Table Of ContentLARGE SIGliALS IN SWITCHING C0:IVERTERS
Thesis by
Robert W. Erickson
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1983
(Submitted November 8, 1982)
t o my pazsnt,i,
Roben*, Str. and P&e,
and 20 my dde, Linda
iuk,
I am indebted to my advisor, Prof. for recruiting me into
the Power Electronics Group and for his strong support during most
of my years a t Caltech. The encouragement and guidance that both he
and my unofficial advisor, Prof. Middlebrook, have provided has been
invaluable, for which I am grateful.
I would like to thank Ms. Janice Tucker for her accurate typing
of the manuscript, and for the extra hours she spent to meet my
dead1 ine.
I appreciate the financial support, by way of Research
Assistantship, provided by the Naval Ocean Systems Center,
the International Business Machines Corporation, and the Office
of Naval Research.
ABSTRACT
The behavior of switching converters under large-signal conditions
is modelled and analyzed. Two major problems are of interest:
the reduction of the distortion which occurs in switching amplifiers,
and the stabilization of switching regulators for all transients which
are expected to occur.
In Part 1, a large-signal model is described which predicts the
simple harmonic distortion generated inherently by switched-mode
amplifiers. The causes of this distortion are identified, and
relatively simple design techniques for its reduction to an acceptable
level are presented. A particularly attractive feature of the method
is the ability to compute harmonic magnitude and phase using linear
ci rcui t models and Bode plots.
The method is extended to account for intermodulation distortion.
Additional effects not described by the model are also discussed,
including crossover distortion and bandwidth limitations.
Although i n its simplest form the buck converter is inherently
linear, considerable distortion may arise when i t is preceded by
an input filter. This problem is solved as an example of the
usefulness and circui t-oriented nature of the method.
In Part 2, a large-signal switching regulator model is derived,
and prominent features of the transient response are determined.
In particular, the various regions of operation are identified in the
state plane, analytical expressions are found for the equilibrium
points of the system, and computer-generated transient wavefoms
are obtained.
As an example, a boost regulator is investigated, and is found
to be stable for small signals but unstable for large transients.
Approximate analytical expressions are found for the waveforms and
salient features of the response, and a number of ways of obtaining
global stability are discussed.
Experimental evidence is presented to verify both the distortion
and stability analyses.
v i
TABLE OF CONTENTS
page
ACKNOWLEDGENEE4TS iii
ABSTRACT i v
PART 1 DISTORTION IN SWITCHING AMPLIFIERS 1
CHAPTER 1 INTRODUCTIOLi 2
CHAPTER 2 AN INTRODUCTION TO SWITCHING AMPLIFIERS 6
Introduction 6
Review of Toplogies and Applications 6
An Elementary Nonlinear Model 15
AiiALYSIS - SIPIPLE HAlU4ONIC DISTORTIOh 2 6
Introduction 26
Series Expansion of State Vector 2 6
Linear Circuit Models 3 1
An Adllitional Simplification 43
Discussion 5 1
3.6 Step-by-step Sumary of the Procedure 59
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CiiAPTER 4 AN EXTENSION INTERMODULATION DISTORTION 62
4.1 Introduction 62
4.2 Mathematical Description 63
4.3 Linear Circuit Models 67
ChAPTER 5 DESIGN EXAMPLES: DISTORTION IN 8UCK AMPLIFIERS 72
5.1 Introduction 72
5.2 Existence of Distortion in Buck Amplifiers 73
5.3 Bridge-Type Buck Amplifier with Input Filter 7 7
5.4 Single-Ended Buck Converter with Input Filter 92
CHAPTER 6 OTHER SOURCES OF HARMONICS
6.1 Introduction
6.2 Bandwidth Limitations
6.3 Switching Ripple and Component Nonidealities
6.4 Crossover Distortion
CHAPTER 7 EXPERIMENTAL VERIFICATION
7.1 Introduction
7.2 Bridge-Type Buck Aii~plifierw ith Input Filter
7.3 Boost Converter
CHAPTER 8 CONCLUSIONS
PART 2 TRANSIENTS IN SWITCHING REGULATORS
CHAPTER 9 INTRODUCTION
9.1 Introduction
9.2 The Large-Signal Stabi 1i ty Problem: An Example
9.3 Outline of Discussion
CHAPTER 10 FUNDAMENTALS OF STATE-PLANE ANALYSIS
10.1 Introduction
10.2 Review of State-Plane Analysis and Piecewi se
Systems
The Switching Regulator Problem in More Detail
CHAPTEi? 11 LARGE-SIGNAL MODELLING
11.1 Introduction
11.2 Unsaturated Region Model
11.3 Saturated Region Model
11.4 Transient Discontinuous Conduction Region Model
CHAPTER 12 LARGE-SIGNAL ANALYSIS
12.1 Introduction
12.2 Equilibrium Points in the Unsaturated Mode
12.3 Equilibrium Points in the Saturated Modes
12.4 Construction of Trajectories
CHAPTER 13 EXAMPLE: AIJALYSIS OF A SIijiPLE BOOST REGULATOR
13.1 Introduction
13.2 Small -Signal Design
13.3 Equi 1i brium Output Voltage Curves
13.4 l4odes of Operation
13.5 Unsaturated Region Waveforms
13.6 0 = Dmin Saturated Region Waveforms
13.7 D = ha, Saturated Region Waveforms
13.8 Transient Discontinuous Conduction Mode Waveforms
13.9 Discussion
ChAPTER 14 EXPERIMENTAL VERIFICATION
CHAPTER 15 CONCLUSIONS
REFERENCES
PART 1
DISTORTION SWITCHING AMPLIFIERS
IN
CHAPTER 1
INTRODUCTION
Switching converters are finding increasing application as motor
controllers, transducer drivers, servo amplifiers, uninterruptible
power supplies, and low-distortion (unity power factor) ac-to-dc power
supplies. In these applications, the converter must reproduce a control
signal at high power levels with reasonable accuracy and efficiency.
Because of their theoretical 100% efficiency, switching converters
would appear to be much better suited for these applications than
conventional class B amplifiers. However, the inherent nonlinearity
i n these converters poses an obstacle to both their analysis and design.
In particular, it i s possible for nonlinearities to cause the
appearance of large, previously unexpected and mysterious amounts of
low and mid-frequency distortion. The analysis of this distortion i n a
design-oriented manner i s the subject of the first part of this thesis.
The technoIogy for the implementation o f switching power supplies
i s currently at a high level and continues to develop. Since both
switching power supplies and switching amplifiers fall into the general
category of "switched-mode power converters", it i s natural to attempt
to adapt this technology to the realization of a power amplifier.
Indeed, i n many instances this may be done with only a few major
modifications [9,10J. However, there i s one fundamental difference
between the amplifier and the power supply: the amplifier must
