102 103 104 105 106-200

-150

-100

-50

0PHASE/deg

Fig. 4.12: Input-to-Output Frequency Response Curves

For ZCS-QRC-Buck Half-Wave

The frequency responses obtained here (Fig. 4.11 and 4.12) give identical results

to [7], [21], which are obtained from the extension of state-space-averaging method. As a

reference, their small signal circuit model is reproduced here in Fig. 4.13 in Pspice

format. A disadvantage with their method [7], [21] is that the model parameters were

derived analytically. This is not desirable, as it deems too mathematical for practical use.

On the contrary, the proposed separation of variables method (SVM) exploits the use of

circuit simulation to obtain model parameters and the frequency responses. This allows a

design engineer to perform the whole modeling process through circuit simulation

approach. In this way, mathematics is avoided, as favored by practitioners.

Hz

Hz


87

R4

1000000M

fs1

Vg0

+-

+-

KiI

G-0.3646

Co

10u

Ro

10.07

Lo

100uH+-

UoI

G0.6768

-+

+-

KvV

E0.3646

-+

+-

KcV

E13.536u

+-

KvI

G0.023

-+

+-

KiV

E-5.788

R2

1000000M

+-

KcI

G0.85275u

R3

1000000M

0

-+

+-

UoV

E0.6768

R1

1000000M

Fig. 4.13: Small Signal AC Model of ZCS-QRC-Buck Half-Wave Based on [21]


88

4.7 Summary

In this chapter, the separation of variables method (SVM) is applied to zero-

current-switching quasi-resonant-converter buck in half wave operation. Two approaches

to obtain model parameters for the fast dynamic model (FDM) were shown. First, the

FDM coefficients were obtained analytically by expanding the appropriate large signal

equations using Taylor series at the nominal operating condition. Then it is shown that

the same FDM coefficients could be obtained through circuit simulation approach. Here,

the steady state operating values are also simulated as opposed to mathematical

derivation. Then frequency responses of the converter are obtained and verified by

comparing against results by [7], [21]. Although the results obtained in SVM and [7],

[21] are identical, the former offers an easier approach as the whole procedure to find

desired frequency responses (and including model parameters or FDM) of the converter

can be done entirely through circuit simulation with little knowledge of the converter’s

operation.

Chapter 5 Extension of SVM Technique to Resonant Converters

89

Chapter 5

Extension of SVM Technique to Resonant Converters

5.1 Introduction

So far, the separation of variables method (SVM) has been successfully applied to

hard-switched converters. This category of converters has distinctive fast variables that

respond almost instantaneous to perturbations and slow variables that respond slowly to

perturbations, over a switching cycle. As a consequence, the converter’s fast dynamic

model (FDM) parameters have constant coefficients. Therefore this method is suitable for

hard-switched converters as discussed in Chapter 3. Here, it may seem that the presence

of resonant tank in quasi-resonant-converters (QRC) would create dynamics in FDM

coefficients. Thus, the method could not be used for quasi-resonant type converters.

However, this was shown otherwise in Chapter 4. This is because quasi-resonant

converters operate similarly to hard-switched converters under discontinuous conduction

mode (DCM) and therefore their resonant tank dynamics do not carry forward to the next

cycle. Hence the transients due to resonant tank settle after one switching cycle. Thus,

implementing SVM for the quasi-resonant type converters pose no difficulties as the

assumption of separation of variables is not violated. However, some power converters

such as series resonant converters with inner feedback control, have very dynamic FDM

coefficients. Ignoring such dynamics results in unacceptable error in the model


90

prediction. In view of this, extending the current SVM technique to include converters

with appreciable FDM dynamics is desirable.

This chapter introduces and applies SVM technique to series resonant converter

(SRC) to obtain small signal transfer functions. Typically, the SRC small signal analysis

based on discrete time model is mathematically quite tedious [11], [23] - [26]. Because of

this, these techniques are left largely to researchers to explore and develop. Practicing

engineers with time constraint to expedite product development to meet short deadlines

would like to use simpler and general techniques like injected-absorbed-current (IAC)

method [14]. However, as pointed out in Chapter 1, the IAC method does not include the

effect of resonant tank dynamic in finding the SRC small signal frequency responses

[14], which may have unwanted consequences. Unlike IAC, the proposed SVM in this

report can be slightly modified to model the effect of SRC resonant tank.

The chapter is divided as follows: Section 5.2 gives a brief overview of SRC

operation with inner-feedback control. Section 5.3 determines the steady state operating

points needed for small signal analysis. Section 5.4, 5.5 and 5.6 outlines the strategy and

techniques of using as well as modifying SVM to model SRC small signal behavior with

diode conduction angle (DCA) control, which is one of the control methods applicable to

SRC. The transfer functions developed are verified through simulation in section 5.7.

Finally, Section 5.8 concludes this chapter.


91

5.2 Basic Operation of Series Resonant Converter (SRC)

The circuit diagram of an SRC (series resonant converter) is depicted in Fig. 5.1.

The power circuit comprises of two equal voltage sources Vd1 and Vd2 coupled with a

half-bridge switch Q1 and Q2 arrangement, a resonant inductor Lr, a resonant capacitor Cr,

an output bridge rectifier, and an output filter capacitor Co with a load resistor Ro. The

control signal ψ determines the instances when switch Q1, and Q2 turn ‘on’ and turn ‘off’

in order to produce an alternating square waveform, vE. The voltage vE together with the

resonant tank elements Lr and Cr resonates with the Ro and Co to establish the output

voltage, vo. The basic operation of the SRC is divided into four modes and can be

explained as below (See Fig. 5.1, 5.2 and 5.3). It is assumed that the SRC is operating

below the resonant region, i.e. the switching frequency is lower than the resonant

frequency of the LC resonant tank. Furthermore, the operation is assumed to be in

continuous conduction mode (CCM), which implies that the switching frequency is above

half the LC resonant frequency. Thus, 0.5fr < fs < fr, where fs = switching frequency and

fr = resonant frequency =rrCLπ2

1 .

Mode 1, Q1 Conduction: The resonant tank current iLr is positive. Both diodes D1

and D2 are not conducting. The conduction period for this switch is denoted as

time tβ or in terms of conduction angle β = tβ/ωr.

Mode 2, D1 Conduction: The current iLr resonates in negative direction and

forces D1 to conduct. D2 is still in blocking state. The period for D1 conduction is

tα. Similarly, α = diode conduction angle = tα/ωr. At the instant tα, Q1 is turned off

while Q2 is turned on.


92

Mode 3, Q2 conduction: This mode has symmetry with respect to Mode 1 (See

Fig. 5.2). The resonant tank current iLr is negative direction. D1 does not conduct

in this interval.

Mode 4, D2 conduction: This mode has symmetry with respect to Mode 2 (See

Fig. 5.3). Current iLr is positive direction. Therefore this forces D2 to conduct. The

conduction period, tα, for D2 is determined by the control method adopted. The

end of this mode completes a switching period.

irect

Co Ro

+

vo

-

io

iC+ vCr -iLr

Lr Cr

ψ

+-

Vd1

+-

Vd2

Q1

Q2

D1

D2

+vE-

OutputRectifier

+vx-

Vd1=Vd2=Vd

4321

od

od

od

od

VVVVVV

VV

−−−−

Modevv xE

Fig. 5.1: Series Resonant Converter in Half-Bridge Switch Arrangement


93

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10-5

-15

-10

-5

0

5

10

15

Fig. 5.2: Resonant Inductor Current of SRC

Referring to Figs. 5.2 and 5.3, the waveforms of the resonant state variables

during steady state condition for modes 1 and 2 are the same as modes 3 and 4 with their

signs reversed. In view of this, the state variables at switching boundary t0, t2, and t4 have

the same magnitudes. Therefore iLr0 = -iLr2 = iLr4 and vCr0 = - vCr2 = vCr4, which suggests

that the SRC state variables possess odd-wave symmetry under steady state operation.

This symmetry characteristic proves to be useful later. It allows analysis to be done for

one-half of the switching period instead of one switching period. Besides the

simplification in analysis, a major advantage of the carrying out analysis over one-half of

Mode 4 Mode 1 Mode 2 Mode 3

to= tα t2 t4t1 t3

iLr0

iLr2

iLr4

Sec

Ampere


94

the switching period is that it doubles the frequency range over which the small signal

model is valid. This is because in averaging techniques, the transfer function prediction

will be valid for up to one-half of the switching frequency. By considering only one-half

of the switching period, the analysis exploits fully the effective doubling of the switching

frequency that typically involved in such bridge converters.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10-5

-400

-300

-200

-100

0

100

200

300

400

Fig. 5.3: Resonant Capacitor Voltage of SRC

A few control methods are possible to realize inner-feedback control in SRC.

Here, inner feedback control refers to the control method adopted to directly determine

the switching instants. Optimal trajectory control, frequency control, diode conduction


to= tα t2 t4t1 t3

vCr0 vCr 4

vCr 2

vCr1

Sec

Volt


95

angle (DCA) control, are just but a few of the inner-feedback control strategies [27] -

[29]. In this thesis, the DCA control is chosen for applying the separation of variables

method (SVM) because under such a control technique, the SRC’s FDM does not settle

fast under perturbed conditions, which may violate the SVM assumption that FDM’s

response is ‘infinitely’ fast. Hence the control technique is chosen for further

investigation. To understand the dynamical effect of DCA, state-plane analysis [27], [28]

may be used to provide a clear and insightful explanation. A brief introduction to state-

plane analysis is given in the next section. Then a DC analysis is performed in Section

5.3. The steady state information is then used to determine the SRC’s AC model, which is

carried out in Section 5.5 and 5.6 employing several versions of SVM methods.

5.2.1 Brief Review of State-Plane Analysis

The state-plane analysis by [27], [28] is a geometrical technique to analyze the

behavior of a particular converter. The resonant state variables are used as the x-y axes

for the two-dimensional state-plane plot. The instantaneous value of the inductor current

iLr is shown at the y-axis whereas the instantaneous value of the capacitor voltage vCr is

shown at the x-axis. The trajectory of the circuit state, as determined by these two state

variables, is a locus, which moves in clockwise trajectory with respect to time at a

resonant angular speed, ωr. A typical steady state SRC trajectory in CCM is illustrated

in Fig. 5.4. Here the subscript ‘N’ is the variables indicate normalized quantities.


96

t0

t1

t2

t3

t4iLr0N

iLr2N

vCr0N

vCr2N

α β

D2

Q1

Q2

D1

-1-Von

-1+Von

1-Von

1+Von

Rd

R

RRd

iLrN

vCrN

vC1N=vCPN

Fig. 5.4: State-Plane Trajectory of SRC under CCM in Below Resonant[28]

The base quantities for normalizing are:-

• Base Voltage = Vd

• Base Current = Vd / Zr

whereby Zr = sqrt(Lr/Cr)


97

The operation of the illustrated trajectory in Fig. 5.4 is as follows. The trajectory

begins at time t0 when switch Q1 begins to conduct. The initial condition for this locus is

the point (iLr0, vCr0). This trajectory also corresponds to mode 1 shown as one of the four

modes in Fig. 5.2 and 5.3. The locus traverses in a circle with radius R and centered at

point (1-VoN, 0). This center point is obtained from the equivalent source of vE, which is

vE1 = Vd – Vo in mode 1. After normalizing this equation with the base voltage Vd, we

obtain vE1N = 1 – VoN. The rest of the center-points for other radiuses are derived

likewise. Mode 1 will end when the locus hits x-axis that also indicates Q1 ceases to

conduct at time t1. Moreover, the locus has also completed β angle (Q1 conduction angle)

during the course of the switch trajectory. Following mode 1, the operation changes to

mode 2 or D1 conduction. The locus now takes on a new radius, Rd, with center at point

(1+VoN, 0). This mode 2 ends following a diode trajectory angle of α. The end of this

mode 2 occurs when switch Q2 is switched on. The instant at which Q2 switches on (or

the end of mode 2) can be determined using the control method adopted. Diode

Conduction Angle (DCA), for instance has a fixed diode angle α depending on the

control voltage. Therefore, the switch Q1 or Q2 turns on after the diode trajectory has

traveled by the controlled angle α. A good review of this and other control methods can

be found in [27]. The loci for Q2 and D2 are similar to Q1 and D1, therefore they are not

explained here. After completing the trajectories for Q2 and D2, the state variables have

completed a switching cycle.


98

5.3 DC Analysis

The parameters of the SRC used for design and verification are given in Table 5.1. The

SRC operates under CCM below resonant frequency under these conditions.

The following values are determined based on steady state analysis.

• iLr0

• vCr0

• vCr1 (vCPN)

• Diode conduction angle, α.

• Switch conduction angle, β.

Table 5.1: Design Parameters of the Series Resonant Converter

with Diode Conduction Angle Control (SRC-DCA)

According to [27], switch radius R is a given parameter for a SRC in CCM

condition. However from design point of view, R is not a useful parameter because it

does not directly give any information about Lr, Cr, and fs value. Nonetheless, from the

trajectory of Fig. 5.4, it can be easily shown that,

Vo = 60V Vi = 200V Ro = 10Ω Io = 3A Cr = 0.053µF Lr = 47.75µH fs = 82kHz ωr = 1/sqrt(LrCr)


99

( ) ( ) ( )RRRf r

s βαω

+=

5.0 (5.1)

Both α(R) and β(R) can be derived from triangle laws, which can be seen from the

geometry in Fig. 5.4. The detailed derivation for α(R) and β(R) and the others are shown

in [27]; therefore they will not be shown here. Thus,

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−−=

oN

oNoN

VRRVV

R2

1acos

2

πα (5.2)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ++−−=

RRVV

R oNoN 1acos

2

πβ (5.3)

From Eqs. (5.1) to (5.3), an explicit equation for R cannot be obtained. Hence, the

switch radius, R, is determined numerically. Fig. 5.5 shows the non-linear relationship

between the switching frequency and switch radius R for the converter under continuous

conduction mode (CCM). It is found that R is 3.036 for fs = 82kHz. Substituting R =

3.036 into Eqs. (5.2) and (5.3) yields,

( ) sorR Rad µα 095.2317.1036.3 == (5.4)

( ) sorR Rad µβ 003.4516.2036.3 == (5.5)


100

Having obtained R, the rest of the variables can be easily found based on simple

geometry. The normalized inductor current, iLr0N, at time t0 is,

( ) 778.1sin == βRiLrON (5.6)

Likewise from the geometry, the normalized capacitor voltage, vCr0N, at time t0 is,

0616.2)1( −=−+−= oNoNCrON VRVv (5.7)

Thus, the actual values are

VVvv dNCrCr 16.20600 −=×= (5.8)

AZVii

r

dNLrLr 9233.500 =×= (5.9)

Finally vCr1 is to be determined, which is the resonant capacitor voltage at time t1 (Fig.

5.3).

oNCPN VRV −+= 1 (5.10)

Hence,

VVCr 6.3431 = (5.11)


101

Fig. 5.5: SRC-DCA: Switching Frequency versus Switch Radius

5.4 Limitation on Small Signal Analysis by SVM Approach

An SRC small signal analysis under DCA control employing SVM technique is

carried out in this section. The methodology to find this small signal transfer function is

the same as the approach taken in the previous chapters. The first and foremost step is to

assume that the SRC circuit with inner feedback control has two distinctive circuit

groups: infinitely fast and infinitely slow sub-circuits. As per this, the resonant state

variables are assumed to be infinitely fast. In Fig. 5.6, the slow variables of the SRC are

replaced with constant sources to generate the FDEC (fast dynamic equivalent circuit).

The output circuit Ro and Co are lumped together as a single equivalent voltage source,

Vo, as before. Also shown in the Fig. 5.6 is the control signal, α, corresponding to


102

previously used ψ. The new symbol indicates a specific mode of control, namely diode

conduction angle or DCA. The fast part of the circuit including nonlinear components is

considered as FDG (fast dynamic group) as shown in Fig. 5.7. Fig. 5.7 shows that the

input current Id and rectifier current Irect are the average fast variables while the average

slow variables are Vd, Vo, and α. Hence, the perturbation of both slow and fast variables

are related by the FDM (fast dynamic model) matrix as given below.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡∆∆

αo

d

rect

d VV

pppppp

II

232221

131211 (5.12)

+ vCr -iLr

Lr Cr

id irect

vo

α

+-

+-

Vd1

+-

Vd2

Q1

Q2

+vE-

Fig. 5.6: FDEC for SRC-DCA


103

FDG+-+-

+-+-

Vd Vo

IdIrect

α

Fig. 5.7: FDG for SRC-DCA

To further simplify the analysis, the average variable ∆Id is dropped from the

FDM (fast dynamic model) Eq. (5.12) because it does not contribute to the development

of ∆Vo/∆α and ∆Vo/∆Vd transfer function. Under different circumstances, if ∆Vo/∆Id is

desired, then ∆Id term should be retained. Hence, this leaves only ∆Irect fast variable to be

determined. The average Irect can be found by averaging the absolute value of the

resonant inductor current over one-half of the switching period or Ts/2. Thus,

( ) ( )

( ) ⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

−−

+

⎥⎦

⎤⎢⎣

⎡−

−+−

=

∫

∫

dtttZ

Vv

dtttZ

Vvtti

TI

t

tr

r

ECr

t

tr

r

ECrrLr

srect

2

1

1

0

121

010

00

sin

sincos2

ω

ωω

(5.13)

Evaluating,

( ) ( )[ ]

( )[ ]⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−−

+

−−

+=

α

ββ

ω

ωωω

tZ

Vv

tZ

Vvti

TI

rr

ECr

rr

ECrr

r

Lr

srect

cos1

cos1sin2

21

100

(5.14)


104

Substituting the steady state values from Eqs. (5.4), (5.5), (5.8), (5.9) and (5.11) into Eq.

(5.14),

AIrect 976.5= (5.15)

The idea now is to find how much Irect moves away from its steady state value Eq.

(5.15), when any of the slow variables Vd, Vo, and α are perturbed in order to obtain p21,

p22, and p23 constant coefficients. Although this step can be easily done through circuit

simulation, a computational approach is taken here. A general-purpose state-plane

program for SRC under DCA control was written in Matlab. This program is shown in

Appendix A. Listed in Table 5.2 are the values of perturbed variables iLr, iCr, etc, for

perturbations in Vd, Vo, and α. These values are in turn used to compute the value of

perturbed Irect using Eq. (5.14).

Table 5.2: Calculation of Perturbed Slow Variables

Thus substituting the values from Table 5.2 into Eq. (5.14), we obtain the perturbed value

of Irect ( = IrectX). Here, the subscript ‘X’ indicates the variable is under perturbed

condition.

iLrX/A vCrX/V vCPX/V αX βX ∆Vd = +1V 6.1537 -208.95 351.74 1.3167R 2.5058R

∆Vo = 0.6V 5.7573 -205.46 339.05 1.3167R 2.5277R

∆α = 5.733 -202.27 337.11 1.329867R 2.5246R 0.013167R


105

A

I

d

VVrectXdd

109.6∆VVVoltage,InputPerturbedtodue

CurrentRectifierPerturbed

d

=+

=∆+

(5.16)

A

I

o

VVrectXoo

911.5∆VVVoltage,OutputPerturbedtodue


o

=+

=∆+

(5.17)

A

IrectX

865.5∆ααAngle,ControlDCAPerturbedtodue


=+

=∆+ αα

(5.18)

Perturbation of Irect then can be found by the following relationships.

rectVVrectXVrect IIIddd

−=∆∆+∆

(5.19)

rectVVrectXVrect IIIooo

−=∆∆+∆

(5.20)

rectVrectXrect IIId

−=∆∆+∆ αα

(5.21)

where Irect, in this case, is the steady state value of the SRC converter as given

in Eq. (5.15).

Substituting Eqs. (5.15), (5.16), (5.17), and (5.18) into Eqs. (5.19), (5.20), and (5.21),

AIdVrect 133.0=∆

∆ (5.22)

AIoVrect 065.0−=∆

∆ (5.23)

AIrect 111.0−=∆∆α

(5.24)


106

Therefore,

133.021 =∆∆

=d

rect

VIp (5.25)

1083.00

22 −=∆∆

=VIp rect (5.26)

4302.823 −=∆

∆=

αrectIp (5.27)

Thus, Eq. (5.12) can be written for SRC under DCA control as

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

−−=∆α

o

d

rect VV

I 4302.81083.0133.0 (5.28)

Based on the FDM Eq. (5.28), a small signal circuit model is constructed and is shown in

Fig. 5.8.

Co Ro

+∆ Vo

-

p21∆Vd p23∆αp22∆Vo

∆Irect

Fig. 5.8: Small Signal Circuit Model of SRC-DCA Using SVM


107

102 103 104 105-60

-50

-40

-30

-20

-10

0GAIN/dB

102

103

104

105

-300

-250

-200

-150

-100

-50

0PHASE/deg

PSpice ResultsSVM Prediction

Fig. 5.9: Input-to-Output Frequency Responses of SRC-DCA via SVM

Fig. 5.9 and 5.10 give the frequency response plots of the input-to-output and control-

to-output transfer functions. Here, the SVM predictions are compared with Pspice results.

The frequency response appears to be accurate up to 4kHz in both cases. A poor model

by low frequency approximation standard, which typically can predict fairly accurately

up to at least one-quarter to one-third of the switching frequency. In the next section, the

reason for this problem is explained and the SVM technique is modified to overcome the

problem.

Hz

Hz


108

102 103 104 105-30

-20

-10

0

10

20

30

40GAIN/dB

102 103 104 105-400

-350

-300

-250

-200

-150

-100

-50

0PHASE/deg

PSpice ResultsSVM Prediction

Fig. 5.10: Control-to-Output Frequency Responses of SRC-DCA via SVM

Hz

Hz


109

5.5 SVM-1 Approach: Dynamic FDM Coefficients

The reason behind the error noted at high perturbation frequency may be explained as

follows. The resonant tank dynamics due to inner-feedback control has been neglected

under SVM approximation. In so doing, the poles-zeros that govern such dynamic

behavior are lost. If these poles-zeros exist at frequencies lower than the switching

frequency then such approximation will produce an erroneous low frequency model. This

aspect was further investigated by plotting the inductor current (sampled at start of each

cycle) variation for perturbation in Vd, Vo, and α. These results are shown in Fig. 5.11(a),

(b) and (c).

The inductor current (and thus the output rectifier current) does not respond

instantaneously to all perturbed slow variables as assumed by Eq. (5.28). In fact, it takes

the inductor current approximately 10 switching cycles to settle down after any

perturbations. Due to this reason, the predicted transfer functions have relative large

error. The resonant tank’s pole-zeros can be extracted from the step response plots

through system identification methodologies for further investigation. System

identification can be very difficult due to complex techniques needed to extract hidden

information regarding the system especially if there is a zero close to a pole. However, in

this work, a simple method based on first order dynamic [30] (See Fig. 5.12) is adopted

to model the effect of the resonant tank dynamics for modifying the SVM technique. The

idea is to model these fast variable dynamics with the simplest model first before

exploring other more complex identification models.


110

0 5 10 15 20 25 30 351.77

1.78

1.79

1.8

1.81

1.82

1.83

1.84

1.85

1.86

(a)

0 5 10 15 20 25 30 351.72

1.73

1.74

1.75

1.76

1.77

1.78

1.79

(b)

0 5 10 15 20 25 30 351.71

1.72

1.73

1.74

1.75

1.76

1.77

1.78

1.79

(c)

Fig. 5.11: Linearized Sampled Inductor Current Dynamics Against Number of

Switching Cycle, due to: (a) Perturbed Vd, (b) Perturbed Vo, and (c) Perturbed α

t/sec

iL(t)/A

iL(t)/A

iL(t)/A

t/sec

t/sec


111

Fig. 5.12: First Order System’s Step Response Curve

An arbitrary first order system’s step response is shown in Fig. 5.12. The system

pole can be easily obtained by inspecting the time when the response reaches 63.2% of its

steady state value A. This time Γ can be converted into the pole’s frequency by the

formula [30],

( )( ) 1

1+Γ

=ssU

sX (5.29)

Where U(s) is the injected step function, s = Laplace Function.

Applying this concept to the step response curves in Fig. 5.11 to find each corresponding

respond time Γ.

• Response due to ∆Vd, sµ5.301 =Γ

• Response due to ∆Vo, sµ3.182 =Γ

• Response due to ∆α, sµ5.303 =Γ

0 Γ

A 0.632A

x(t)

t


112

Thus substituting these Γs into Eq. (5.29) and considering only perturbation terms (i.e.

the DC terms are removed),

( )( ) 32787

32787+

+=∆∆

ssVsI

d

rect (5.30)

( )( ) 54645

54645+

−=∆∆

ssVsI

o

rect (5.31)

( )( ) 32787

32787+

−=∆

∆ss

sI rect

α (5.32)

So these poles in Eqs. (5.30) to (5.32) are taken as missing from the gain-phase

plots in Fig. 5.9 and 5.10. Thus, in this method, Eq. (5.28) is corrected intuitively by

multiplying with the perturbation terms in Eqs. (5.30) to (5.32). Note that in multiplying,

the ‘-ve’ sign in Eqs. (5.30) to (5.32) are dropped as Eq. (5.28) has already taken care of

it.

Hence the corrected FDM matrix is,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

+−⎟

⎠⎞

⎜⎝⎛

+−⎟

⎠⎞

⎜⎝⎛

+=∆

αo

d

rect VV

sssI

32787327874302.8

54645546451083.0

3278732787133.0

(5.33)


113

Here, it may be seen that the FDM matrix coefficients are no longer constants but possess

dynamic terms.

And the small signal equivalent circuit for the SRC-DCA based on Eq. (5.33) is,

Co Ro

+∆ Vo

-

p21 32787 ∆Vd

Irect

s+32787p22 54645 ∆Vo

s+54645p23 32787 ∆α

s+32787

Fig. 5.13: Small Signal Circuit Model of the SRC-DCA

with Corrected FDM Coefficients

The new plots for input to output and control to output perturbation responses are

shown in Fig. 5.14 and 5.15 along with the uncorrected model. Here, the SVM

(separation of method) that incorporate FDM (fast dynamic model) dynamics is given the

name Separation of Variables Method Type 1 or SVM-1. The SVM-1 prediction with

corrected terms offers much better accuracy than the uncorrected SVM approach. The

prediction goes well almost up to the switching frequency before the prediction starts to

deviate from the simulated results. Judging from the steps required to plot these gain-

phase curves, the technique is remarkably simple and yet effective in predicting gain-

phase behavior almost up to the switching frequency. As compared to the existing

techniques such as [11], [23], [24], [26], the SVM-1 method with corrected terms offer a

much simpler alternative to model small signal behavior of an SRC.


114

102 103 104 105-60

-40

-20

0GAIN/dB

102 103 104 105-300

-200

-100

0PHASE/deg

PSpice Results SVM Prediction SVM-1 Prediction

Fig. 5.14: Input-to-Output Frequency Responses of SRC-DCA via SVM and SVM-1

Scrutinizing the SVM-1 gain-phase plots reveals that in a certain frequency region

(for example around 3kHz in Fig. 5.14), there still is a discrepancy between predicted and

simulated results. The reason is likely due to keeping Vo constant while perturbing Vd

and α in the suggested modified technique (SVM-1). Theoretically, this is not correct

because the change in output rectifier current due to Vd and α perturbations causes

changes in Vo in turn. If this Vo dynamical effect were considered in the step response,

then the SVM technique will become very complex due to the interaction between the

FDM and the output circuit. This in turn destroys the main purpose of our seeking a

simple method for small signal analysis.

Hz

Hz


115

102 103 104 105-40

-20

0

20

40GAIN/dB

102 103 104 105-400

-300

-200

-100

0PHASE/deg

PSpice Results SVM Prediction SVM-1 Prediction

Fig. 5.15: Control-to-Output Frequency Responses of SRC-DCA

via SVM and SVM-1

However, it is still possible to model exactly the resonant tank dynamics together

with the output voltage interaction exactly by incorporating sampled-data approach into

the SVM. This proposed method is called Separation of Variables Method Type 2 or

SVM-2 and is presented in the next section.

Hz

Hz


116

5.6 SVM-2 Approach: Resonant Tank Fast Dynamic Model (RFDM)

In this section, another modified SVM (separation of variables method) is

proposed, which is given the name SVM-2 (separation of variables method type 2). Here,

SVM-2 method is developed to produce a better model prediction than the SVM-1 model.

In SVM-1 analysis, ∆Vd, ∆Vo, and ∆α are treated as slow variables while ∆Irect is treated

the fast variable. An analysis of the SVM-1 strategy suggested that the resonant tank

dynamics were not fully represented in the SVM-1 method. In view of this, perturbed

resonant state variables ∆iLr and ∆vCr, which are sampled at start of every cycle, maybe

treated as slow variables and therefore are included in the FDM matrix (of SVM-2). The

inclusion of these state variables ∆iLr and ∆vCr turn the FDM into a discrete time system.

This is not desirable because FDM should be a continuous model so that circuit

simulation approach can be taken. To remedy this problem, a low frequency

approximation is made as before so that switching ripple and resonant tank frequency can

be ignored. Therefore, this allows the variables in the FDM, which are sampled data, to

be joined from one cycle to the next cycle and so forth to form smooth, continuous

variables. This results in a new continuous FDM equation for the SRC with DCA control,

[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

=∆

αo

d

Cr

Lr

rect

VVvi

aaaaaI 3534333231 (5.34)


117

where a31 to a35 are constants that relates the perturbation of (continuous) input

variables ∆iLr, ∆vCr, ∆Vd, ∆Vo, and ∆α to the perturbation (continuous) of output

variable ∆Irect.

These resonant state variables may be viewed as slow variables because, under

the influence of inner-feedback control, they can contribute poles-zeros in the low

frequency region. However, Eq. (5.34) is not complete because physically it is not very

meaningful. Perturbed resonant tank state variables are internal characteristic of the SRC

under DCA control that cannot be externally injected. This characteristic has to be

appropriately modeled by analyzing the resonant tank dynamics in sampled-data manner

and then coupled to Eq. (5.34). This is the same as obtaining FDM for the resonant tank

and a special name is given to it as the Resonant Tank Fast Dynamic Model (RFDM). In

RFDM, the responses of the state variables at one-half cycle later, (See Fig. 5.16)

∆iLr(tk+1) and ∆vCr(tk+1), have to be analyzed by perturbing slow variables, ∆Vd(tk),

∆Vo(tk), and ∆α(tk) at time t0. In addition, it is also recognized that state variables

∆iLr(tk+1), and ∆vCr(tk+1) also depend on initial conditions ∆iLr(tk) and ∆vCr(tk). Based on

these, the sampled-data resonant tank model can be written as Eq. (5.35) (RFDM) based

on Fig. 5.16. The figure is essentially the same as Figs. 5.2 and 5.3 with the sampling

instants clearly shown.


118

( )( )

( )( )( )( )

( ) ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡∆∆

+

+

k

ko

kd

kCr

kLr

kCr

kLr

ttVtVtvti

aaaaaaaaaa

tvti

α

2524232221

1514131211

1

1 (5.35)

where a11 to a25 are constants.

Note: tk+1 – tk = one-half switching period = Ts/2,

tk = t0 ( = tα = start of Mode 1 in Figs. 5.2 and 5.3)

tk+1 = t2 ( = start of Mode 3 in Figs. 5.2 and 5.3)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10-5

-15

-10

-5

0

5

10

15

(a) (continue next page)


tk tk+1 tk+2

iLr(k)

iLr(k+2)

tk+1/2 tk+3/4

iLr(k+1)

sec


119

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10-5

-400

-300

-200

-100

0

100

200

300

400

(b)

Fig. 5.16: Resonant Tank Behavior with

(a) Discrete Time Inductor Current (b) Discrete Time Capacitor Voltage

With the resonant tank model ready, Eq. (5.35) can be coupled to Eq. (5.34).

Physically, how the two equations complement each other can be seen in Fig. 5.17.

Nevertheless it is worth noting that Eq. (5.34) is a continuous-time model while Eq.

(5.35) is a discrete-time model. For the moment, such mismatch of signals is assumed to

pose no problem in order to not to cloud the SVM-2 concept. Converting sampled-data to

continuous signal can be easily done by approximation but this discussion is put on hold

until the following section.

Mode 1 Mode 2 Mode 3

tk tk+1 tk+1tk+1/2 tk+3/4

vCr(tk) vCr(tk+2)

vCr(tk+1)

sec


120

∆ iLr

∆ vCr

RFDM

FDM ∆ irect Co Ro

+∆ Vo

-

∆ Vd

∆ Vo

∆ α

Fig. 5.17: Interconnection of FDM and RFDM

with Injected Variables and External Circuit

5.6.1 RFDM and FDM Coefficients

This subsection illustrates techniques to find resonant tank fast dynamic model

(RFDM) and fast dynamic model (FDM) coefficients. To find RFDM coefficients, the

same Matlab program (Appendix A) is used to simulate the resonant tank behavior for

one-half of the switching period. This computer algorithm is based on the resonant tank

equations shown in Eqs. (5.36) and (5.37). Fig. 5.16 can be used as a reference to

understand these equations.


121

( )kkrr

EkCrkLr tt

ZVv

i −−

−= ++

+ 5.02)5.0(

)1( sinω (5.36)

( ) ( )kkrEkCrkCr ttVvv −−= ++ 12)()1( cosω (5.37)

where,

( ) ( )kkrr

EkCrkkrkLrkLr tt

ZVv

ttii −−

−−= +++ 5.01)(

5.0)()5.0( sincos ωω (5.38)

( ) ( ) ( )kkrEkCrkkrrkLrkCr ttVvttZiv −−+−= +++ 5.01)(5.0)()5.0( cossin ωω (5.39)

The above equations are obtained from [27]. The first step to obtain RFDM

coefficients is to inspect and measure both inductor current and capacitor voltage at time

tk+1 after perturbation of a slow variable at time tk. The perturbation magnitude in this

application example is typically set at 1% of the steady state value. An output voltage Vo

that is 60V at steady state, for instance, should have ∆Vo = 0.6V perturbation. When there

is a correlation between a slow variable and resonant state variables, the latter will

deviate from steady state values. The resonant inductor current in Fig. 5.18, for instance,

deviates from its steady state behavior when each of the slow variables iLr(tk), vCr(tk), Vd,

Vo, and α is perturbed. The amount of deviation iLr(tk+1) is measured after Ts/2.


122

2 3 4 5 6 7 8 9 10

x 10-6

-8

-6

-4

-2

0

2

4

6

8

10

12

2 3 4 5 6 7 8 9 10

x 10-6

-8

-6

-4

-2

0

2

4

6

8

10

12

(a) (b)

2 3 4 5 6 7 8 9 10

x 10-6

-8

-6

-4

-2

0

2

4

6

8

10

12

2 3 4 5 6 7 8 9 10

x 10-6

-6

-4

-2

0

2

4

6

8

10

12

(c) (d)

2 3 4 5 6 7 8 9 10

x 10-6

-8

-6

-4

-2

0

2

4

6

8

10

12

(e)

Fig. 5.18: Response of the resonant inductor current after one-half period under

(a) Perturbed resonant inductor current, ∆iLr(k) (b) Perturbed resonant capacitor

voltage, ∆vCr(k) (c) perturbed input voltage, ∆Vd(k) (d) perturbed output voltage,

∆Vo(k) and (e) perturbed control angle, ∆α(k)

Solid Line – Steady State Dotted Line – Perturbed Response

∆iLr(k+1) ∆iLr(k+1)

∆iLr(k+1)

∆iLr(k+1) ∆iLr(k+1)

sec

sec

sec sec

sec


123

The perturbed average output rectifier current ∆Irect can be determined from its steady

state averaged output rectifier current over one-half of the switching cycle, which is given

below.

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+= ∫∫+

+

+ 2/1

4/12

4/1

12 k

kLr

k

kLr

srect dttidtti

TI (5.40)

where,

( ) ( ) ( )tZ

Vvtiti r

r

EkCrrkLrLr ωω sincos 1)(

)(1

−−= 5.0; +≤≤ kk ttt (5.41)

( ) ( )tZ

Vvti r

r

EkCrLr ωsin2)5.0(

2

−−= + 15.0; ++ ≤≤ kk ttt (5.42)

Note that the method so far introduced is somewhat similar to those in [23] which

use discrete time methods to determine the small signal transfer functions. The proposed

SVM-2 method so far differs from the these methods in that the present method is not

fully discrete-time based with resonant tank response being modeled as discrete system

and the other slow variables (Vd, Vo, and α) being modeled as continuous system.

However, as mentioned in the introduction Chapter 1, such combination of both

continuous and discrete system based modeling is also not new as the concept was

already proposed by [11]. Nevertheless, the steps taken to model the discrete events of

the resonant state variables by [11] are different from that SVM-2 in which the latter does

not involved averaging of perturbed resonant state variables, which can be considered

less tedious. But this difference may not be that significant. Sections 5.6.1 to 5.6.4


124

modify the SVM-2 analysis further so that the matrix coefficients can be easily

determined using simulation-based approach, if necessary. This is viewed as an important

contribution in this chapter.

Again, to find these perturbations, the same Matlab program is used (Appendix A).

The results are tabulated in Table 5.3. The values ∆iLr(tk+1) and ∆vCr(tk+1) are computed

using Eqs. (5.36) and (5.37) and are compared with the steady state results. As mentioned

earlier, if there is a relationship, perturbed input variable will cause output variables (of

the RFDM) iLr(tk+1) and vCr(tk+1) to deviate from their steady state points. These resonant

state variables under a perturbed condition after one-half of the switching period (tk+1) are

given as iLrX(tk+1) and vCrX(tk+1) in Table 5.3. Note that the subscript ‘X’ here indicates

perturbed condition. Therefore the differences between perturbed states and steady states

taken at tk+1 are ∆iLr(tk+1) = iLrX(tk+1) - iLr(tk+1) and ∆vCr(tk+1) = vCrX(tk+1) - vCr(tk+1).

Likewise, the output variable of the FDM, ∆Irect, is calculated using Eq. (5.40) in a similar

manner, which is ∆Irect = Irectx - Irect. Here, the output state variables, iLr(tk+1), vCr(tk+1), and

Irect at steady state are found (from numerical computation) as follows.

• iLr(tk+1) = -5.9208A

• vCr(tk+1) = 206.098V

• Irect = 5.9730A


125

Table 5.3: Perturbation of Variables by Numerical Computation

Table 5.3 presents Matlab numerical results for perturbed output quantities

iLrX(tk+1), vCrX(tk+1) and IrectX(tk+1) under separate input perturbations at time tk, ∆iLr(tk),

∆vCr(tk), ∆Vd, ∆Vo, and ∆α. The discrete time symbol is dropped for the last three terms

because they are approximated as constants. In this table, they are five columns. The first

column from left shows the input state variable values at start of switching cycle or tk.

The second column from left presents perturbation of the same variables. The third

column shows numerical results for perturbed output variable, ∆iLrX(tk+1), after one-half

of the switching period under five separate input perturbations, ∆iLr(tk), ∆vCr(tk), ∆Vd,

∆Vo, and ∆α. Likewise for column four and five for vCrX(tk+1) and IrectX(tk+1). Each

deviation is compared to its steady state value and then the appropriate difference is

computed.

State at tk Perturbation at tk iLrX(tk+1)/A vCrX(tk+1)/V IrectX/A

iLr(k) = 5.9202A ∆iLr(k) = 0.059202A -5.9545 206.3500 5.9963

vCr(k) = -206.16V ∆ vCr(k) = -2.0616V -5.9748 206.5183 6.0102 Vd = 100V ∆ Vd = +1V -5.9470 207.3017 5.9911 Vo = 60V ∆ Vo = 0.6V -5.8664 206.2746 5.9544 α = 1.317R ∆α = 0.1317R -5.9045 203.7535 5.9741


126

Thus,

5692.0059202.0

9208.59545.5

)(

)1()1(11 −=

+−=

∆

−= ++

kLr

kLrkLrX

iii

a (5.43a)

02619.00616.2

9208.59748.5

)(

)1()1(12 =

−+−

=∆

−= ++

kCr

kLrkLrX

vii

a (5.43b)

0262.01

9208.59470.5

)(

)1()1(13 −=

++−

=∆

−= ++

kd

kLrkLrX

Vii

a (5.43c)

09067.06.0

9208.58664.5

)(

)1()1(14 −=

+−=

∆

−= ++

ko

kLrkLrX

Vii

a (5.43d)

4958.11317.0

9208.59405.5)1()1(15 −=

+−=

∆

−= ++

αkLrkLrX ii

a (5.43e)

2566.4059202.0

098.20635.206

)(

)1()1(21 =

−=

∆

−= ++

kLr

kCrkCrX

ivv

a (5.43f)

2039.006098.2

098.2065183.206

)(

)1()1(22 −=

−−

=∆

−= ++

kCr

kCrkCrX

vvv

a (5.43g)

2037.11

098.2063017.207)1()1(23 =

+−

=∆

−= ++

d

kCrkCrX

Vvv

a (5.43h)

2943.06.0

098.2062746.206)1()1(24 =

−=

∆

−= ++

o

kCrkCrX

Vvv

a (5.43i)

1781317.0

098.2067535.203)1()1(25 −=

−=

∆

−= ++

αkCrkCrX vv

a (5.43j)

3936.0059202.0

9730.59963.5

)(31 =

−=

∆−

=kLr

rectrectX

iIIa (5.43k)

018050.006098.2

9730.50102.6

)(32 −=

−−

=∆

−=

kCr

rectrectX

vIIa (5.43l)


127

0181.01

9730.59911.533 =

+−

=∆

−=

d

rectrectX

VIIa (5.43m)

031.06.0

9730.59544.534 −=

−=

∆−

=o

rectrectX

VIIa (5.43n)

00835.01317.0

9730.59741.535 =

−=

∆−

=α

rectrectX IIa (5.43o)

Here a11 to a25 are the RFDM coefficients and a31 to a35 are the FDM coefficients.

5.6.2 Practical Method to Compute RFDM Coefficients.

Eq. (5.35) represents the resonant tank fast dynamic Model (RFDM). When a

slow variable (in this RFDM) is perturbed at time tk, the output state variables respond at

time tk+1. The time lapsed is one-half of the switching period. However, the RFDM and

FDM coefficients, particularly coefficient a11, a12, a21, and a22, are very difficult to find by

circuit simulation. This is because of the difficulty of introducing perturbation iLr(tk) and

vCr(tk), etc. In order to overcome this problem, Eq. (5.35) is manipulated such that ∆iLr

and ∆vCr are eliminated from the ‘right-hand-side’ (R.H.S.). Since this equation

represents a discrete-time model of first order, it is possible to remove them by back

substitution that will result in a second order model. Rewriting Eq. (5.35) as two

equations,


128

( ) ( ) ( ) ( ) ( ) ( )kkokdkCrkLrkLr tatVatVatvatiati α∆+∆+∆+∆+∆=∆ + 15141312111

(5.44)

( ) ( ) ( ) ( ) ( ) ( )kkokdkCrkLrkCr tatVatVatvatiatv α∆+∆+∆+∆+∆=∆ + 25242322211

(5.45)

From Eq. (5.44),

( ) ( ) ( ) ( ) ( ) ( )12

151413111

atatVatVatiatitv kkokdkLrkLr

kCrα∆−∆−∆−∆−∆

=∆ + (5.46)

Also, from Eq. (5.46), considering the next discrete time interval,

( ) ( ) ( ) ( ) ( ) ( )12

11511411311121 a

tatVatVatiatitv kkokdkLrkLrkCr

++++++

∆−∆−∆−∆−∆=∆

α

(5.47)

Substituting Eq. (5.46) and (5.47) into Eq. (5.45),

( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )kkokd

kkokdkLrkLr

kLr

kkokdkLrkLr

tatVatVaa

tatVatVatiatia

tiaa

tatVatVatiati

α

α

α

∆+∆+∆+

⎥⎦

⎤⎢⎣

⎡ ∆−∆−∆−∆−∆+

∆=

∆−∆−∆−∆−∆

+

+++++

252423

12

15141311122

21

12

1151141131112

(5.48)


129

In FDEC (fast dynamic equivalent circuit), slow variables are to be treated as constants;

therefore we assume that ∆Vo(tk+1) = ∆Vo(tk) = ∆Vo. After simplification,

( ) ( ) ( ) ( ) ( )[ ]

[ ][ ] α∆+−+

∆+−+∆+−=

∆−−∆+−∆ ++

1522152512

1422142412

1322132312

22112112122112

aaaaaVaaaaa

Vaaaaatiaaaatiaati

o

d

kLrkLrkLr

(5.49)

In a similar manner, the following equation of ∆vCr variable can be obtained.

( ) ( ) ( ) ( ) ( )[ ]

[ ][ ] α∆+−+

∆+−+∆+−=

∆−−∆+−∆ ++

2525112115

2424112114

2323112113

22112112122112

aaaaaVaaaaa

Vaaaaatvaaaatvaatv

o

d

kCrkCrkCr

(5.50)

In Eqs (5.49) and (5.50), the only coefficients to be determined are a11, a12, a21, and a22.

The rest of the coefficients can be obtained based on the method in Section 5.6.1.

In order to determine a11, a12, etc using a circuit simulation approach, Vd, Vo, and

α in the FDEC (Fig. 5.6) are perturbed separately as before. The perturbation of the iLr

and vCr at tk+1 and tk+2 instants are noted from the simulation. These are then used to

obtain ∆iLr(tk+2), ∆iLr(tk+1), ∆vCr(tk+2), and ∆vCr(tk+1). Note that ∆iLr(tk) and ∆vCr(tk) will be

zero. Then using Eq. (5.49) for each of the ∆Vd, ∆Vo, and ∆α separately, linear equations

connecting a11, a12, and a22 may be obtained. These are then solved for a11, a12, and a22.

Likewise, Eq. (5.50) is used to obtain the remaining coefficient a21.


130

5.6.3 Odd Symmetry Problem of RFDM

For the purpose of analysis, both the FDM and the RFDM matrices can be

combined as shown by Eq. (5.51). The matrix A1 is derived by analyzing the SRC

behavior for the 1st half of the switching period, after perturbation. However, for the 2nd

half of the switching period, certain coefficients will have reversed signs due to the

alternating nature of iLr and vCr waveform. Eq. (5.52) describes this behavior.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

3534333231

2524232221

1514131211

1

aaaaaaaaaaaaaaa

A (5.51)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−−−−−

=

3534333231

2524232221

1514131211

2

aaaaaaaaaaaaaaa

A (5.52)

The peculiar observation has been discussed in [24], [25]. They explained the

behavior to be due to the odd symmetry phenomenon that exists in the SRC. They have

also rigorously solved the problem through mathematical manipulation. The idea is to

convert the odd symmetry wave to even symmetry wave [24], [25] so that the FDM (an

even symmetry function) and RFDM are coherent. First, Fig. 5.19(a) illustrates that an

arbitrary odd symmetry waveform behavior when f1 f2. Then in Fig. 5.19(b), when

f1 -f2, an even symmetry waveform is obtained. Although |f1| equals |f2|, two different

symbols are used for clarity purpose. Here, this observation indicates that by multiplying


131

f2 in Fig. 5.19(a) with a negative sign, we converted an odd symmetry wave to an even

symmetry wave. Therefore, the RFDM coefficients can be converted likewise to an even

symmetry matrices by multiplying them with ‘negative one’ as given below.

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−=

3534333231

2524232221

1514131211

111aaaaaaaaaaaaaaa

A (5.53)

(a)

(b)

Fig. 5.19: Arbitrary Periodic Waveform (a) Odd Symmetry (b) Even Symmetry

-f2(t)

0 1 2

t

f1(t) f2(t)

0 1 2

t

f1(t)


132

There is, however, a possible way to resolve this problem without any

multiplication of signs. Perturbed responses can be measured at time tk+2 that is after one

switching cycle instead of one-half of switching cycle. This technique, nonetheless,

yields less accurate results as shown in the gain-phase plots in Fig. 5.25 and 5.26. Only

results for this analysis (one switching cycle instead of half-switching cycle) are shown.

This is because the analysis reduces the sampling frequency to fs (instead of fs/2), thus

limiting the accuracy of the small signal frequency response at higher perturbation

frequencies. The derivation is not shown because it uses the same approach as before.

Also, due to its limitation in predicting the frequency response accurately, this is not

deemed as a useful technique.

5.6.4 Small Signal Analysis by SVM-2 Approach

One final step before constructing small signal equivalent circuit model is to

convert the RFDM described by Eq. (5.35) from a discrete time model to a continuous

time model. This ensures that the RFDM can be coupled to the continuous FDM

described by Eq. (5.34).


133

Applying a z-transformation on Eq. (5.35),

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡∆∆

αo

d

Cr

Lr

Cr

Lr

VVvi

aaaaaaaaaa

vziz

2524232221

1514131211 (5.54)

Simplifying,

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡∆∆

αo

d

Cr

Lr

Cr

Lr

VVvi

aaaaaaaaaa

zvi

2524232221

15141312111 (5.55)

Alternatively,

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=⎥⎦

⎤⎢⎣

⎡∆∆

αo

d

Cr

Lr

Cr

Lr

VVvi

aaaaaaaa

az

azvi

25242321

15141312

22

11

00

10

01

(5.56)

where, 2sTS

ez = , jωs = , 1j −= , and Ts = switching Period.


134

By substituting 2sTS

ez = , the discrete time relationship can be converted into

continuous time one. Since 2sTS

ez = can be expanded in terms of power series and

because only low frequency region is of interest in this analysis, the ‘z’ term can be

approximated as a first order expansion,

ssTz +≈1 (5.57)

Thus, from Eq. (5.55),

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∆∆∆∆∆

⎥⎦

⎤⎢⎣

⎡+

=⎥⎦

⎤⎢⎣

⎡∆∆

αo

d

Cr

Lr

sCr

Lr

VVvi

aaaaaaaaaa

sTvi

2524232221

1514131211

11 (5.58)

This is the low frequency continuous system equation for ∆iLr and ∆vCr. Now,

Eqs. (5.58) and (5.34) can be combined to construct the small signal SRC circuit model

(Fig. 5.20) similar to that conceptually portrayed earlier in Fig. 5.17. Notice that the

variables ∆iLr, ∆vCr, ∆Vd, ∆Vo, and ∆α in the RFDM are all continuous variables, though

the model (RFDM) has been derived based on discrete system approach. Fig. 5.20 is

realized in a Pspice circuit format as shown in Fig. 5.21, which is then used to simulate

and obtain the transfer functions. To obtain the small signal responses from the RFDM

and FDM, the proposed SVM–2 requires only a single step, which is simulation of the


135

circuit in Fig. 5.20 and 5.21. In [11], on the other hand, many complex steps are used

such as to absorb resonant tank state variables into their model. This results in an

equivalent circuit without resonant state variables terms. Unlike [11], SVM-2 proposed

does not try to absorb these resonant state variables but instead let them remain in the

model, resulting in a faster and shorter approach to obtain transfer functions.

- +

a11∆ iLr a12∆ vCr a13∆Vd a14∆V0 a15∆ α



- + - + - + - + 2fs/(2fs+s)

2fs/(2fs+s)∆ iLr

∆ vCr

+∆ Vo

-

RoCo

∆ Ιrect

FFDM

RFDM

+∆ Vd

-

+∆ α

-

Fig. 5.20: Small Signal Equivalent Circuit Model

for SRC-DCA using FDM and RFDM


136

-++

-

a25

214

R8 1G

Co 20uF

-++

-

Ex2

R6 1G

0

+-

a32

-0.016

Vd 1

+-

a14

-0.13445

+-

a15

-2.2827

a31 0.3483

R7 1G

a11 0.427

R1 1G

+-

Ex1

-1

-++

-

a22

0.1924

-++

-

E0

E1 ELAPLAC E 82000/(82000+s)

OUT+ OUT- IN+

IN-

+-

a34

-0.0462

E2 ELAPLAC E 82000/(82000+s)

OUT+ OUT- IN+

IN-

-++

-

a23

-1.56

Vx 0Vdc

-++

-

a24

0.165

+-

a21 -2.21

R3 1G

Alpha 0

R4 1G

R9 1G

R4 1G

+-

ax

1

R5 1G R2

1G

Ro 10 +

-

a35

-1.32

R5 1G

+-

a33

0.034

+-

a12

-0.02045

+-

a13

0.073

Fig. 5.21: PSPICE Realization of the Model in Fig. 5.20


137

5.7 Verification and Comparison of SVM Techniques for SRC

Through Pspice simulation of the actual circuit scheme, the frequency response of

the series resonant converter (SRC) with diode conduction angle (DCA) control was

obtained. Here, the data will be used to verify the validity of the three proposed model

predictions (SVM, SVM-1, and SVM-2 methods). An actual SRC with DCA control is

developed and discussed in the next Subsection 5.7.1. A bandpass LC notch filter is also

developed in order to extract the perturbation frequency response information. This data

is then plotted against the results obtained through the abovementioned methods in

Section 5.7.2 for verification.

5.7.1 Pspice Circuit Simulation for SRC-DCA

A Pspice format of a series resonant converter (SRC) with diode conduction angle

(DCA) is developed in this section, as shown in Fig. 5.22, for verifying model predictions

developed by SVM methods. The design parameters used here is listed in Table 5.1 as

seen in the segment ‘POWER CIRCUIT’ in Fig. 5.22. The ‘INPUT VOLTAGE’ in this

schematic functions as the alternating vE source, which is discussed earlier in Section 5.2.

The output voltage of this SRC power circuit is across the load ‘Ro’ of which its steady

state value is determined by the reference voltage ‘V_ref’ source in the ‘DCA

CONTROL CIRCUIT’ segment. This control circuit takes the resonant inductor ‘Lr’

current information through ‘Current Sense’ or H1 and then computes the diode


138

conduction angle (DCA) in the same segment, so that the next appropriate switch can be

turned on at the appropriate moment. By injecting perturbation frequencies separately

into ‘V_inject1’ and ‘V_inject2’, the circuit gives input-to-output and control-to-output

frequency responses. To measure these frequency responses, the output voltage at ‘Ro’ is

tapped by the ‘BANDPASS FILTER’ (a notch type). The small signal perturbation

response extraction at a specific perturbation frequency, fp, is then done by selecting a

combination of inductor ‘L_f’ value and capacitor ‘C_f’ that satisfies the following

equation, fCfL

f p __21

×=

π.

U1

10us1

2

+-

1

1n(ReferenceVoltage)

Lr

47.75uH

1G

C_f

15V

+

1

ESUM

IN1+IN1-

IN2+IN2-

OUT+

OUT-

1k

V_inject2

POWER CIRCUIT

10V

-+

+-E10

-+

+-

1G

DCA CONTROLCIRCUIT

15V

1k

ESUM

IN1+IN1-

IN2+IN2-

OUT+

OUT-

FrequencyResponseMeasurement Node

Ro

10

1

BANDPASSFILTER

ESUM

IN1+IN1-

IN2+IN2-

OUT+

OUT-

EMULT

IN1+IN1-

IN2+IN2-

OUT+

OUT-

6.1uF1n

(Control Signal Perturbation)

Co

20uF

-+

+-

-10

+ - H1

1000

Vref 2.213

1k+

5k

15

-15

+

1

1k

INPUT VOLTAGE

-+

+-

-0.1

-+

+-

Cr

0.053uF

V_inject150V

(Input VoltagePerturbation)

10V

L_f1k

1k

10-10

2

-+

+-

0.1

-1

1k+

6.1uF

0

1k

+-

-1

(Current Sense)

1k

U2

10us1

2

2k 10

-10

Fig. 5.22: PSpice Realization of SRC with DCA Control


139

5.7.2 Simulation Results

In this section, various frequency response plots of the SRC-DCA obtained

through SVM, SVM-1, and SVM2 methods are compared against prediction from Pspice

simulation. The input-to-output frequency response curves are shown in Fig. 5.23 while

the control-to-output frequency response curves are shown in Fig. 5.24. Here, it can be

seen that the most accurate plots are obtained through SVM-2 method and the worst

predictions are from the SVM method. This is expected because the SVM-2 employs

sampled-data approach to model the SRC behavior, including the inner feedback control

strategy (or the effect of resonant tank dynamics). The effect of resonant tank dynamics

may be viewed as a result due to the interaction between the resonant tank and the inner

feedback control, which is inherent to most SRC converters. Therefore, as pointed out in

Section 5.4, neglecting this dynamical effect will result in large model prediction error as

portrayed by these plots. The SVM-1 model prediction, which has shown to be quite

promising, is able to predict quite accurately close to the switching frequency. A

drawback is, however, that the model prediction shows inconsistent results at low

frequency region. This deviation, nevertheless, does not get worse that rapidly as

perturbation frequencies get higher. This is noticeable in Fig. 5.23 and 5.24. The input-

to-output gain, for instance, (See Fig. 5.23) begins to deviate from around 4kHz with an

approximated error of 2dB, then the error maintain relatively constant as the frequencies

increased. The error comes close to 3dB at 10kHz, which is still acceptable for control

design. But, finally, the error becomes larger to about 6dB at 40kHz. The reason for this

model prediction error has been discussed earlier in Section 5.5. The error, however,


140

seems to be smaller for the control-to-output frequency response for both gain and phase

curves. Although SVM-2 is more accurate than SVM-1, the latter is deemed more

important as the approach is much simpler for practical use. A summary is given in Table

5.4 for these observations.

102 103 104 105-60

-40

-20

0GAIN/dB

102 103 104 105-300

-200

-100

0PHASE/deg

PSpice ResultsSVM-2 SVM-1 SVM

Fig. 5.23: Input-to-Output Frequency Responses of the SRC-DCA

using SVM, SVM-1, and SVM-2 Methods

Hz

Hz


141

102 103 104 105-40

-20

0

20

40GAIN/dB

102 103 104 105-400

-300

-200

-100

0PHASE/deg

PSpice ResultsSVM-2 SVM-1 SVM

Fig. 5.24: Control-to-Output Frequency Response of the SRC-DCA

using SVM, SVM-1, and SVM-2 Methods

The SVM-2 technique based on sampling data at every half-cycle and full-cycle

are compared in Fig. 5.25 and 5.26. As can be seen and was pointed out in Section 5.6.3,

the SVM-2 based on half-cycle modeling is not as accurate as the full-cycle modeling.

Hz

Hz


142

Table 5.4: Comparison of Different SVM Techniques for SRC - DCA

102 103 104 105-60

-40

-20

0GAIN/dB

102 103 104 105-300

-200

-100

0PHASE/deg

PSpice ResultsHalf-Cycle Full-Cycle

Fig. 5.25: Input-to-Output Frequency Response of the SRC-DCA


SVM SVM-1 SVM-2 Accuracy of Model Prediction Worst Acceptable Best Bandwidth of Prediction Very Low Close to ½ Sampling Frequency Difficulty of Technique Easiest Easy Very Difficult


143

102 103 104 105-40

-20

0

20

40GAIN/dB

102 103 104 105-400

-300

-200

-100

0PHASE/deg

PSpice ResultsHalf-Cycle Full-Cycle

Fig. 5.26: Control-to-Output Frequency Response of the SRC-DCA


5.8 Summary

The proposed SVM has been demonstrated, in Chapter 3 and 4, to be a very

effective tool for converters that exhibit extreme characteristics of fast and slow

variables. In this chapter, however, the SVM is applied to a case where the fast variables

Hz

Hz


144

are not “infinitely fast”. Many resonant converters fall under this category. Here, a series

resonant converter with diode conduction angle type converter has been used as an

example to explain the application of SVM to such converters and control methods.

When the SVM was directly applied to this converter and control, a relatively large error

in small signal performance prediction was obtained. This was attributed to the fact that

poles from the effect of resonant tank in SRC were not considered in SVM approach. As

a remedy to the problem, these poles were estimated from step response curves. The

constant terms of the FDM were multiplied by these poles to correct the transfer

functions. This method, named as SVM-1, has been shown to give reasonably accurate

results up to large perturbation frequencies. Furthermore the approach is very simple and

intuitive, which can be very appealing to practicing engineers. More accurate transfer

functions can also be obtained by incorporating sampled-data approach into SVM. This

method has been named SVM-2. Even though the procedure shown for SVM-22 is more

tedious and complicated than SVM-1 technique, it is still less mathematical and easier

than the discrete – time analysis contemporaries such as [11]. One of the benefits seen in

SVM-2 as compared to another similar approach by [11] is that the SVM-2 involves less

steps and less calculations. The SVM-2 approach has been shown to result in very

accurate modeling of the SRC-DCA even up to half of the effective sampling frequency.

Chapter 6 Conclusions and Future Work

145

Chapter 6

Conclusions and Future Work

6.1 Summary of Work

This report has presented a novel approach, which has been named the Separation

of Variables Method (SVM), for determining the small signal performance of a DC-DC

converter. The method is intuitive and relatively easy to apply. It is applicable to hard-

switched type converters in continuous-conduction-mode (CCM) and discontinuous-

conduction-mode (DCM), quasi-resonant type converters, and resonant type converters

with inner feedback control.

6.2 Basic Separation of Variables (SVM) Technique

(a) The Method

The proposed separation of variables method divides a converter into a slow and a

fast sub-circuit. The method assumes the fast sub-circuit variables react rapidly to

perturbations while the slow sub-circuit variables maintain almost as constants. The

resultant of this assumption is the Fast Dynamic Equivalent Circuit (FDEC) model


146

where all the slow components are replaced by constant sources. Through this, all the

constant sources are perturbed separately to obtain the Fast Dynamic Model (FDM) that

characterizes the behavior of the fast sub-circuit. After obtaining the fast dynamic model

of the converter, a small signal equivalent circuit model can be constructed by coupling

the inputs and outputs of the fast dynamic model with the once removed slow

components. Here, the small signal equivalent circuit model are simulated for obtaining

frequency responses of the converter or analyzed mathematically for obtaining transfer

functions via basic linear circuit theory.

(b) Merits

The separation of variables method has the following merits:

(i) Conceptually easy: The method involves separating a converter into two

distinctive sub-circuits based on their response speed.

(ii) Easy to apply and use: The method is suitable for practicing engineers, as

it is mainly procedure-based approach.

(iii) Simplified analysis: The method allows order reduction of a converter up

to several magnitudes.

(iv) Fast Way to Determine Model Parameters: The method allows a

converter’s model parameters to be obtained through circuit simulation.

(v) Small signal equivalent circuit model: The end result of the analysis is an

equivalent circuit model that permits the use of circuit simulation

approach to obtain frequency responses of the converter.


147

(c) Work Done

The followings are the work done:

(i) Application examples to continuous-conduction-mode operation of hard-

switched converters (buck, boost, and buck-boost converters). The model

parameters of these converters were obtained through analytical and

circuit simulation approaches, which in turn used for small signal model

construction. The model predictions were verified against state-space-

averaging [1] models.

(ii) Application examples to discontinuous-conduction-mode operation of

hard-switched converters (buck, boost, and buck-boost converters). It has

been shown, under such operation condition, the inductor current is a fast

variable and therefore it cannot be replaced by a constant current source.

Model parameters obtained here are shown to be the same as [15]. The

transfer functions derivation is not covered as they can be obtained in the

same way as the continuous conduction mode cases.

(iii) Application example to quasi-resonant-type converters. A zero-current-

switching quasi-resonant buck converter in half wave mode is used for this

analysis. Both analytical and circuit simulation approach to obtain model

parameters were presented. The model predictions of this converter are

identical to those that obtained through extension of state-space- averaging

[7], [21] method.


148

6.3 Extension of SVM Technique to Resonant Converters

(a) The Problem

When separation of variables method was applied to a series-resonant-converter

with diode-conduction-angle control, it was found that there was a large deviation in the

model prediction. This is because the effect of resonant tank dynamics is ignored in the

method. Since the interaction between the resonant tank variables and the inner feedback

control strategy is inherent to the resonant converter, ignoring this behavior will result in

erroneous model prediction.

(b) Solutions Proposed

The effect of resonant tank dynamics of the series resonant converter gave rise to

a fast dynamic model whose coefficients contain dynamic terms. In view of this, two

solutions are proposed to account for this effect: the separation of variables method type

1 (SVM-1) and separation of variables method type 2 (SVM-2)

The dynamical behavior of the fast dynamic model is identified through step

response in the SVM-1 technique. The poles and zeros information are then extracted.

Small signal dynamic terms are constructed from these poles and zeros and then

multiplied, as correction factors, to the FDM constant coefficients. This produces a model

prediction that is accurate close to one-half of the sampling frequency. However, there is


149

still a relatively small error in the low frequency region. This is because during the step

response identifications, the output voltage was held constant, which is not correct.

For a more accurate modeling of the effect of resonant tank dynamics, the SVM-2

is proposed. This method uses sampled data approach to characterize the resonant tank

variables behavior. This results in a resonant tank fast dynamic model (RFDM), whose

coefficients are constant that are sampled every half cycle. The SVM-2 method also

requires the resonant tank variables to appear as input variables in both RFDM and the

usual FDM (fast dynamic model). This result in a discrete time FDM. However, by

approximating a smooth curve to join these discrete variables, the discrete FDM turns

into a continuous model. The discrete RFDM is also converted to a continuous model and

then coupled to the FDM in order to construct a small signal model by connecting the

FDM inputs and outputs to external components.

(c) Work Done

The SVM, SVM-1, and SVM-2 methods were applied to a series-resonant-

converter (SRC) with diode-conduction-angle (DCA) control. These results were

compared against the frequency responses of the Pspice circuit simulation of the actual

series-resonant-converter under diode-conduction-angle control. This circuit simulation

was performed by injecting sinusoidal perturbation frequencies separately into the input

voltage and control angle during its steady state operation. Then the converter’s

frequency responses were extracted through a notch band-pass filter.


150

6.4 Overall Summary of SVM Methods

A general approach to implement SVM, SVM-1, and SVM-2 methods has been

presented in this thesis. The scopes and limitations of each method were also discussed in

their chapters respectively. Each SVM, SVM-1, and SVM-2 methods is summarized

below.

(a) The Separation of Variables Method (SVM)

This is the easiest technique to use among the proposed three methods. However,

the SVM method is only applicable for converters with fast variables that respond almost

instantaneously to perturbations and with slow variables that behave as constants under

these perturbations. This results in a fast dynamic model (FDM) with constant

coefficients, which is used for small signal equivalent circuit construction. For converters

with dynamic FDM coefficients, this method cannot be applied.

(b) The Separation of Variables Method Type 1 (SVM-1)

The SVM-1 method is a slight modified version of the SVM method. Besides

having the same methodology as the SVM method, it also models the dynamical behavior

of the fast dynamic model (FDM). In this way, poles and zeros introduced by the fast

sub-circuit are included in the analysis. This method is suitable for converters with FDM,


151

whose coefficients are not constants but contains dynamics. The model prediction is close

to one-half of the sampling frequency. A limitation of this method, however, is that there

exists a slight error in the model prediction that begins to deviate from low frequency

perturbations.

(c) The Separation of Variables Method Type 2 (SVM-2)

The SVM-2 is the most difficult technique of all. It employs sampled data

technique to model the poles and zeroes of the dynamical FDM. The model prediction of

this method gives accurate frequency response of a converter up to almost half of the

sampling frequency. One of the merits of this technique is that the frequency responses of

the circuit model can be obtained through circuit simulation. This eases a design engineer

modeling work.

(d) Comparison of SVM, SVM-1, and SVM-2 Methods to Others

A major advantage of all these methods is that they produce small signal

equivalent circuit model as the end results. This allows a circuit simulation approach to

be taken to find the frequency responses of converters. This simulation-based technique

is deemed useful and important because it has more practical usage for design engineers

with little time to digest and implement complex techniques. A summary of comparison

of SVM methods with other contemporaries is presented in Table 6.1.


152

Table 6.1: Comparison of Different Small Signal Analysis Methods

SD SSA ESSA PSM EQV CKT IAC SVM SVM-1 SVM-2

CCM Yes Yes No Yes No Yes Yes Yes Yes 1 Hard-Switched

Converters DCM Yes Yes No Yes No Yes Yes Yes Yes

2 Quasi-Resonant Converters Yes No Yes No No Yes No Yes Yes

3 Resonant Converters Yes No No No Yes Yes No Yes Yes

4 Order Reduction for Slowly Varying Variables No No No No No One Several Several Several

5 Difficulty in Obtaining Model Parameters

Most Difficult Difficult Difficult Difficult Very

Difficult Simple Most Simple Simple Difficult

6 Difficulty in Implementing the Method

Most Difficult Difficult Difficult Easy Easy Very

SimpleVery

Simple Very

Simple Very

Simple

7 Direct Results in Equivalent Circuit Model No No No Yes No Yes Yes Yes Yes

Abbreviations: SD = General Sampled Data Approaches [18], [25], [26], etc. SSA = State Space Averaging Method [1] ESSA = Extension of State Space Averaging Method [7] PSM = Pulse Width Modulated Switch Modeling Method [10] EQV CKT = Equivalent Circuit Modeling of Resonant Converters [11] IAC = Injected Absorbed Current Method [14], [15] SVM = Separation of Variables Method SVM-1 = Separation of Variables Method Type 1 SVM-2 = Separation of Variables Method Type 2

Even though sampled data approach such as [18], [25], [26] produces very

accurate model, the methodology involved is very complicated and is mostly left for

researchers to explore.


153

The state-space-averaging (SSA) method [1], as pointed out earlier, can be

difficult to implement because it is quite mathematically intensive. This approach is

limited to hard-switched converters only.

The extension of state-space-averaging (ESSA) method [7] resolves one of the

limitations of state-space-averaging. Here, the ESSA method extends its application to

quasi-resonant type converters. Still, the method is quite mathematically cumbersome.

The pulse-width-modulated switch modeling (PSM) method [10] employs the

idea of replacing the three terminals point encompassing a switch and a diode by a linear

circuit network. After this is done, circuit simulation or circuit analysis approaches are

either used to obtain frequency responses. This method is limited to hard-switched type

and quasi-resonant type converters. Again, this method is also mathematically intensive.

The equivalent circuit modeling of resonant converters [11] is developed for

determining a small signal circuit model of the converter. Although circuit simulation can

be used here for obtaining results, the methodology to determine the equivalent circuit

model parameters is very complex.

Although the injected-absorbed-current (IAC) [14], [15] method has proven to be

conceptually easy and has shown to be a general method for small signal modeling, it has

several limitations. These are resolved through SVM methods.


154

The SVM and SVM-1 methods have shown to be the easiest among the

contemporaries. The method was applied to hard-switched type, quasi-resonant type, and

resonant type (with inner feedback control) converters.

The SVM-2 method has more complicated procedures than SVM, SVM-1 and

IAC methods. However, it has been shown that model parameters for SVM-2 can be

determined through circuit simulation, thus avoiding complicated mathematics.

Furthermore, the implementation of SVM-2 has also shown to be much easier than [11].

This is deemed as an important contribution for practical small signal discrete-time

modeling.

6.5 Future Work

The future works needed are:

(1) Mathematical justification of the three methods (SVM, SVM-1, and SVM2). Here

the coverage for the scopes and limits of these methods would be understood

better. Possible unification of the three methods is also suggested.

(2) More application examples such parallel resonant converters with inner feedback

control. In addition, new experimental results to verify these techniques are

suggested.

References

155

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Converter Power Stages”, in IEEE Power Electronics Specialists’ Conference , 1976.

[2] V. A. Caliskan, G. C Verghese and A. M. Stankovic, “Multifrequency Averaging of

DC/DC Converters”, in IEEE Transactions on Power Electronics, Vol. 14, No. 1, Jan

1999.

[3] R. Tymerski, “Application of Time-Varying Transfer Function for Exact Small

Signal Analysis”, in IEEE Transactions on Power Electronics, Vol. 9, No. 2, Jan 1994.

[4] S. R. Sanders and et al., “Generalized Averaging Method for Power Conversion”, in

IEEE Transactions on Power Electronics, Vol. 6, No. 2, Apr. 1991

[5]. B. Johansson and M. Lenells, “Possibilities of Obtaining Small Signal Models of DC-

to-DC Power Converters by Means of System Identification”, XXX

[6]. J. Y. Choi and et al., “System Identifications of Power Converters Based on a Black

Box Approach”, in IEEE Transactions of Circuits and Systems – I: Fundamental Theory

and Applications, Vol 45., No. 11, Oct 1998.

[7] A. F. Witulski and R. W. Erickson, “Extension of State Space Averaging and Beyond”,

in IEEE Transactions on Power Electronics, Vol. 5, No. 1, 1990.

[8] L. C. Lim, “Control Analysis of a Soft-Switched DC-DC Converter with PWM

Control”, Undergraduate’s Final Year Project, National University of Singapore, 93/94.

[9] A. S. Kislvoski, “On the Role of Physical Insight in Small Signal Analysis of

Switching Power Converters”, Applied Power Electronics Conference and Exposition,

1993, APEC '93, Conference Proceedings 1993, Eighth Annual.

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[10] V. Vorperian, R. Tymerski, and F. C. Lee, “Equivalent Circuit Models for Resonant

and PWM Switches”, in IEEE Transactions on Power Electronics, Vol. 4, No. 2, 1989.

[11] A. F. Witulski, A. F. Hernandez, and R. W. Erickson, “Small Signal Equivalent Circuit

Modeling of Resonant Converter”, in IEEE Transactions on Power Electronics, Vol. 6,

No. 1, Jan. 1991.

[12] M. G. Kim and M. J. Youn, “A Discrete Time Domain Modeling and Analysis of

Controlled Series Resonant Converter”, in IEEE Transactions on Industrial

Electronics, Vol. 30, No. 1, Feb. 1991.

[13] V. Vorperian, “Approximate Small Signal Analysis of the Series and Parallel

Resonant Converters”, in IEEE Transactions on Power Electronics, Vol. 4, No. 1, Jan.

1989.

[14] A. Kislovski and R. Redl, “Generalization of the Injected-Absorbed-Current

Dynamic Analysis Method of DC-DC Switching Power Cells”, ISCAS 1992 Record.

[15] A. Kislovski and R. Redl, “Dynamic Analysis of Switching Mode DC-DC

Converters”, Van Nostrand Reinhold, New York, 1991.

[16] M. Hissem and et. al, “Application of Current-Injected Modelling Approach to Quasi

Resonant Converters”, in IEEE Transactions on Power Electronics, Vol. 4, No. 4, Apr.

Jul. 1997

[17] A. Kislovski, “On the Use of Duality in the Dynamic Analysis of Switching Power

Converters”, Proc. INTELEC 1992, 14th International.

[18] J. G. Kassakian, M. F. Schlecht, and G. C. Verghese., “Principles of Power

Electronics”, Addison-Wesley, 1992.

[19] M. H. Rashid, “SPICE for Power Electronics and Electric Power”, International

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[20] J. Keown, “OrCad Pspice and Circuit Analysis”, 4th Ed, Prentice Hal, 2001.

[21] R. W. Erickson, “Fundamentals of Power Electronics,” Chapman & Hall, 1997.

[22] K. H. Liu and F. C. Lee, “Zero Voltage Switching Technique in DC/DC Converters”,

in IEEE Transactions on Power Electronics, Vol. 5, No. 3, Jan. 1990.

[23] I. Batarseh and K. Siri, “Generalized Approach to the Small Signal Modeling of DC-

to-DC Resonant Converters”, IEEE Transactions on Aerospace and Electronics

Systems, Vol. 29, No. 3, Jul. 1993.

[24] K. Siri, S. J. Fang, and C. Q. Lee, “State-Plane Approach to Frequency Response of

Resonant Converters”, IEE Proceedings of Electronics Circuit and Systems, Oct. 1991.

[25] M. G. Kim and M. J. Youn, “A Discrete Time Modeling and Analysis of Controlled

Series Resonant Converter”, IEEE Transactions on Industrial Electronics, Vol. 38, No.

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[26] M. E. Elbuluk, G. C. Verghese and J. G. Kassakian, “Sampled Data Modeling and

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Appendix A

158

Appendix A

Matlab Simulation of State Plane Trajectory of

Series Resonant Converter With Diode Conduction

Angle Control

In this section, a Matlab program is written to simulate a series resonant converter

resonant tank state trajectories under diode conduction angle control. The source code is

given below. In this program, there are five main sections. Section 1 initializes all

variables. Section 2 finds the steady state values of the resonant tank state variables.

Section 3 finds the state plane trajectory parameters [28]. Section 4 defines the number of

cycle needed for this simulation. Note that if step response identification is required, then

the number of cycles should be large so it can cover the new steady state value. However,

if a cycle-to-cycle perturbation for discrete time analysis is needed, then the number of

cycle should be at least 2. Finally Section 5 computes and plots all the switching instants

of the resonant tank state variables. The general idea is to find the switching instant when

the diode conduction reaches control α. Each of these state variables at these switching

instants are saved in the array IP(1,k), VP(1,k), IQ(1,k) and VQ(1,k), where Q = number

of switching cycles, IP and VP are upper quadrant switching instants for resonant tank

current and voltage, and IQ and VQ are lower quadrant switching instants for the same

variables. Note that the symbols used in this source code may differ from those used in

the previous chapters.

Appendix A

159

% SRC DCA % SECTION 1: INITIALIZATION %=========================== clear; pi = 3.142; % Pi constant Vo = 60; % Output Voltage Vi = 100; % Input Voltage Ro = 10; % Load Resistance Io = 3; % Load Current fs = 82e+3; % Switching Frequency -- NOT DETERMINED HERE Cr = 0.053e-6; % Resonant Capacitor Lr = 47.75e-6; % Resonant Inductor Zr =sqrt(Lr/Cr); ws = 2*pi*fs; % Switching Angular Frequency wn = 1/sqrt(Lr*Cr); % Resonant Angular Frequency Von = Vo/Vi; % Normalised Output Voltage Vdn = 1; % Normalized Input Voltage. % SECTION 2: OBTAINING STEADY STATE VALUES OF RESONANT TANK VARIABLES % ================================================================== I = 1.777; % Inductor Current at start of cycle TEMP1 = I^2/(1-Von^2); U = -Von*(sqrt(1+TEMP1) + 1); % Capacitor Voltage at start of cycle DAR = atan(I/abs(U - (-1-Von))); DA = 180*DAR/pi; % Diode Conduction Angle TDA = DA/wn; % Diode Conduction Time % SECTION 3: COMPUTE TRAJECTORY PARAMETERS %========================================= RQX = abs(U)+1-Von; RQY = I; RQS = sqrt(RQX^2+RQY^2);% Steady State Switch Radius Value RDS = RQS-2*Von; % Steady State Diode Radius Value alpha = DAR; % Steady State Control Angle beta = pi - acos((RQS^2+4-RDS^2)/(4*RQS));% Steady State Switch Angle RQ1(1,1) = RQS; % Upper Quadrant Switch Radius for Cycle 1 RQ2(1,1) = RQS; % Lower Quadrant Switch Radius for Cycle 1 RD1(1,1) = RDS; % Upper Quadrant Diode Radius for Cycle 1 RD2(1,1) = RDS; % Lower Quadrant Diode Radius for Cycle 1 IP(1,1) = RDS*sin(alpha);% Lr Current at start of 1st half of Cycle 1 VP(1,1) = -RDS*cos(alpha) - 1 - Von;% Cr Voltage at start of 1st half Cycle 1 IQ(1,1) = -RDS*sin(alpha); % Lr Current at start of 2nd half of Cycle 1 VQ(1,1) = RDS*cos(alpha) + 1 + Von; % Cr Voltage at start of 2nd half of Cycle 1 beta1(1,1) = pi - acos((RQ1(1,1)^2+4-RD2(1,1)^2)/(4*RQ1(1,1))); %Switch angle for 1st half of Cycle 1 beta2(1,1) = beta1(1,1); %Switch angle for 2nd half of Cycle 1 Period(1,1) = (2*alpha + beta1(1,1) + beta2(1,1))*1e+6/wn; %Period of Switch Cycle 1 t = 0:0.01e-6:10e-6; % SECTION 4: DEFINE NUMBER OF CYCLES % ================================== for k = 2:1:5; % SECTION 5: TRAJECTORIES COMPUTATION % ===================================

Appendix A

160

IP(1,k) = RD2(1,k-1)*sin(alpha); VP(1,k) = - RD2(1,k-1)*cos(alpha) + ( - Vdn - Von ); plot(VP(1,k),IP(1,k),'o'); % Plot Starting Point P % SUB-SECTION 5.1: PERTURBATION INJECTIONS % ======================================== if k == 3 % For perturbation %IP(1,k) = IP(1,k) + IP(1,k)*0.01; %VP(1,k) = VP(1,k) + VP(1,1)*0.01; %Vdn = Vdn*1.01; %Von = Von*1.01; alpha = alpha*1.01; end %==========END of SUB-SECTION 5.1========= % Calculate Switch 1 Radius and Beta Angle RQ1(1,k) = sqrt( IP(1,k)^2 + ( Vdn - Von - VP(1,k))^2); beta1(1,k) = pi - atan( IP(1,k) / ( Vdn - Von - VP(1,k)) ); VCPN1 = RQ1(1,k) + Vdn - Von; % Plot Switch 1 Trajectory betaU = 0: 0.01*beta1(1,k) : beta1(1,k); IQU = 0; % Reset VQU = 0; % Reset IQU = RQ1(1,k)*sin(beta1(1,k) - betaU); VQU = RQ1(1,k)*cos(beta1(1,k) - betaU) + ( Vdn - Von ); plot(VQU,IQU,'-'); % Calculate Diode 1 Radius RD1(1,k) = RQ1(1,k) - 2*Von; % Plot Diode 1 Trajectory alphaU = 0: 0.01*alpha : alpha; IDU = 0; % Reset VDU = 0; % Reset IDU = -RD1(1,k)*sin(alphaU); VDU = RD1(1,k)*cos(alphaU) + ( Vdn + Von ); plot(VDU,IDU,'-'); % Find Switching Point at Lower Quadrant IQ(1,k) = -RD1(1,k)*sin(alpha); VQ(1,k) = RD1(1,k)*cos(alpha) + ( Vdn + Von ); plot(VQ(1,k),IQ(1,k),'x'); % Calculate Switch 2 Radius and Beta Angle RQ2(1,k) = sqrt( IQ(1,k)^2 + ( VQ(1,k) - ( - Vdn + Von ) )^2 ); beta2(1,k) = pi - atan( -IQ(1,k) / ( VQ(1,k) - ( - Vdn + Von )) ) ; % Plot Switch 2 Trajectory betaL = 0: 0.01*beta2(1,k) : beta2(1,k); IQL = 0; % Reset VQL = 0; % Reset IQL = -RQ2(1,k)*sin(beta2(1,k) - betaL); VQL = -RQ2(1,k)*cos(beta2(1,k) - betaL) + ( - Vdn + Von ); plot(VQL,IQL,'-'); % Calculate Diode 2 Radius RD2(1,k) = RQ2(1,k) - 2*Von; % Plot Diode 2 Trajectory alphaL = 0: 0.01*alpha : alpha;

Appendix A

161

IDL = 0; % Reset VDL = 0; % Reset IDL = RD2(1,k)*sin(alphaL); VDL = -RD2(1,k)*cos(alphaL) + ( - Vdn - Von ); plot(VDL,IDL); % Find Switching Point at upper Quadrant IP(1,k+1) = RD2(1,k)*sin(alpha); VP(1,k+1) = -RD2(1,k)*cos(alpha) + ( - Vdn - Von ); plot(VP(1,k+1),IP(1,k+1),'x'); %%%%%%%% end %%%%%%%% hold off;

small signal modeling of dc-dc power converters … · 2018-01-09 · fig. 3.13: buck-boost...

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