Fundamentals of Power Electronics (Cơ bản về Điện tử công suất) - Robert W. Erickson, Dragan Maksimović

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Robert W. Erickson Dragan Maksimovi´c Department of Electrical, Computer, Department of Electrical, Computer, and Energy Engineering and Energy Engineering University of Colorado Boulder University of Colorado Boulder Boulder, CO, USA Boulder, CO, USA ISBN 978-3-030-43879-1 ISBN 978-3-030-43881-4 (eBook) https://doi.org/10.1007/978-3-030-43881-4 1st edition: © Springer Science+Business Media Dordrecht 1997 2nd edition: © Kluwer Academic Publishers 2001 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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Preface The objective of the First and Second Editions was to serve as a textbook for introductory power electronics courses where the fundamentals of power electronics are defined, rigorously pre- sented, and treated in sufficient depth so that students acquire the knowledge and skills needed to design practical power electronic systems. An additional goal was to contribute as a reference book for engineers who practice power electronics design, and for students who want to develop their knowledge of the area beyond the level of introductory courses. In this Third Edition, the basic objectives and philosophy of the earlier editions have not been changed. Since we wrote the Second Edition, the field of power electronics has grown tremen- dously, including new significant commercial applications such as electric vehicles, wireless power transfer, and utility microgrids. Technical growth includes the commercialization of wide bandgap power semiconductors, widespread digital control of switching converters, and matu- ration of converter modeling. Our university power electronics curriculum has evolved as well, in content as well as in organization. This edition is a response to these changes, and represents a significant revision relative to the previous edition. As of 2020, at the University of Colorado we offer a sequence of three core graduate courses in power electronics. The first course, Introduction to Power Electronics, covers basic converter analysis, converter controllers, and magnetics. In the Third Edition, this material is presented in Chaps. 1–12, at the level and in the order covered in this class. Our second course, Modeling and Control of Power Electronics Systems, covers more advanced topics of power converter applica- tions, control, and design-oriented analysis. This material is covered in detail in Chaps. 13–21 in the Third Edition; this portion of the text represents a major revision of technical material and coverage. Our third course, Resonant and Soft Switching Phenomena in Power Electronics, relies primarily on supplementary notes rather than this textbook. Chapters 22 and 23 of the Third Edition cover a summary of a portion of this third course. The coverage of power semiconductor devices in Chap. 4 has been bolstered and updated. The discussion of power diode switching has been significantly expanded, leading into aver- aged modeling of diode-induced switching loss. New material on wide bandgap devices and on MOSFET gate drivers has been added. The discussion of switching loss mechanisms has been updated and reorganized, and the MCT section is removed. The Third Edition adopts a more mature viewpoint of averaging, based on the trapezoidal moving average defined in Eq. (7.3). The waveforms of the averaged model become true con- tinuous quantities, with the approximations and logical steps clearly defined. New material in vii

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viii Preface Chap. 7 includes a section on the averaging operator, and a new treatment of how the small- ripple approximation works with the trapezoidal moving average. Additionally, the logical flow of Chap. 7 has been significantly revised to conform to how we now teach this material in our on-campus courses, and new material on state-space averaging has been added. This new view- point of averaging then is followed throughout the remainder of the book. Of most note, this viewpoint leads to the current-programmed control model of Tan and Middlebrook. The current- programmed control Chap. 18 has been significantly revised and updated accordingly. The high- frequency effects of sampling are discussed as well, in connection with current-programmed control and also with ac modeling of the discontinuous conduction mode. The previous treatment of stability and phase margin would leave some students with mis- conceptions; to alleviate this, we have introduced a new section on Nyquist stability. Instructors may choose whether there is time to cover this material in a power electronics course, but the explanation is available as a reference in the text. The origin of the phase margin text is rigor- ously explained, and special cases such as conditionally stable systems or those with multiple crossover frequencies are adjudicated. A new section in the chapter on input filters has been added, which relies on the Nyquist stability criterion to determine the exact stability boundary in the presence of an input filter. An all-new Part IV Advanced Modeling, Analysis, and Control Techniques has been orga- nized to follow the logical flow of our advanced converter control course, and incorporates new chapters on null double injection techniques (Middlebrook’s feedback theorem and extra ele- ment theorem) and on digital control of switching converters. The topics of circuit averaging, average switch modeling, and averaged simulation are consolidated into a single logical chap- ter. New examples of the extra element theorem include solution of the SEPIC averaged switch model, and damping the internal resonances of the SEPIC. Chapter 18 on current-programmed control has been significantly revised and reorganized. As noted above, it now employs the model of Tan and Middlebrook, using the trapezoidal mov- ing average. New sections on simulation, sampling and high-frequency dynamics, and input filters are incorporated into the chapter. A new section on average current-mode control has also been added. The new Chap. 19 on digital control of switching converters extends the analog control techniques of earlier chapters, to address the relevant issues of digital controllers. Quantiza- tion, sampling, and controller delays are modeled. The Z-transform is employed to model the discrete-time portion of the feedback loop, with the Laplace transform used as usual for the remaining analog system. Digital compensator design and realization is then addressed. This text has evolved from courses developed over thirty-five years of teaching power elec- tronics at the University of Colorado. These courses, in turn, were heavily influenced by our previous experiences as graduate students at the California Institute of Technology, under the direction of Profs. Slobodan ´Cuk and R. D. Middlebrook, to whom we are grateful. We would also like to thank the many readers of the First and Second Editions, students, and instructors who offered their comments and suggestions, or who pointed out errata. We have attempted to incorporate these suggestions wherever possible. Boulder, CO, USA Robert W. Erickson Boulder, CO, USA Dragan Maksimovi´c

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction to Power Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Several Applications of Power Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Elements of Power Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Part I Converters in Equilibrium 2 Principles of Steady-State Converter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Volt-Second and Charge Balance, Small-Ripple Approximation . . . . . . . . . . . . . 18 2.3 Boost Converter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Cuk Converter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .´ 30 2.5 Estimating the Output Voltage Ripple in Converters Containing Two-Pole Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Steady-State Equivalent Circuit Modeling, Losses, and Efficiency. . . . . . . . . . . . . 43 3.1 The DC Transformer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Inclusion of Inductor Copper Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Construction of Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.1 Inductor Voltage Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.2 Capacitor Current Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.3 Complete Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.4 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 How to Obtain the Input Port of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 Example: Inclusion of Semiconductor Conduction Losses in the Boost Converter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ix

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x Contents 4 Switch Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Switch Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.1 Single-Quadrant Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.2 Current-Bidirectional Two-Quadrant Switches . . . . . . . . . . . . . . . . . . . . . 72 4.1.3 Voltage-Bidirectional Two-Quadrant Switches . . . . . . . . . . . . . . . . . . . . . 75 4.1.4 Four-Quadrant Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.1.5 Synchronous Rectifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Introduction to Power Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.1 Breakdown Voltage, Forward Voltage, and Switching Speed . . . . . . . . . 79 4.2.2 Transistor Switching Loss with Clamped Inductive Load . . . . . . . . . . . . 80 4.3 The Power Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.1 Introduction to Power Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.2 Discussion: Power Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.3 Modeling Diode-Induced Switching Loss . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3.4 Boost Converter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4 Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) . . . . . . . . . . . . 99 4.4.1 Introduction to the Power MOSFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.4.2 Wide-Bandgap FETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4.3 MOSFET Gate Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.5 Minority-Carrier Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.1 Bipolar Junction Transistor (BJT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.2 Insulated-Gate Bipolar Transistor (IGBT) . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.5.3 Thyristors (SCR, GTO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.6 Additional Sources of Switching Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.6.1 Device Capacitances, and Leakage, Package, and Stray Inductances . . . 122 4.6.2 Inducing Switching Loss in Other Elements . . . . . . . . . . . . . . . . . . . . . . . 124 4.6.3 Efficiency vs. Switching Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.7 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5 The Discontinuous Conduction Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.1 Origin of the Discontinuous Conduction Mode, and Mode Boundary . . . . . . . . 135 5.2 Analysis of the Conversion Ratio M(D, K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.3 Boost Converter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.4 Summary of Results and Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6 Converter Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.1 Circuit Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.1.1 Inversion of Source and Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.1.2 Cascade Connection of Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.1.3 Rotation of Three-Terminal Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1.4 Differential Connection of the Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.2 A Short List of Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.3 Transformer Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.1 Full-Bridge and Half-Bridge Isolated Buck Converters . . . . . . . . . . . . . . 181 6.3.2 Forward Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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Contents xi 6.3.3 Push-Pull Isolated Buck Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.3.4 Flyback Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.3.5 Boost-Derived Isolated Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.3.6 Isolated Versions of the SEPIC and the ´Cuk Converter . . . . . . . . . . . . . . 201 6.4 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Part II Converter Dynamics and Control 7 AC Equivalent Circuit Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.2 The Basic AC Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7.2.1 Averaging the Inductor and Capacitor Waveforms . . . . . . . . . . . . . . . . . . 221 7.2.2 The Average Inductor Voltage and the Small-Ripple Approximation . . . 222 7.2.3 Discussion of the Averaging Approximation . . . . . . . . . . . . . . . . . . . . . . . 223 7.2.4 Averaging the Capacitor Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.2.5 The Average Input Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.2.6 Perturbation and Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.2.7 Construction of the Small-Signal Equivalent Circuit Model . . . . . . . . . . 230 7.2.8 Discussion of the Perturbation and Linearization Step . . . . . . . . . . . . . . . 232 7.2.9 Results for Several Basic Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.2.10 Example: A Nonideal Flyback Converter . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.3 Modeling the Pulse-Width Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.4 The Canonical Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.4.1 Development of the Canonical Circuit Model . . . . . . . . . . . . . . . . . . . . . . 245 7.4.2 Example: Manipulation of the Buck–Boost Converter Model into Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7.4.3 Canonical Circuit Parameter Values for Some Common Converters . . . 250 7.5 State-Space Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.5.1 The State Equations of a Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 7.5.2 The Basic State-Space Averaged Model . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.5.3 Discussion of the State-Space Averaging Result . . . . . . . . . . . . . . . . . . . . 256 7.5.4 Example: State-Space Averaging of a Nonideal Buck–Boost Converter 259 7.5.5 Example: State-Space Averaging of a Boost Converter with ESR . . . . . 264 7.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 8 Converter Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.1 Review of Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.1.1 Single-Pole Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 8.1.2 Single Zero Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8.1.3 Right Half-Plane Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 8.1.4 Frequency Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.1.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 8.1.6 Quadratic Pole Response: Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.1.7 The Low-Q Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

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xii Contents 8.1.8 The High-Q Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.1.9 Approximate Roots of an Arbitrary-Degree Polynomial . . . . . . . . . . . . . 304 8.2 Analysis of Converter Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8.2.1 Example: Transfer Functions of the Buck–Boost Converter . . . . . . . . . . 309 8.2.2 Transfer Functions of Some Basic CCM Converters . . . . . . . . . . . . . . . . 315 8.2.3 Physical Origins of the RHP Zero in Converters . . . . . . . . . . . . . . . . . . . . 316 8.3 Graphical Construction of Impedances and Transfer Functions . . . . . . . . . . . . . . 317 8.3.1 Series Impedances: Addition of Asymptotes . . . . . . . . . . . . . . . . . . . . . . . 318 8.3.2 Series Resonant Circuit Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 8.3.3 Parallel Impedances: Inverse Addition of Asymptotes . . . . . . . . . . . . . . . 322 8.3.4 Parallel Resonant Circuit Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 8.3.5 Voltage Divider Transfer Functions: Division of Asymptotes . . . . . . . . . 325 8.4 Graphical Construction of Converter Transfer Functions . . . . . . . . . . . . . . . . . . . 327 8.5 Measurement of AC Transfer Functions and Impedances . . . . . . . . . . . . . . . . . . . 332 8.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 9 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.2 Effect of Negative Feedback on the Network Transfer Functions . . . . . . . . . . . . 350 9.2.1 Feedback Reduces the Transfer Functions from Disturbances to the Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 9.2.2 Feedback Causes the Transfer Function from the Reference Input to the Output to Be Insensitive to Variations in the Gains in the Forward Path of the Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 9.3 Construction of 1/(1 + T) and T/(1 + T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 9.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 9.4.1 The Phase Margin Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 9.4.2 The Nyquist Stability Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 9.4.3 The Relationship Between Phase Margin and Closed-Loop Damping Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 9.4.4 Transient Response vs. Damping Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 373 9.4.5 Load Step Response vs. Damping Factor . . . . . . . . . . . . . . . . . . . . . . . . . 375 9.5 Regulator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 9.5.1 Lead (PD) compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 9.5.2 Lag (PI) Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 9.5.3 Combined (PID) Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 9.5.4 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 9.6 Measurement of Loop Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 9.6.1 Voltage Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 9.6.2 Current Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 9.6.3 Measurement of Unstable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.7 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

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Contents xiii Part III Magnetics 10 Basic Magnetics Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 10.1 Review of Basic Magnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 10.1.1 Basic Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 10.1.2 Magnetic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 10.2 Transformer Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 10.2.1 The Ideal Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 10.2.2 The Magnetizing Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 10.2.3 Leakage Inductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 10.3 Loss Mechanisms in Magnetic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 10.3.1 Core Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 10.3.2 Low-Frequency Copper Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 10.4 Eddy Currents in Winding Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 10.4.1 Introduction to the Skin and Proximity Effects . . . . . . . . . . . . . . . . . . . . . 426 10.4.2 Leakage Flux in Windings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 10.4.3 Foil Windings and Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 10.4.4 Power Loss in a Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 10.4.5 Example: Power Loss in a Transformer Winding . . . . . . . . . . . . . . . . . . . 436 10.4.6 Interleaving the Windings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 10.4.7 PWM Waveform Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 10.5 Several Types of Magnetic Devices, Their B–H Loops, and Core vs. Copper Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.5.1 Filter Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.5.2 AC Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 10.5.3 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 10.5.4 Coupled Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 10.5.5 Flyback Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 10.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 11 Inductor Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 11.1 Filter Inductor Design Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 11.1.1 Maximum Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 11.1.2 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 11.1.3 Winding Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 11.1.4 Winding Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 11.1.5 The Core Geometrical Constant Kg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 11.2 The Kg Method: A First-Pass Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 11.3 Multiple-Winding Magnetics Design via the Kg Method . . . . . . . . . . . . . . . . . . . 465 11.3.1 Window Area Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 11.3.2 Coupled Inductor Design Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 11.3.3 First-Pass Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 11.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 11.4.1 Coupled Inductor for a Two-Output Forward Converter . . . . . . . . . . . . . 474

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xiv Contents 11.4.2 CCM Flyback Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 11.5 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 12 Transformer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 12.1 Transformer Design: Basic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 12.1.1 Core Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 12.1.2 Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 12.1.3 Copper Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 12.1.4 Total Power Loss vs. ΔB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 12.1.5 Optimum Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 12.2 A First-Pass Transformer Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 12.2.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 12.3.1 Example 1: Single-Output Isolated ´Cuk Converter . . . . . . . . . . . . . . . . . . 492 12.3.2 Example 2: Multiple-Output Full-Bridge Buck Converter . . . . . . . . . . . . 496 12.4 AC Inductor Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 12.4.1 Outline of Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 12.4.2 First-Pass AC Inductor Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 501 12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Part IV Advanced Modeling, Analysis, and Control Techniques 13 Techniques of Design-Oriented Analysis: The Feedback Theorem . . . . . . . . . . . . 509 13.1 Introduction to Part IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 13.2 The Feedback Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 13.2.1 Basic Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 13.2.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 13.3 Example: Op Amp PD Compensator Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 13.4 Example: Closed-Loop Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 13.5 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 14 Circuit Averaging, Averaged Switch Modeling, and Simulation . . . . . . . . . . . . . . . 547 14.1 Circuit Averaging and Averaged Switch Modeling . . . . . . . . . . . . . . . . . . . . . . . . 548 14.1.1 Obtaining a Time-Invariant Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 14.1.2 Circuit Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 14.1.3 Perturbation and Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 14.1.4 Indirect Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 14.2 Additional Configurations of Switch Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 14.3 Simulation of Averaged Circuit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 14.3.1 Simulation Model of the Ideal CCM Averaged Switch Network . . . . . . 568 14.3.2 Averaged Switch Modeling and Simulation of Conduction Losses . . . . 569 14.3.3 Inclusion of Switch Conduction Losses in Simulations . . . . . . . . . . . . . . 571 14.3.4 Example: SEPIC DC Conversion Ratio and Efficiency . . . . . . . . . . . . . . 572

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Contents xv 14.3.5 Example: Transient Response of a Buck–Boost Converter . . . . . . . . . . . 575 14.4 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 15 Equivalent Circuit Modeling of the Discontinuous Conduction Mode. . . . . . . . . . 585 15.1 Introduction to DCM Converter Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 15.2 DCM Averaged Switch Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 15.3 Small-Signal AC Modeling of the DCM Switch Network . . . . . . . . . . . . . . . . . . 600 15.3.1 Example: Control-to-Output Frequency Response of a DCM Boost Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 15.4 Combined CCM/DCM Averaged Switch Simulation Model . . . . . . . . . . . . . . . . 608 15.4.1 Example: CCM/DCM SEPIC Frequency Responses . . . . . . . . . . . . . . . . 611 15.4.2 Example: Loop Gain and Closed-Loop Responses of a Buck Voltage Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 15.5 High-Frequency Dynamics of Converters in DCM . . . . . . . . . . . . . . . . . . . . . . . . 618 15.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 16 Techniques of Design-Oriented Analysis: Extra Element Theorems . . . . . . . . . . . 625 16.1 Extra Element Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 16.1.1 Basic Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 16.1.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 16.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 16.2 EET Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 16.2.1 A Simple Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 16.2.2 An Unmodeled Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 16.2.3 SEPIC Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 16.2.4 Damping the SEPIC Internal Resonances . . . . . . . . . . . . . . . . . . . . . . . . . 644 16.3 The n-Extra Element Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 16.3.1 Introduction to the n-EET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 16.3.2 Procedure for DC-Referenced Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 653 16.4 n-EET Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 16.4.1 Two-Section L–C Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 16.4.2 Bridge-T Filter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 16.5 Frequency Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 16.5.1 Example: Damped Input Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 16.5.2 Other Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 17 Input Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 17.1.1 Conducted EMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 17.1.2 The Input Filter Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 17.2 Effect of an Input Filter on Converter Transfer Functions . . . . . . . . . . . . . . . . . . 679 17.2.1 Modified Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 17.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 17.2.3 Impedance Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684

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xvi Contents 17.3 Buck Converter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 17.3.1 Effect of Undamped Input Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 17.3.2 Damping the Input Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 17.4 Design of a Damped Input Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 17.4.1 Rf –Cb Parallel Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 17.4.2 Rf –Lb Parallel Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 17.4.3 Rf –Lb Series Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 17.4.4 Cascading Filter Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 17.4.5 Example: Two Stage Input Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 17.5 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 17.5.1 Modified Phase Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 17.5.2 Closed-Loop Input Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 17.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 17.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 18 Current-Programmed Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 18.1 A Simple First-Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 18.1.1 Simple Model via Algebraic Approach: Buck–Boost Example . . . . . . . 729 18.1.2 Averaged Switch Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 18.2 Oscillation for D > 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 18.3 A More Accurate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 18.3.1 Current-Programmed Controller Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 18.3.2 Small-Signal Averaged Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 18.4 Current-Programmed Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 18.4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 18.4.2 Current-Programmed Transfer Functions of the CCM Buck Converter . 755 18.4.3 Results for Basic Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758 18.4.4 Addition of an Input Filter to a Current-Programmed Converter . . . . . . 760 18.5 Simulation of CPM Controlled Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 18.5.1 Simulation Model for CPM Controlled Converters in CCM . . . . . . . . . . 764 18.5.2 Combined CCM/DCM Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . 765 18.5.3 Simulation Example: Frequency Responses of a Buck Converter with Current-Programmed Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 18.6 Voltage Feedback Loop Around a Current-Programmed Converter . . . . . . . . . . 769 18.6.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 18.6.2 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770 18.7 High-Frequency Dynamics of Current-Programmed Converters . . . . . . . . . . . . 772 18.7.1 Sampled-Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 18.7.2 First-Order Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 18.7.3 Second-Order Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 18.8 Discontinuous Conduction Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 18.9 Average Current-Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 18.9.1 System Model and Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 18.9.2 Design Example: ACM Controlled Boost Converter . . . . . . . . . . . . . . . . 791 18.10 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

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Contents xvii 19 Digital Control of Switched-Mode Power Converters . . . . . . . . . . . . . . . . . . . . . . . . 805 19.1 Digital Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 19.1.1 A/D and DPWM Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 19.1.2 Sampling and Delays in the Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . 810 19.2 Introduction to Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812 19.2.1 Integration in Continuous Time and in Discrete Time . . . . . . . . . . . . . . . 812 19.2.2 z-Transform and Frequency Responses of Discrete-Time Systems . . . . . 814 19.2.3 Continuous Time to Discrete Time Mapping . . . . . . . . . . . . . . . . . . . . . . 817 19.3 Discrete-Time Compensator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822 19.3.1 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 19.3.2 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 19.4 Digital Controller Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827 19.4.1 Discrete-Time Compensator Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 828 19.4.2 Quantization Effects, Digital Pulse-Width Modulators and A/D Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 19.5 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838 Part V Modern Rectifiers and Power System Harmonics 20 Power and Harmonics in Nonsinusoidal Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 849 20.1 Average Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 20.2 Root-Mean-Square (RMS) Value of a Waveform . . . . . . . . . . . . . . . . . . . . . . . . . 853 20.3 Power Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 20.3.1 Linear Resistive Load, Nonsinusoidal Voltage . . . . . . . . . . . . . . . . . . . . . 854 20.3.2 Nonlinear Dynamic Load, Sinusoidal Voltage . . . . . . . . . . . . . . . . . . . . . 855 20.4 Power Phasors in Sinusoidal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858 20.5 Harmonic Currents in Three-Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 20.5.1 Harmonic Currents in Three-Phase Four-Wire Networks . . . . . . . . . . . . 859 20.5.2 Harmonic Currents in Three-Phase Three-Wire Networks . . . . . . . . . . . 861 20.5.3 Harmonic Current Flow in Power Factor Correction Capacitors . . . . . . . 862 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 21 Pulse-Width Modulated Rectifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 21.1 Properties of the Ideal Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868 21.2 Realization of a Near-Ideal Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870 21.2.1 CCM Boost Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872 21.2.2 Simulation Example: DCM Boost Rectifier . . . . . . . . . . . . . . . . . . . . . . . 876 21.2.3 DCM Flyback Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878 21.3 Control of the Current Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880 21.3.1 Average Current Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 21.3.2 Current-Programmed Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 21.3.3 Critical Conduction Mode and Hysteretic Control . . . . . . . . . . . . . . . . . . 889 21.3.4 Nonlinear Carrier Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892 21.4 Single-Phase Converter Systems Incorporating Ideal Rectifiers . . . . . . . . . . . . . . 895 21.4.1 Energy Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895

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xviii Contents 21.4.2 Modeling the Outer Low-Bandwidth Control System . . . . . . . . . . . . . . . 900 21.5 RMS Values of Rectifier Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 21.5.1 Boost Rectifier Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906 21.5.2 Comparison of Single-Phase Rectifier Topologies . . . . . . . . . . . . . . . . . . 908 21.6 Modeling Losses and Efficiency in CCM High-Quality Rectifiers . . . . . . . . . . . 910 21.6.1 Expression for Controller Duty Cycle d(t) . . . . . . . . . . . . . . . . . . . . . . . . 913 21.6.2 Expression for the DC Load Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913 21.6.3 Solution for Converter Efficiency η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915 21.6.4 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916 21.7 Ideal Three-Phase Rectifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 21.8 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 Part VI Resonant Converters 22 Resonant Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933 22.1 Sinusoidal Analysis of Resonant Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938 22.1.1 Controlled Switch Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938 22.1.2 Modeling the Rectifier and Capacitive Filter Networks . . . . . . . . . . . . . . 940 22.1.3 Resonant Tank Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942 22.1.4 Solution of Converter Voltage Conversion Ratio M = V/Vg . . . . . . . . . . 943 22.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944 22.2.1 Series Resonant DC–DC Converter Example . . . . . . . . . . . . . . . . . . . . . . 944 22.2.2 Subharmonic Modes of the Series Resonant Converter . . . . . . . . . . . . . . 946 22.2.3 Parallel Resonant DC–DC Converter Example . . . . . . . . . . . . . . . . . . . . . 947 22.3 Soft Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951 22.3.1 Operation of the Full Bridge Below Resonance: Zero-Current Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951 22.3.2 Operation of the Full-Bridge Above Resonance: Zero-Voltage Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954 22.4 Load-Dependent Properties of Resonant Converters . . . . . . . . . . . . . . . . . . . . . . . 957 22.4.1 Inverter Output Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 22.4.2 Dependence of Transistor Current on Load . . . . . . . . . . . . . . . . . . . . . . . . 960 22.4.3 Dependence of the ZVS/ZCS Boundary on Load Resistance . . . . . . . . . 965 22.4.4 Another Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 22.4.5 LLC Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972 22.4.6 Results for Basic Tank Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973 22.5 Exact Characteristics of the Series and Parallel Resonant Converters . . . . . . . . . 976 22.5.1 Series Resonant Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 22.5.2 Parallel Resonant Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 22.6 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988

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Contents xix 23 Soft Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 23.1 Soft-Switching Mechanisms of Semiconductor Devices . . . . . . . . . . . . . . . . . . . 996 23.1.1 Diode Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 23.1.2 MOSFET Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 23.1.3 IGBT Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 23.2 The Zero-Current-Switching Quasi-Resonant Switch Cell . . . . . . . . . . . . . . . . . . 1003 23.2.1 Waveforms of the Half-Wave ZCS Quasi-Resonant Switch Cell . . . . . . 1005 23.2.2 The Average Terminal Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009 23.2.3 The Full-Wave ZCS Quasi-Resonant Switch Cell . . . . . . . . . . . . . . . . . . 1014 23.3 Resonant Switch Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 23.3.1 The Zero-Voltage-Switching Quasi-Resonant Switch . . . . . . . . . . . . . . . 1017 23.3.2 The Zero-Voltage-Switching Multiresonant Switch . . . . . . . . . . . . . . . . . 1019 23.3.3 Quasi-Square-Wave Resonant Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020 23.4 Soft Switching in PWM Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025 23.4.1 The Zero-Voltage Transition Full-Bridge Converter . . . . . . . . . . . . . . . . . 1025 23.4.2 The Auxiliary Switch Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 23.4.3 Auxiliary Resonant Commutated Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 23.5 Summary of Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 Appendices RMS Values of Commonly Observed Converter Waveforms . . . . . . . . . . . . . . . . . . . . . . 1037 A.1 Some Common Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 A.2 General Piecewise Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040 Magnetics Design Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 B.1 Pot Core Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044 B.2 EE Core Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 B.3 EC Core Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 B.4 ETD Core Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047 B.5 PQ Core Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 B.6 American Wire Gauge Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

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1 Introduction 1.1 Introduction to Power Processing The field of power electronics is concerned with the processing of electrical power using elec- tronic devices [1–7]. The key element is the switching converter, illustrated in Fig. 1.1. In gen- eral, a switching converter contains power input and control input ports, and a power output port. The raw input power is processed as specified by the control input, yielding the conditioned out- put power. One of several basic functions can be performed [2]. In a dc–dc converter, the dc input voltage is converted to a dc output voltage having a larger or smaller magnitude, possibly with opposite polarity or with isolation of the input and output ground references. In an ac-dc rectifier, an ac input voltage is rectified, producing a dc output voltage. The dc output voltage and/or ac input current waveform may be controlled. The inverse process, dc–ac inversion, in- volves transforming a dc input voltage into an ac output voltage of controllable magnitude and frequency. Ac-ac cycloconversion involves converting an ac input voltage to a given ac output voltage of controllable magnitude and frequency. Control is invariably required. It is nearly always desired to produce a well-regulated output voltage, in the presence of variations in the input voltage and load current. As illustrated in Fig. 1.2, a controller block is an integral part of any power processing system. High efficiency is essential in any power processing application. The primary reason for this is usually not the desire to save money on one’s electric bills, nor to conserve energy, in spite of the nobility of such pursuits. Rather, high efficiency converters are necessary because construction of low-efficiency converters, producing substantial output power, is impractical. The efficiency of a converter having output power Pout and input power Pin is Power Switching Power Fig. 1.1 The switching converter, a input converter output basic power processing block Control input © Springer Nature Switzerland AG 2020 1 R. W. Erickson, D. Maksimovi´c, Fundamentals of Power Electronics, https://doi.org/10.1007/978-3-030-43881-4_1

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