INTRODUCTION
`TO POWER
`ELECTRONICS
`
`11
`
`Ii
`
`I
`
`I
`
`I
`
`I
`
`Daniel W. Hart
`
`Valparaiso University
`Valparaiso, Indiana
`
`PRENTICE HALL, Upper Saddle River, New Jersey 07 458
`
`LGE-1014 / Page 1 of 22
`LGE v. Fundamental
`
`

`

`Hart, Daniel W.
`Introduction to power electronics I Daniel W. Hart.
`p. cm
`Includes bibliographical references and index.
`ISBN 0-02-351182-6
`1. Power electronics.
`TK7881.15.H37 1997
`62 l.31 '7--<lc20
`
`I. Title.
`
`96-41825
`CIP
`
`Acquisitions editor: Eric Svendsen
`Production editor: Barbara Kraemer
`Editor-in-chief: Marcia Horton
`Managing Editor: Bayani Mendoza Deleon
`Director of production and manufacturing: David W. Riccardi
`Copy editor: Patricia Daly
`Cover designer: Bruce Kenselaar
`Manufacturing buyer: Julia Meehan
`Editorial assistant: Andrea Au
`
`a© 1997 by Prentice-Hall, Inc
`
`::_
`-
`
`Simon & Schuster/A Viacom Company
`Upper Saddle River, NJ 07458
`
`All rights reserved. No part of this book may be
`reproduced, in any form or by any means,
`without permission in writing from the publisher.
`
`The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
`development, research, and testing of the theories and programs to determine their effectiveness. The author and
`publisher make no warranty of an kind, expressed or. implied, with regard to these programs or the documentation
`contained in this book. The author and publisher shall not be liable for incidental or consequential damages
`in connection with, or arising out of, the furnishing, performance, or use of these programs.
`
`Printed in the United States of America
`
`10 9 8 7 6 5 4 3 2
`
`ISBN 0-02-351182-6
`
`Prentice Hall International (UK) Limited, London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada, Inc., Toronto
`Prentice-Hall Hispanoamericana, S.A., Mexico
`Prentice-Hall of India Private Limited, New Delhi
`Prentice-Hall of Japan, Inc., Tokyo
`Simon & Schuster Asia Pte. Ltd., Singapore
`Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
`
`TRADEMARK INFORMATION
`PSpice® is a registered trademark of
`MicroSim Corporation
`
`LGE-1014 / Page 2 of 22
`
`

`

`CONTENTS
`
`CHAPTER 1
`
`INTRODUCTION
`
`1.1
`
`1.2
`
`1.3
`
`1.4
`
`1.5
`
`1.6
`
`Introduction 1
`
`Converter Classification 2
`
`Electronic Switches 3
`The Diode, 3
`Thyristors, 4
`Transistors, 5
`
`Switch Selection 7
`
`Spice and PSpice 9
`
`Switches in PSpice 9
`The Voltage-controlled Switch, 9
`Transistors, 10
`Diodes, 12
`SCRs, 12
`Convergence Problems in PSpice, 13
`
`1.7
`
`Comparing Simulation Results in PSpice 14
`
`Bibliography 14
`
`Problems 15
`
`- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
`
`1
`
`v
`
`LGE-1014 / Page 3 of 22
`
`

`

`CHAPTER 2 POWER COMPUTATIONS
`
`17
`
`2.1
`
`2.2
`
`2.3
`
`2.4
`
`2.5
`
`2.6
`
`2.7
`
`2.8
`
`Introduction 17
`
`Power and Energy 17
`Instantaneous Power, 17
`Energy, 18
`Average Power, 18
`
`Inductors And Capacitors 20
`
`Energy Recovery 23
`
`Effective (Root Mean Square) Values 29
`
`Apparent Power And Power Factor 36
`Apparent Power, S, 36
`Power Factor, 37
`
`Power Computations For Sinusoidal Ac Circuits 37
`
`power Computations For Nonsinusoidal
`Periodic Waveforms 38
`Fourier Series, 39
`Average Power, 40
`Nonsinusoidal Source and Linear Load, 40
`Sinusoidal Source and Nonlinear Load, 42
`
`2.9
`
`Power Computations Using PSpice 45
`
`2.10
`
`Summary 52
`
`Bibliography 53
`
`Problems 54
`
`CHAPTER 3 HALF-WAVE RECTIFIERS: THE BASICS OF ANALYSIS
`
`59
`
`3.1
`
`3.2
`
`3.3
`
`3.4
`
`3.5
`
`3.6
`
`3.7
`
`Introduction 59
`
`Resistive Load 60
`Creating a de Component Using an Electronic Switch, 60
`
`Resistive-Inductive Load 61
`
`PSpice Simulation 65
`Using Simulation Software for Numerical Computations, 65
`
`R-L Source Load 69
`Supplying Power to a de Source from an ac Source, 69
`
`Inductor-Source Load 72
`Using Inductance to Limit Current, 72
`
`The Freewheeling Diode 74
`Creating a de Current, 74
`Reducing Load Current Harmonics, 79
`
`LGE-1014 / Page 4 of 22
`
`

`

`3.8
`
`3.9
`
`Half-Wave Rectifier with a Capacitor Filter 80
`Creating a de Voltage from an ac Source, 80
`
`The Controlled Half-Wave Rectifier 87
`R-LLoad, 89
`R-L Source Load, 91
`
`3.10
`
`PSpice Solutions for Controlled Rectifiers 92
`
`3.11 Commutation 95
`The Effect of Source lnductance, 95
`
`3.l2
`
`Summary 97
`
`Problems 98
`
`CHAPTER 4 FULL-WAVE AND THREE-PHASE
`RECTIFIERS: CONVERTING AC TO DC
`
`104
`
`4.1
`
`4.2
`
`4.3
`
`4.4
`
`4.~
`
`Introduction 104
`
`Single-Phase Full-Wave Rectifiers 104
`The Bridge Rectifier, 107
`The Center-tapped Transformer Rectifier, 107
`Resistive Load, 108
`R-L Load, 108
`Source Harmonics, 111
`PSpice Simulati9n, 112
`R-L Source Load, 114
`Capacitance Output Filter, 117
`Voltage Doublers, 119
`L-C Filtered Output, 120
`Continuous Current for L-C Filtered Output, 121
`Discontinuous Current for L-C Filtered Output, 122
`
`Controlled Full-Wave Rectifieri; 125
`Resistive Load, 126
`R-L Load, Discontinuous Current, 127
`R-L Load, Continuous Current, 129
`PSpic(! Simulation of Controlled
`Fitll-wave Rectifiers, 132
`Controlled Rectifier with R-L Source Load, 133
`Controller] Single-phase Converter
`Operating as an Inverter, 136
`
`Three-Phase Rectifiers 137
`
`Controlled Three-Phase Rectifiers 142
`PSpice Simulation of Controlled Three-Phase Rectifiers, 143
`Twelve-pulse Rectifiers, 145
`The Three-phase Converter Operating as an Inverter, 147
`
`4.6
`
`De Power Transmission 149
`
`Contents
`
`vii
`
`LGE-1014 / Page 5 of 22
`
`

`

`4.7
`
`Commutation: Effect of Source Inductance 153
`Single-phase Bridge Rectifier, 153
`Three-phase Rectifier, 154
`
`4.8
`
`Summary 156
`
`Problems 157
`
`CHAPTER 5 AC VOLTAGE CONTROLLERS:
`AC TO AC CONVERTERS
`
`162
`
`5 .1
`
`5.2
`
`5.3
`
`5.4
`
`5.5
`
`5.6
`
`Introduction 162
`
`The Single-Phase Ac Voltage Controller 162
`Basic Operation, 162
`Single-phase Controller with a Resistive Load, 164
`Single-phase Controller with an R-L Load, 167
`PSpice Simulation of Single-phase AC Voltage Controllers, 171
`
`Three-Phase Voltage Controllers 173
`Y-connected Resistive Load, 173
`Y-connected R-L Load, 177
`Delta-connected Resistive Load, 179
`
`Induction Motor Speed Control 181
`
`Static Var Control 182
`
`Summary 182
`
`Problems 183
`
`CHAPTER 6 DC-DC CONVERTERS
`
`185
`
`6.1
`
`6.2
`
`6.3
`
`6.4
`
`6.5
`
`6.6
`
`6.7
`
`6.8
`
`viii
`
`Linear Voltage Regulators 185
`
`A Basic Switching Converter 186
`
`The Buck Converter 187
`Voltage and Current Relationships, 187
`Output Voltage Ripple, 192
`
`Design Considerations 194
`
`The Boost Converter 196
`Voltage and Current Relationships, 196
`Output Voltage Ripple, 199
`
`The Buck-Boost Converter 201
`Voltage and Current Relationships, 201
`Output Voltage Ripple, 204
`
`The Cuk Converter 205
`
`Nonideal Effects on Converter Performance 211
`Switch Voltage Drops, 211
`
`Contents
`
`LGE-1014 / Page 6 of 22
`
`

`

`Capacitor Resistance: Effect on Ripple, 211
`Inductor Resistance, 212
`Switching Losses, 214
`
`6.9
`
`6.10
`
`Discontinuous-Current Operation 215
`Buck Converter with Discontinuous Current, 215
`Boost Converter with Discontinuous Current, 218
`
`PSpice Simulation Of De-de Converters 221
`A Switched ?Spice Model, 221
`An Averaged Circuit Model, 225
`
`6.11
`
`Summary 231
`
`Bibliography 231
`
`Problems 232
`
`CHAPTER 7 DC POWER SUPPLIES
`
`236
`
`7 .1
`
`7.2
`
`7.3
`
`7.4
`
`7.5
`
`7.6
`
`7.7
`
`7.8
`
`7 .9
`
`Introduction 236
`
`Transformer Models 236
`
`The Flyback Converter 238
`
`The Forward Converter 246
`
`The Double-Ended Forward Converter 253
`
`The Push-Pull Converter 255
`
`Full-Bridge And Half-Bridge De-de Converters 259
`
`Current-Fed Converters 262
`
`Multiple Outputs 266
`
`7 .10 Converter Selection 267
`
`7 .11
`
`7.12
`
`PSpice Simulation of De Power SuppJies 267
`Flyback Converter, 268
`Forward Converter, 269
`Push-Pull Converter, 270
`Current-fed Converter, 271
`
`Power Supply Control 272
`Control Loop Stability, 273
`Small Signal Analysis, 273
`Switch Transfer Function, 274
`Filter Transfer Function, 274
`Pulsewidth Modulation Transfer Function, 275
`Error Amplifier with Compensation, 276
`Design of a Compensated Error Amplifier, 279
`?Spice Simulation of Feedback Control, 282
`PWM Control Circuits, 285
`
`Contents
`
`ix
`
`LGE-1014 / Page 7 of 22
`
`

`

`Bibliography 287
`
`Problems 287
`
`CHAPTER 8
`
`INVERTERS: CONVERTING AC TO DC
`
`291
`
`8.1
`
`8.2
`
`8.3
`
`8.4
`
`8.5
`
`8.6
`
`8.7
`
`8.8
`
`8.9
`
`Introduction 291
`
`The Full-Bridge Converter 291
`
`The Square-Wave Inverter 293
`
`Fourier Series Analysis 297
`
`Total Harmonic Distortion 298
`
`PSpice Simulation of Square-Wave inverters 299
`
`Amplitilde And Harmonic Control 302
`
`The Half-Bridge Inverter 308
`
`Pulse-Width-Modulated Output 308
`Bipolar Switching, 309
`Unipolar Switching, 309
`
`8.10
`
`PWM Definitions and Considerations 311
`
`8.11
`
`8.12
`
`8.13
`
`8.16
`
`PWM Harmonics 312
`Bipolar Switching, 312
`Unipolar Switching, 316
`
`Simulation.of Pulse-Width-Modulated Inverters 317
`Bipolar PWM, 317
`Unipolar PWM, 320
`
`Three-Phase Inverters 326
`The Six-step lnverter; 326
`PWM Three-phase Inverters, 329
`
`PSpice Simulation of Three-Phase Inverters 331
`Six-step Three-phase Inverters, 331
`PWM Three-phase Inverters, 332
`
`8.15
`
`Induction Motor Speed Control 332
`
`8.16
`
`Summary 334
`
`Bibliography 334
`
`Problems 335
`
`CHAPTER 9 RESONANT CONVERTERS
`
`Introduction 338
`
`A Resonant Switch Converter: Zero-Current Switching 339
`
`9.1
`
`9.2
`
`x
`
`338
`
`Contents
`
`LGE-1014 / Page 8 of 22
`
`

`

`9.3
`
`9.4
`
`9.5
`
`9.6
`
`9.7
`
`9.8
`
`9.9
`
`A Resonant Switch Converter: Zero-Voltage Switching 345
`
`The Series Resonant Inverter 351
`Switching Losses, 353
`Amplitude Control, 353
`
`The Series Resonant De-de Converter 357
`Basic Operation, 357
`Operation for W 5 2: w 0 , 357
`Operation for w0 < W 5 < w0 , 363
`Operation for w5 < w 0 /2, 364
`Variations on the Series Resonant De-De Converter, 365
`
`The Parallel Resonant De-de Converter 365
`
`The Series-Parallel De-de Converter 368
`
`Resonant Converter Comparison 371
`
`The Resonant De Link Converter 372
`
`9.10
`
`Summary 375
`
`Bibliography 376
`
`Problems 376
`
`CHAPTER 10 DRIVE AND SNUBBER CIRCUITS
`
`380
`
`10.1
`
`Introduction 380
`
`10.2 MOSFET Drive Circuits 380
`
`10.3
`
`. 10.4
`
`10.5
`
`10.6
`
`10.7
`
`10.8
`
`Bipolar Transistor Drive Circuits 386
`
`Thyristor Drive Circuits 391
`
`Transistor Snubber Circuits 392
`
`Energy Recovery Snubber Circuits 400
`
`Thyristor Snubber Circuits
`
`400
`
`Summary 401
`
`Bibliography 401
`
`Problems 401
`
`APPENDIX A FOURIER SERIES
`FOR SOME COMMON WAVEFORMS 403
`
`APPENDIX B STATE-SPACE AVERAGING 408
`
`INDEX 415
`
`Contents
`
`xi
`
`LGE-1014 / Page 9 of 22
`
`

`

`I
`
`~
`!r ,_
`I b
`
`I
`
`I)
`y
`~ :·
`e
`
`'
`
`2
`
`POWER
`COMPUTATIONS
`
`2.1 INTRODUCTION
`
`Power computations are essential in analyzing and designing power electronics circuits.
`Basic power concepts are reviewed in this chapter, with particular emphasis on power cal(cid:173)
`culations for circuits with nonsinusoidal voltages and currents. Extra treatment is given to
`some special cases that are encountered frequently in power electronics. Power computa(cid:173)
`tions using the circuit simulation program PSpice are demonstrated.
`
`2.2 POWER AND ENERGY
`
`Instantaneous Power
`
`The instantaneous power for any device is computed from the voltage a~ross it and the cur(cid:173)
`rent in it. Instantaneous power is
`
`I p(t) = v(t)i(t). I
`
`(2-1)
`
`17
`
`LGE-1014 / Page 10 of 22
`
`

`

`I'
`
`I
`
`''
`' I
`
`This relationship is valid for any device or circuit. Instanta~eous power is generally a
`time-varying quantity. If the passive sign convention illustrated in Fig. 2- la is observed, the
`device is absorbing power if p(t) is positive at a specified value of time t. The device is sup(cid:173)
`plying power if p(t) is negative. Sources freque~tly have an assumed current direction con(cid:173)
`sistent with supplying power. With the con~e~tion of Fig. 2-1 b, a positive p(t) indicates that
`the source is supplying power.
`
`+
`
`v(t)
`
`+
`
`v(t)
`
`(a)
`
`(b)
`
`(a) Passive sign convention:
`Figure 2.1
`p(t) > 0 indicates power is being absorbed.
`(b) p (t) > 0 indicates power is being
`supplied by the sourct<.
`
`Energy
`
`Energy, or work, is the integral of instantaneous power. Observing the passive sign conven(cid:173)
`tion, the energy absorbed by a component in the time interval from t 1 to t2 is
`
`w =
`
`ft2
`
`t1
`
`p(t)dt.
`
`(2-2)
`
`If v(t) is in volts and i(t) is in amperes, power has units of watts and energy has units of
`joules.
`
`Average Power
`
`Periodic voltage and current functions produce a periodic instantaneous power function.
`Average power is the time average of p(t) over one or more periods. Average power P is
`computed from
`
`1 fto+T
`1 f to+T
`P = -
`p(t)dt = -
`Ti0
`Tt0
`
`v(t)i(t)dt
`
`(2-3)
`
`where T is the period of the power waveform. Combining Eqs. 2-3 and 2-2, power is also
`computed from energy per period:
`
`CWl
`~
`
`(2-4)
`
`Average power is sometimes called real power or active power, especially in ac cir(cid:173)
`cuits. The term power usually means average power. The total average power absorbed in a
`circuit equals the total average power supplied.
`
`18
`
`Power Computations
`
`Chap. 2
`
`LGE-1014 / Page 11 of 22
`
`

`

`+ VcE -
`
`v,
`
`Figure 6.1 A basic linear regulator.
`
`While this may be a simple way of converting a de supply voltage to a lower de volt(cid:173)
`age and regulating the output, the low efficiency of this circuit is a serious drawback for
`power applications. The power absorbed by the load is V0 I £, and the power absorbed by the
`transistor is VcEI v assuming a small base current. The power loss in the transistor makes
`this circuit inefficient. For example, if the output voltage is one-quarter of the input voltage,
`the load resistor absorbs one-quarter of the source power, which is an efficiency of 25%.
`The transistor absorbs the other 75% of the power supplied by the source. Lower output
`voltages result in even lower efficiencies.
`
`6.2 A BASIC SWITCHING CONVERTER
`
`An efficient alternative to the linear regulator is the switching converter. In a switching con(cid:173)
`verter circuit, the transistor operates as an electronic switch by being completely on or com(cid:173)
`pletely off (saturation or cutoff for a BJT). This circuit is also known as a de chopper.
`Assuming the switch is ideal in Fig. 6-2, the output is the same as the input when the
`switch is closed, and the output is zero when the switch is open. Periodic opening and clos(cid:173)
`ing of the switch results in the pulse output shown in Fig. 6-2c. The average or de compo(cid:173)
`nent of the output is
`
`v,
`
`(a)
`
`(b)
`
`~1.~
`
`~Open
`
`0
`
`186
`
`DT
`T
`I· (1-D)T ·I
`(c)
`
`(a) A basic de-de switching
`Figure 6.2
`converter. (b) Switching equivalent.
`( c) Output voltage.
`
`De-de Converters
`
`Chap. 6
`
`LGE-1014 / Page 12 of 22
`
`

`

`}IDT
`IIT
`V0 = T
`V0 (t)dt = T
`V,dt = V,D.
`
`0
`
`0
`
`(6-1)
`
`The de component of the output is controlled by adjusting the duty ratio D, which is the
`fraction of the period that the switch is closed:
`
`D==-
`
`ton
`ton
`= -= t f
`ton + toff
`T
`on
`
`(6-2)
`
`where f is the switching frequency in hertz. The de component of the output will be less
`than or equal to the input for this circuit.
`The power absorbed by the ideal switch is zero. When the switch is open, there is no
`current in it; when the switch is closed, there is no voltage across it. Therefore, all power is
`absorbed by the load, and the energy efficiency is 100%. Losses will occur in a real switch
`because the voltage across it will not be zero when it is on, and the switch must pass through
`the linear region when making a transition from one state to the other.
`
`6.3 THE BUCK CONVERTER
`
`Controlling the de component of a pulsed output of the type in Fig. 6-2c may be sufficient
`for some applications, but often the objective is to produce an output that is purely de. One
`way of obtaining a de output from the circuit of Fig. 6-2a is to insert a low-pass filter after
`the switch. Figure 6-3a shows an inductor-capacitor (L-C) low-pass filter added to the basic
`converter. The diode provides a path for the inductor current when the switch is opened and
`is reverse biased when the switch is closed. This circuit is called a buck converter or a down
`converter because the output voltage is less than the input.
`
`Voltage and Current Relationships
`
`If the low-pass filter is ideal, the output voltage is the average of the input voltage to the fil(cid:173)
`ter. The input to the filter, vx in Fig. 6-3a, is V, when the switch is closed and is zero when
`the switch is open, provided that the inductor current remains positive, keeping the diode
`on. If the switch is closed periodically at a duty ratio D, the average voltage at the filter
`input is V,D, as seen by Eq. 6-1.
`This analysis assumes that the diode remains forward biased for the entire time that
`the switch is open, implying that the inductor current remains positive. An inductor current
`that remains positive throughout the switching period is known as continuous current. Con(cid:173)
`versely, discontinuous current is characterized by the inductor current returning to zero dur(cid:173)
`ing each period.
`Another way of analyzing the operation of the buck converter of Fig. 6-3a is to exam(cid:173)
`ine the inductor voltage and current. This analysis method will prove useful for designing
`the filter and for analyzing circuits that are presented later in this chapter.
`
`Sec. 6.3
`
`The Buck Converter
`
`187
`
`LGE-1014 / Page 13 of 22
`
`

`

`Vs
`
`irs
`
`Vs
`
`(a)
`
`VL= Vs- V0
`+
`
`+
`, Vx =Vs
`
`(b)
`
`(c)
`
`(a) Buck de-de converter.
`Figure 6.3
`(b) Equivalent for switch closed.
`(c) Equivalent for switch open.
`
`The buck converter (and de-de converters in general) has the following properties
`when operating in the steady state:
`
`1. The inductor current is periodic:
`
`2. The average inductor voltage is zero (see Chapter 2, Section 2.3):
`1 ft+T
`VL = T t
`
`vL(A.)dA. = 0.
`
`''3. The average capacitor current is zero (see Chapter 2, Section 2.3):
`1 ft+T
`le = T t
`
`ic(A.)dA. = o.
`
`(6-3)
`
`(6-4)
`
`(6-5)
`
`4. The power supplied by the source is the same as the power delivered to the load. For
`nonideal components, the source also supplies the losses:
`
`188
`
`De-de Convarters
`
`Chap.6
`
`LGE-1014 / Page 14 of 22
`
`

`

`Ps = P0
`(ideal)
`Ps = P0 +losses
`
`(nonideal).
`
`(6-6)
`
`Analysis of the buck converter of Fig. 6-3a begins by making these assumptions:
`
`1. The circuit is operating in the steady state.
`2. The inductor current is continuous (always positive).
`3. The capacitor is very large, and the output voltage is held constant at voltage V0 • This
`restriction will be relaxed later to show the effects of finite capacitance.
`4. The switching period is T; the switch is closed for time DT and open for time
`(1 - D)T.
`5. The components are ideal.
`
`The key to the analysis for determining the output V0 is to examine the inductor current and
`inductor voltage first for the switch closed and then for the switch open. The net change in
`inductor current over one period must be zero for steady-state operation. The average
`inductor voltage is zero.
`
`Analysis for the switch closed. When the switch is closed in the buck con-
`verter circuit of Fig. 6-3a, the diode is reverse biased and Fig. 6-3b is an equivalent circuit.
`The voltage across the inductor is
`
`Rearranging,
`
`diL V, - V0
`- = - - - (switchclosed).
`dt
`L
`
`Since the derivative of the current is a positive constant, the current increases linearly, as
`shown in Fig. 6-4b. The change in current while the switch is closed is computed by modi(cid:173)
`fying the preceding equation:
`
`(6-7)
`
`Analysis for the switch open. When the switch is open, the diode becomes
`forward biased to carry the inductor current, and the equivalent circuit of Fig. 6-3c applies.
`The voltage across the inductor when the switch is open is
`
`diL
`VL =-Vo= Ldt.
`
`Sec. 6.3
`
`The Buck Converter
`
`189
`
`LGE-1014 / Page 15 of 22
`
`

`

`DT
`
`T
`
`ic
`
`Figure 6.4 Buck converter waveforms.
`(a) Inductor voltage. (b) Inductor current.
`( c) Capacitor current.
`
`Rearranging,
`
`diL = -V,,
`dt
`L
`
`(switch open).
`
`The derivative of current in the inductor is a negative constant, and the current decreases
`linearly, as shown in Fig. 6-4b. The change in inductor current when the switch is open
`IS
`
`!!:..iL _
`!!:..t
`
`_ V0
`!!:..iL
`(1 -D)T
`L
`
`(!!:..iL)open = -( { )o -D)T.
`
`(6-8)
`
`Steady-state operation requires that the inductor current at the end of the switching
`cycle be the same as that at the beginning, meaning that the net change in inductor current
`over one period is zero. This requires
`
`Using Eqs. 6-7 and 6-8,
`
`c~ ~ V,, )vr -(i' )(1 -D)T = 0.
`
`190
`
`De-de Converters
`
`Chap. 6
`
`LGE-1014 / Page 16 of 22
`
`

`

`Solving for V0 ,
`
`(6-9)
`
`which is the same result as Eq. 6-1. The buck converter produces an output which is less
`than or equal to the input.
`An alternative derivation of the output voltage is based on the inductor voltage, as
`shown in Fig. 6-4a. Since the average inductor voltage is zero for periodic operation,
`VL = (V, - V0 )DT + (-V0 )(l - D)T = 0.
`Solving the preceding equation for V0 yields the same result as Eq. 6-9, V0 = V,D.
`Note that the output voltage depends only on the input and the duty ratio D. If the
`input voltage fluctuates, the output voltage can be regulated by adjusting the duty ratio
`appropriately. A feedback loop is required to sample the output voltage, compare it to a ref(cid:173)
`erence, and set the duty ratio of the switch accordingly.
`The average inductor current must be the same as the average current in the load
`resistor, since the average capacitor current must be zero for steady-state operation:
`
`Vo
`h=IR= R.
`
`(6-10)
`
`Since the change in inductor current is known from Eqs. 6-7 and 6-8, the maximum and
`minimum values of the inductor current are computed as
`
`fl.iL
`lmax=h+l
`= V0 + l[V0 (1 _ D)T] = V [_!_ + (1 - D)]
`
`0 R
`
`2/j
`
`R
`
`2 L
`
`fl.iL
`/min =h-l
`= V0 _ l[V0 (1 _ D)T] = V [_!_- (1 - D)]
`
`0 R
`
`2Lf
`
`R2L
`
`(6-11)
`
`(6-12)
`
`where f = 1 / T is the switching frequency in hertz.
`For the preceding analysis to be valid, continuous current in the inductor must be ver(cid:173)
`ified. An easy check for continuous current is to calculate the minimum inductor current
`from Eq. 6-12. Since the minimum value of inductor current must be positive for continu(cid:173)
`ous current, a negative minimum calculated from Eq. 6-12 is not allowable due to the diode
`and indicates discontinuous current. The circuit will operate for discontinuous inductor cur(cid:173)
`rent, but the preceding analysis is not valid. Discontinuous current operation is discussed
`later in this chapter.
`Equation 6-12 can be used to determine the combination of L and f that will result in
`continuous current. Since I min = 0 is the boundary between continuous and discontinuous
`current,
`
`Sec. 6.3
`
`The Buck Converter
`
`191
`
`LGE-1014 / Page 17 of 22
`
`

`

`-·--"--·----·- --- --- ----------------~
`If the desired switching frequency is established,
`·
`
`(1 - D)R
`Lmin =
`2J
`
`--------
`-----------"- - - - - ----
`where Lmin is the minimum iJ:lductancerequired forcontiriilous current.
`
`Output Voltage Ripple
`
`(6-13)
`
`(6-14)
`
`In the preceding analysis, the capacitor was assumed to be very large to keep the output
`voltage constant. In practice, the output voltage cannot be kept perfectly constant with a
`finite capacitance. The variation in output voltage, or ripple, is computed from the voltage(cid:173)
`current relationship of the capacitor. The current in the capacitor is
`
`shown in Fig. 6-5a.
`While the capacitor current is positive, the capacitor is charging. From the definition
`of capacitance,
`
`Q=CV0
`LiQ = CLi V,,
`
`LiV: = LiQ
`c.
`
`0
`
`ic
`
`(b)
`
`Figure 6.5 Buck converter waveforms.
`(a) Capacitor current. (b) Capacitor ripple
`voltage.
`
`192
`
`De-de Converters
`
`Chap.6
`
`LGE-1014 / Page 18 of 22
`
`

`

`The change irt charge, AQ, is the area of the triangle above the time axis:
`
`resulting in
`
`Using Eq. 6-S for Aiv
`
`AV:= TAiL
`SC.
`
`0
`
`AV:= j_ Vo(l - D)T = Vo(l - D)
`SLCJ2
`SC L
`
`0
`
`(6-15)
`
`111 this equation, A V0 is the peak-to-peak ripple voltage at the output, as shown in Fig. 6-Sb.
`It is also useful to express the ripple as a fraction of the output voltage:
`
`(6-16)
`
`if the ripple is not large, the assumption of a constant output is reasonable and the preced(cid:173)
`ing analysis is essentially valid.
`Since the converter components are assumed to be ideal, the power supplied by the
`source must be the same as the power absorbed by the load resistor:
`
`Ps =Po
`
`V.ls =Volo
`
`(6-17)
`
`or
`
`Vo= ls
`v.
`Io
`Note that the preceding relationship is similar to the voltage-current relationship for a trans(cid:173)
`former in ac applications. Therefore, the buck converter circuit is equivalent to a de trans(cid:173)
`former.
`
`Example 6-1 Buck Converter
`The buck de-de converter of Fig. 6-3a has the following parameters:
`
`V,= 50V
`D=0.4
`L=400µH
`C= lOOµF
`f ='=20kHz
`R=200.
`
`Assuming ideal components, calculate (a) the output voltage V0 , (b) the maximum and mini(cid:173)
`mum inductor current, and ( c) the output voltage ripple.
`
`Sec. 6.3
`
`The Buck Converter
`
`193
`
`LGE-1014 / Page 19 of 22
`
`

`

`Solution
`(a) The inductor current is assumed to be continuous, and the output voltage is
`computed from Eq. 6-9:
`
`(b) Maximum and minimum inductor currents are computed from Eqs. 6-11and6-12:
`
`V0 = Y,D = (50)(0.4) = 20 V.
`
`lmax = v{ ~+ 12~:]
`
`= 20[.l +
`]
`1 - 0.4
`20 2c4oo)c10r62oc10)3
`= 1 + 1~5
`= 1.75 A
`
`[ 1 1 ~DJ
`
`/min= Vo R- 2LJ
`
`1.5
`= 1 - 2 = 0.25 A.
`
`The average inductor current is 1 A, and D.iL = 1.5 A. Note that the minimum inductor cur(cid:173)
`rent is positive, verifying that the assumption of continuous current was valid.
`(c) The output voltage ripple is computed from Eq. 6-16:
`
`D.V0
`Vo
`
`1 - 0.4
`l-D
`sc4oo)c1or6ooo)oor6c20000)2
`8LCJ2
`= 0.00469 = 0.469%.
`
`Since the output ripple is sufficiently small, the assumption of a constant output voltage
`was reasonable.
`
`6.4 DESIGN CONSIDERATIONS
`
`Most buck converters are designed for continuous-current operation. The choice of switch(cid:173)
`ing frequency and inductance tb give continuous current is given by Eq. 6-13, and the out(cid:173)
`put ripple is described by Eq. 6-16. Note that as the switching frequency increasei, the
`minimum size of the inductor to produce continuous current and the minimum size of the
`capacitor to limit output ripple both decrease. Therefore, high switching frequencies are
`desirable to reduce the size of both the inductor and the capacitor.
`The trade-off for high switching frequencies is increased power loss in the switches,
`which is discussed later in this chapter and in Chapter 10. Increased power loss for the
`switches decreases the converter's efficiency, and the larger heat sink required for the transis(cid:173)
`tor switch offsets the reduction in size of the inductor and capacitor. Typical switching fre(cid:173)
`quencies are in the 20-kHz to 50-kHz range, although frequencies in the hundreds of kilohertz
`are not uncommon. As switching devices improve, switching frequencies will increase.
`The inductor wire must be rated at the rms current, and t)le core should not saturate for
`peak inductor current. The capacitor must be selected to limit the output ripple to the design
`specifications, to withstand peak output voltage, and to carry the required rms current.
`
`194
`
`De-de Converters
`
`Chap. 6
`
`LGE-1014 / Page 20 of 22
`
`

`

`The switch arid diode must withstand maximum voltage stress when off and maxi(cid:173)
`mum current when on. The temperature ratings must not be exceeded, possibly requiring a
`heat sink.
`
`Example 6-2 Buck Converter Design
`Design a buck converter to produce an output voltage of 18 V across a 10-0 load resistor. The
`output voltage ripple must not exceed 0.5%. The de supply is 48 V. Design for continuous
`inductor current. Specify the duty ratio, the sizes of the inductor and capacitor, the peak volt(cid:173)
`age rating of each device, and the rms current in the inductor and capacitor.
`
`Solution The duty ratio for continuous-current operation is determined from Eq. 6-9:
`
`D = ~ = !~ = 0.375.
`
`The switching frequency and inductor size must be selected for continuous-current operation. Let
`the switching frequency arbitrarily be 40 kHz, which is well above the audio range and is low
`enough to keep switching losses small. The minimum inductor size is determined from Eq. 6-14:
`
`. = (1 - D)R = (1 - 0.375) 10 = 78
`Lmm
`2(40000)
`2f
`
`H
`µ
`.
`
`Let the inductor be 25% larger than the minimum to ensure that inductor current is continuous:
`
`L = 1.254,in = (1.25)(78 µH) = 97 .5 µH.
`
`Average inductor current and the change in current are determined from Eqs. 6-10 and 6-7:
`
`(v -v)
`
`)
`1
`(
`tl.iL= ~ DT= 97.5(1or6 (0.375) 40000 =2.88A.
`
`h=Va=18=18A
`10
`.
`R
`
`( 48 - 18 )
`
`The maximum and minimum inductor currents are determined from Eqs. 6-11 and 6-12:
`
`il"
`I max= h + ~L = 1.8 + 1.44 = 3.24A
`
`.
`AiL
`I min= h -Z = 1.8 - 1.44 = 0.36A.
`
`The inductor must be rated for rms current, which is computed as in Chapter 2 (see Exam~
`2-8). For the offset triangular wave,
`
`IL,rms =
`
`The capacitor is selected using Eq. 6-16:
`
`1-D
`
`c
`
`1 - 0.375
`8(97.5)(10r6<.005)(40000)2
`
`= lOO F
`µ
`·
`
`Peak capacitor current is tl.iL/ 2 = 1.44 A, and rms capacitor current for the triangular wave(cid:173)
`form is 1.44 / J3 = 0.83 A.
`
`Sec. 6.4
`
`Design Considerations
`
`195
`
`LGE-1014 / Page 21 of 22
`
`

`

`The maximum voltage across the switch and diode is Y,., or 48 V. The inductor voltage
`when the switch is closed is Vs - V,, = 48 - 18 = 30 V. The inductor voltage when the Switch
`is open is V0 = 18 V. Therefore, the inductor must withstand 30 V. The capacitor must be rated
`for the 18-v output.
`
`6.5 THE BOOST CONVERTER
`
`The boost converter is shown in Fig. 6-6. This is another switching converter that operates
`by periodically. opening and closing an electronic switch. It is called a boost converter
`because the output voltage is larger than the input.
`
`Voltage and Current Relationships
`
`The analysis assumes the following:
`
`1. Steady-state conditions exist.
`2. The switching period is T, and the switch is closed for time DT and open for (1 - D)T.
`3. The inductor current is continuous (always positive).
`4. The capacitor is very large, and the output voltage is held constant at voltage V0
`5. The components ate ideal.
`
`•
`
`~ic
`
`,,
`
`Figure 6.6 The boost converter:
`(a) Circuit. (b) Equivalent for the switch
`dosed. (c) Equivalent for the switch open.
`
`De-de Converters
`
`Chap.6
`
`VL= Vs
`+
`-
`
`(a)
`
`(b)
`
`(c)
`
`Vs
`
`Vs
`
`196
`
`LGE-1014 / Page 22 of 22
`
`