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开关电源方面的经典参考书，特别是变压器和电感器的计算和选型。能提供重要依据。

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TRANSFORMERS AND INDUCTORS FOR POWER ELECTRONICS TRANSFORMERS AND INDUCTORS FOR POWER ELECTRONICS THEORY, DESIGN AND APPLICATIONS W. G. Hurley National University of Ireland, Galway, Ireland W. H. W€olﬂe Convertec Ltd, Wexford Ireland This edition ﬁrst published 2013 # 2013 John Wiley & Sons Ltd. Registered ofﬁce John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial ofﬁces, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as expressly permitted by law, without either the prior written permission of the Publisher, or authorization through payment of the appropriate photocopy fee to the Copyright Clearance Center. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. MATLAB1 is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB1 software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB1 software. Library of Congress Cataloging-in-Publication Data Hurley, William G. Transformers and inductors for power electronics: theory, design and applications / W.G. Hurley, W.H. W€olﬂe. pages cm Includes bibliographical references and index. ISBN 978-1-119-95057-8 – ISBN 978-1-118-54464-8 – ISBN 978-1-118-54466-2– ISBN 978-1-118-54467-9 – ISBN 978-1-118-54468-6 1. Electric transformers–Design and construction. 2. Electric inductors–Design and construction. I. W€olﬂe, Werner H. II. Title. TK2551.H87 2013 621.3104–dc23 2012039432 ISBN 978-1-119-95057-8 Set in 10/12pt Times-Roman by Thomson Digital, Noida, India To Our Families Contents About the Authors xiii Acknowledgements xv Foreword xvii Preface xix Nomenclature xxiii Chapter 1 Introduction 1 1.1 Historical Context 1 1.2 The Laws of Electromagnetism 4 1.2.1 Ampere’s Magnetic Circuit Law 4 1.2.2 Faraday’s Law of Electromagnetic Induction 5 1.3 Ferromagnetic Materials 7 1.4 Losses in Magnetic Components 10 1.4.1 Copper Loss 10 1.4.2 Hysteresis Loss 11 1.4.3 Eddy Current Loss 13 1.4.4 Steinmetz Equation for Core Loss 14 1.5 Magnetic Permeability 14 1.6 Magnetic Materials for Power Electronics 16 1.6.1 Soft Magnetic Materials 17 1.6.2 The Properties of some Magnetic Materials 19 1.7 Problems 21 References 21 Further Reading 21 SECTION I INDUCTORS 23 Chapter 2 Inductance 25 2.1 Magnetic Circuits 25 2.2 Self and Mutual Inductance 30 2.3 Energy Stored in the Magnetic Field of an Inductor 34 viii 2.3.1 Why Use a Core? 2.3.2 Distributed Gap 2.4 Self and Mutual Inductance of Circular Coils 2.4.1 Circular Filaments 2.4.2 Circular Coils 2.5 Fringing Effects around the Air Gap 2.6 Problems References Further Reading Chapter 3 Inductor Design 3.1 The Design Equations 3.1.1 Inductance 3.1.2 Maximum Flux Density 3.1.3 Winding Loss 3.1.4 Optimum Effective Permeability 3.1.5 Core Loss 3.1.6 The Thermal Equation 3.1.7 Current Density in the Windings 3.1.8 Dimensional Analysis 3.2 The Design Methodology 3.3 Design Examples 3.3.1 Example 3.1: Buck Converter with a Gapped Core 3.3.2 Example 3.2: Forward Converter with a Toroidal Core 3.4 Multiple Windings 3.4.1 Example 3.3: Flyback Converter 3.5 Problems References Further Reading SECTION II TRANSFORMERS Chapter 4 Transformers 4.1 Ideal Transformer 4.1.1 No Load Conditions 4.1.2 Load Conditions 4.1.3 Dot Convention 4.1.4 Reﬂected Impedance 4.1.5 Summary 4.2 Practical Transformer 4.2.1 Magnetizing Current and Core Loss 4.2.2 Winding Resistance 4.2.3 Magnetic Leakage 4.2.4 Equivalent Circuit 4.3 General Transformer Equations Contents 35 38 39 39 40 48 51 53 54 55 55 55 55 56 57 58 58 59 61 61 64 64 69 74 75 84 89 89 93 95 96 97 98 99 100 101 102 102 105 105 107 109 Contents ix 4.3.1 The Voltage Equation 109 4.3.2 The Power Equation 112 4.3.3 Winding Loss 113 4.3.4 Core Loss 114 4.3.5 Optimization 114 4.4 Power Factor 116 4.5 Problems 121 References 122 Further Reading 122 Chapter 5 Transformer Design 123 5.1 The Design Equations 124 5.1.1 Current Density in the Windings 124 5.1.2 Optimum Flux Density unlimited by Saturation 125 5.1.3 Optimum Flux Density limited by Saturation 126 5.2 The Design Methodology 128 5.3 Design Examples 129 5.3.1 Example 5.1: Centre-Tapped Rectiﬁer Transformer 129 5.3.2 Example 5.2: Forward Converter 134 5.3.3 Example 5.3: Push-Pull Converter 140 5.4 Transformer Insulation 146 5.4.1 Insulation Principles 147 5.4.2 Practical Implementation 147 5.5 Problems 148 Further Reading 155 Chapter 6 High Frequency Effects in the Windings 159 6.1 Skin Effect Factor 160 6.2 Proximity Effect Factor 163 6.2.1 AC Resistance in a Cylindrical Conductor 165 6.3 Proximity Effect Factor for an Arbitrary Waveform 171 6.3.1 The Optimum Thickness 174 6.4 Reducing Proximity Effects by Interleaving the Windings 182 6.5 Leakage Inductance in Transformer Windings 184 6.6 Problems 187 References 193 Further Reading 193 Chapter 7 High Frequency Effects in the Core 197 7.1 Eddy Current Loss in Toroidal Cores 197 7.1.1 Numerical Approximations 200 7.1.2 Equivalent Core Inductance 201 7.1.3 Equivalent Core Resistance 202 7.2 Core Loss 204 7.3 Complex Permeability 209 7.4 Laminations 212 x 7.5 Problems References Further Reading SECTION III ADVANCED TOPICS Chapter 8 Measurements 8.1 Measurement of Inductance 8.1.1 Step Voltage Method 8.1.2 Incremental Impedance Method 8.2 Measurement of the B-H Loop 8.3 Measurement of Losses in a Transformer 8.3.1 Short-Circuit Test (Winding/ Copper Loss) 8.3.2 Open-Circuit Test (Core/ Iron Loss) 8.3.3 Core Loss at High Frequencies 8.3.4 Leakage Impedance at High Frequencies 8.4 Capacitance in Transformer Windings 8.4.1 Transformer Effective Capacitance 8.4.2 Admittance in the Transformer Model 8.5 Problems References Further Reading Chapter 9 Planar Magnetics 9.1 Inductance Modelling 9.1.1 Spiral Coil in Air 9.1.2 Spiral Coil on a Ferromagnetic Substrate 9.1.3 Spiral Coil in a Sandwich Structure 9.2 Fabrication of Spiral Inductors 9.2.1 PCB Magnetics 9.2.2 Thick Film Devices 9.2.3 LTCC Magnetics 9.2.4 Thin Film Devices 9.2.5 Summary 9.3 Problems References Further Reading Chapter 10 Variable Inductance 10.1 Saturated Core Inductor 10.2 Swinging Inductor 10.3 Sloped Air Gap Inductor 10.4 Applications 10.4.1 Power Factor Correction 10.4.2 Harmonic Control with Variable Inductance Contents 214 216 216 219 221 221 222 223 225 227 228 229 232 235 237 238 239 244 245 245 247 248 249 253 261 265 265 267 270 271 274 275 298 299 301 303 309 312 315 315 317 Contents xi 10.4.3 Maximum Power Point Tracking 323 10.4.4 Voltage Regulation 329 10.5 Problems 331 References 335 Further Reading 335 Appendix A 337 Index 341 About the Authors William Gerard Hurley was born in Cork, Ireland. He received the B.E. degree in Electrical Engineering from the National University of Ireland, Cork in 1974, the M.S. degree in Electrical Engineering from the Massachusetts Institute of Technology, Cambridge MA, in 1976 and the PhD degree at the National University of Ireland, Galway in 1988. He was awarded the D.ENG degree by the National University of Ireland in 2011. He worked for Honeywell Controls in Canada from 1977–1979, and for Ontario Hydro from 1979–1983. He lectured in electronic engineering at the University of Limerick, Ireland from 1983 to 1991 and is currently Professor of Electrical Engineering at the National University of Ireland, Galway. He is the Director of the Power Electronics Research Centre there. He served on the faculty at the Massachusetts Institute of Technology as a Visiting Professor of Electrical Engineering in 1997–1998. Prof. Hurley has given invited presentations on magnetics in Mexico, Japan, Singapore, Spain, the Czech Republic, Hong Kong, China and USA. His research interests include high frequency magnetics, power quality, and renewable energy systems. He received a Best Paper Prize for the IEEE Transactions on Power Electronics in 2000. Prof. Hurley is a Fellow of the IEEE. He has served as a member of the Administrative Committee of the Power Electronics Society of the IEEE and was General Chair of the Power Electronics Specialists Conference in 2000. Werner Hugo W€olﬂe was born in Bad Schussenried, Germany. He graduated from the University of Stuttgart in Germany in 1981 as a Diplom-Ingenieur in Electronics. He completed a PhD degree in Electrical Engineering at the National University of Ireland, Galway in 2003. He worked for Dornier Systems GmbH from 1982–1985 as a Development Engineer for power converters in space craft applications. From 1986–1988 he worked as a Research and Development Manager for industrial AC and DC power. Since 1989 he has been Managing Director of Convertec Ltd. in Wexford, Ireland, a company of the TRACOPOWER Group. Convertec develops high reliability power converters for industrial applications. He is currently an Adjunct Professor in Electrical Engineering at the National University of Ireland, Galway. Acknowledgements We would like to acknowledge Prof. John Kassakian, M.I.T. for his continued support for our magnetics work for many years. We are indebted to the numerous staff and students of the National University of Ireland, Galway, past and present who have contributed to this work. A special thanks to Dr Eugene Gath, University of Limerick for his mathematical input to the optimisation problems. The contributions of Dr Ningning Wang, Tyndall Institute and Dr Jian Liu, Volterra to the planar magnetics material is much appreciated. A special word of gratitude goes to PhD students Dr Maeve Duffy, Dr John Breslin who contributed to many of the ideas in this text. Their PhD theses form the foundations upon which this book is based. We appreciate the many insights and ideas that arose in discussions with Joe Madden, Enterprise Ireland; Prof. Dean Patterson, University of Nebraska-Lincoln; Prof. Ron Hui, University of Hong Kong; Prof. Dave Perreault, M.I.T.; Prof. Charles Sullivan, Dartmouth College; Dr Arthur Kelley and Prof Cian O ’Mathuna, University College Cork. We acknowledge the reviewers for their thorough efforts: Dr Noel Barry, National Maritime College of Ireland, Cork; Dr Ziwei Ouyang, Danish Technical University; Dr Kwan Lee, Hong Kong University and Jun Zhang, NUI, Galway. The graphics were prepared by Longlong Zhang, Zhejiang University and Francois Lemarchand, University of Nantes. Designs and solutions were provided by Ignacio Lope, University of Zaragoza. The references were assembled by Migle Makelyte, NUI, Galway. The measurements were performed by Slawomir Duda, Convertec Ltd.; Robin Draye, Universite Paul Sabatier, Toulouse and Lionel Breuil, University of Nantes. Dr Padraig O ’Cathain wrote the equations in Latex. Credit for the cover design goes to Dee Enright and John Breslin. Two individuals converted diverse notes into a cohesive manuscript and deserve special mention and thanks: Mari Moran who edited the whole document and Francois Lemarchand who completed the graphics, wrote the MATLAB programs and organised the references. We are grateful for the support of the Wiley staff in Chichester who guided us in the process of preparing the manuscript for publication. This work was supported by the Grant-in-Aid Publications Fund at the National University of Ireland, Galway and the Scholarly Publication Grants Scheme of the National University of Ireland. Finally we would like to acknowledge the support of our families: our wives (Kathleen and Ingrid) and sons and daughters (Deirdre, Fergus, Yvonne, Julian and Maureen) who have all inspired our work. Foreword It’s too big! It’s too hot! It’s too expensive! And the litany goes on, recognizable to those of us who have designed inductors and transformers, the bane of power electronics. In writing this book, Professor Hurley and Doctor W€olﬂe have combined their expertise to produce a resource that, while not guaranteeing freedom from pain, at least provides substantial anaesthesia. Ger Hurley has been engaged in research, teaching and writing about magnetic analysis and design for almost 40 years, since his time as a graduate student at MIT completing his thesis on induction heating under my supervision. And Werner W€olﬂe brings to this text, in addition to his extensive industrial experience, the beneﬁt of having been Prof. Hurley’s student. So, in some very small way, I take some very small credit for this book. Today’s demands on power electronics are unprecedented and, as their application moves ever further into the commodity marketplace (solar PV converters, EV and hybrid drives, home automation, etc.), the emphases placed on cost and efﬁciency are driving a sharp focus on the high-cost transformers and inductors in these products. As we venture into design domains, where electroquasistatics no longer obtains, and where the contradictory demands of efﬁciency and size reduction create an engineering confrontation, we need the guidance that this book provides. While many books have been written to aid the engineer in the design of magnetics, they almost exclusively present design rules and formulas without exposing the underlying physics that governs their use. Hurley and W€olﬂe, too, provide formulas and rules, but the emphasis is on understanding the fundamental physical phenomena that lead to them. As we move to higher frequencies, new geometries, new materials and new manufacturing technologies, we can no longer simply ﬁnd an appropriate formula, go to a catalogue to select a pot core, C-core or E-core, and begin winding. An understanding of electromagnetic fundamentals, modelling and analysis is now critically important to successful design – an understanding that Hurley and W€olﬂe convey most effectively. With its comprehensive scope and careful organization of topics, covering fundamentals, high-frequency effects, unusual geometries, loss mechanisms, measurements and application examples, this book is a ‘must have’ reference for the serious power electronics engineer pursuing designs that are not too big, not too hot and not too expensive. Hurley and W€olﬂe have produced a text that is destined to be a classic on all our shelves, right next to ‘The Colonel’s’ book1. A remarkable achievement. John G. Kassakian Professor of Electrical Engineering The Massachusetts Institute of Technology 1 McLyman, Colonel W.T. (1978) Transformer and Inductor Design Handbook. Marcel Dekker, Inc., New York. Preface The design of magnetic components such as transformers and inductors has been of interest to electronic and electrical engineers for many years. Traditionally, treatment of the topic has been empirical, and the ‘cook-book’ approach has prevailed. In the past, this approach has been adequate when conservative design was acceptable. In recent years, however, space and cost have become premium factors in any design, so that the need for tighter designs is greater. The power supply remains one of the biggest components in portable electronic equipment. Power electronics is an enabling technology for power conversion in energy systems. All power electronic converters have magnetic components in the form of transformers for power transfer and inductors for energy storage. The momentum towards high-density, high-efﬁciency power supplies continues unabated. The key to reducing the size of power supplies is high-frequency operation, and the bottleneck is the design of the magnetic components. New approaches are required, and concepts that were hitherto unacceptable to the industry are gaining ground, such as planar magnetics, integrated magnetics and matrix conﬁgurations. The design of magnetic components is a compromise between conﬂicting demands. Conventional design is based on the premise that the losses are equally divided between the core and the winding. Losses increase with frequency, and high-frequency design must take this into account. Magnetic components are unique, in that off-the-shelf solutions are not generally available. The inductor is to the magnetic ﬁeld what the capacitor is to the electric ﬁeld. In the majority of applications, the capacitor is an off-the-shelf component, but there are several reasons for the lack of standardization in inductors and transformers. In terms of duality, the voltage rating is to the capacitor what the current rating is to the inductor. Dielectric materials used in capacitor manufacture can be chosen so that voltage rating greatly exceeds the design speciﬁcation without incurring extra cost. In this way, a spectrum of voltage ratings can be covered by a single device. On the other hand, the current ﬂow in an inductor gives rise to heat loss, which contributes to temperature rise, so that the two speciﬁcations are interlinked. This, in turn, determines the size of the conductors, with consequential space implications. Magnetic components are usually the most bulky components in a circuit, so proper sizing is very important. Returning to the duality analogy, the dielectric material in a capacitor is to the electric ﬁeld what ferromagnetic material in a magnetic component is to the magnetic ﬁeld. In general, dielectrics are linear over a very large voltage range and over a very wide frequency range. However, ferromagnetic materials are highly non-linear and can be driven into xx Preface saturation with small deviations from the design speciﬁcations. Furthermore, inductance is a frequency-dependent phenomenon. Dielectric loss does not contribute to temperature rise in a critical way, whereas magnetic core loss is a major source of temperature rise in an inductor. The totality of the above factors means that magnetic component design is both complex and unique to each application. Failure mechanisms in magnetic components are almost always due to excessive temperature rise, which means that the design must be based on both electrical and thermal criteria. A good designer must have a sound knowledge of circuit analysis, electromagnetism and heat transfer. The purpose of this book is to review the fundamentals in all areas of importance to magnetic component design and to establish sound design rules which are straightforward to implement. The book is divided into four sections, whose sequence was chosen to guide the reader in a logical manner from the fundamentals of magnetics to advanced topics. It thus covers the full spectrum of material by providing a comprehensive reference for students, researchers and practising engineers in transformer and inductor design. The Introduction covers the fundamental concepts of magnetic components that serve to underpin the later sections. It reviews the basic laws of electromagnetism, as well as giving a historical context to the book. Self and mutual inductance are introduced and some important coil conﬁgurations are analyzed; these conﬁgurations form the basis of the practical designs that will be studied later on. The concepts of geometric mean distance and geometric mean radius are introduced to link the formulas for ﬁlaments to practical coils with ﬁnite wires such as litz wires. In Section I, the design rules for inductor design are established and examples of different types of inductors are given. The single coil inductor, be it in air or with a ferromagnetic core or substrate, is the energy storage device. A special example is the inductor in a ﬂyback converter, since it has more than one coil. This treatment of the inductor leads on to the transformer in Section II, which has multiple coils and its normal function is to transfer energy from one coil to another. Section II deals with the general design methodology for transformers, and many examples from rectiﬁers and switched mode power supplies are given. Particular emphasis is placed on modern circuits, where non-sinusoidal waveforms are encountered and power factor calculations for non-sinusoidal waveforms are covered. In a modern power converter, the transformer provides electrical isolation and reduces component stresses where there is a large input/output conversion ratio. The operation of the transformer at high frequency reduces the overall size of the power supply. There is an inverse relationship between the size of a transformer and its frequency of operation, but losses increase at high frequency. There is skin effect loss and proximity effect loss in the windings due to the non-uniform distribution of the current in the conductors. The core loss increases due to eddy currents circulating in the magnetic core and also due to hysteresis. General rules are established for optimizing the design of windings under various excitation and operating conditions – in particular, the type of waveforms encountered in switching circuits are treated in detail. A simple, straightforward formula is presented to optimize the thickness of a conducting layer in a transformer winding. Finally, Section III treats some advanced topics of interest to power supply designers. The authors feel that the book would be incomplete without a section on measurements, a topic that is often overlooked. Advances in instrumentation have given new impetus to accurate Preface xxi measurements. Practitioners are well aware of the pitfalls of incorrect measurement techniques when it comes to inductance, because of the non-linear nature of hysteresis. Planar magnetics have now become mainstream. The incorporation of power supplies into integrated circuits is well established in current practice. This book is of interest to students of electrical engineering and electrical energy systems – graduate students dealing with specialized inductor and transformer design and practising engineers working with power supplies and energy conversion systems. It aims to provide a clear and concise text based on the fundamentals of electromagnetics. It develops a robust methodology for transformer and inductor design, drawing on historical references. It is also a strong resource of reference material for researchers. The book is underpinned by a rigorous approach to the subject matter, with emphasis on the fundamentals, and it incorporates both depth and breadth in the examples and in setting out up-to-date design techniques. The accompanying website www.wiley.com/go/hurley_transformers contains a full set of instructors’ presentations, solutions to end-of-chapter problems, and digital copies of the book’s ﬁgures. Prof. W. G. Hurley and Dr Werner W€olﬂe National University of Ireland, Galway, Ireland March 2013 Nomenclature The following is a list of symbols used in this book, and their meanings. A Ac Ag AL Am Ap At Aw a a1, a2 Bmax Bo Bsat b Ceff D d d1, d2 F f G, g GMD g(x) h hc h1, h2 Iˆ Idc In In(x), Kn(x) I’rms Irms Jo Average or geometric mean radius Cross-sectional area of magnetic core Cross-sectional area of the gap Inductance per turn Effective cross-sectional area of magnetic circuit Product of window winding area Â cross-sectional area Surface area of wound transformer Bare wire conduction area Transformer turns ratio Inside and outside radii of a coil Maximum ﬂux density Optimum ﬂux density Saturation ﬂux density Winding dimension: see Figure 6.4 Effective capacitance of a transformer Duty cycle Thickness of foil or layer Height of ﬁlaments or coil centres above ferromagnetic substrate Magnetomotive force, mmf Frequency in hertz Maximum and minimum air gap lengths Geometric mean distance between coils Air-gap length at x Winding dimension: see Figure 2.14 Coefﬁcient of heat transfer by convection Coil heights in axial direction Peak value of the current waveform Average value of current RMS value of the nth harmonic of current Modiﬁed Bessel functions of the ﬁrst and second kind, respectively RMS value of the derivative of the current waveform RMS value of the current waveform Current density xxiv Nomenclature J(r) J0(x), J1(x) Kc K(f), E(f) Ki Kt Kv k ka, kc, kw kf ki kp kpn ks ku L Leff Ll Lm Ls lc M MLT m N n Pcu Pfe Po Pv p R R Rac Rdc Reff Rd Ru r1, r2 ro s T Ta Tmax t tr Current density at radius r Bessel functions of the ﬁrst kind Material parameter Complete elliptic integrals of the ﬁrst and second kind, respectively Current waveform factor 48.2 Â 103 Voltage waveform factor Coupling coefﬁcient Dimensionless constants (see Equations 3.25, 3.26 and 3.27) Core stacking factor Am/Ac Deﬁned in Figure 7.28 Power factor Ratio of the AC resistance to DC resistance at nth harmonic frequency Skin-effect factor Window utilization factor Self-inductance Effective inductance Leakage inductance Magnetizing inductance Additional coil inductance due to ferromagnetic substrate Magnetic path length of core Mutual inductance Mpean length of a turn (jvm0s) Number of turns in coil Harmonic number Copper or winding loss Iron or core loss Output power Power loss per unit volume Number of layers Average or geometric mean radius Reluctance AC resistance of a winding with sinusoidal excitation DC resistance of a winding Effective AC resistance of a winding, with arbitrary current waveform DC resistance of a winding of thickness d0 Thermal resistance Inside and outside radii of a coil Radius of bare wire Substrate separation in sandwich structure Period of a waveform Ambient temperature Maximum operating temperature Substrate thickness Rise time (0–100%) Nomenclature xxv Vrms RMS value of the voltage waveform VA Voltampere rating of winding Vc Volume of core Vo DC output voltage Vs DC input voltage Vw Volume of winding hvi Average value of voltage over time t Wa Window winding area of core Wc Electrical conduction area Wm Stored energy in a magnetic ﬁeld w Winding dimension: see Figure 6.4 Z Impedance Zi Internal impedance of a conductor z Axial separation a, b Material constants a20 Temperature co-efﬁcient of resistivity at 20C D Ratio d/d0 DB Flux density ripple DT Temperature rise DV Output voltage ripple d Skin depth d0 Skin depth at fundamental frequency dn Skin depth at the nth harmonic frequency f Flux f(k) Deﬁned in Equation 9.49 f0 Deﬁned in Equation 9.58 g Ratio of iron loss to copper loss L Deﬁned in Equation 9.36 l Flux linkage m Static or absolute permeability m0 Magnetic permeability of free space 4p Â 10–7 H/m meff Effective relative permeability mi Initial permeability minc Incremental permeability mopt Optimum value of effective relative permeability mr Relative permeability mrs Complex relative permeability h Porosity factor r20 Electrical resistivity at 20 C rw Electrical resistivity s Electrical conductivity t Time for ﬂux to go from zero to its maximum value C (5p2–1)/15 v Angular frequency (rad/s) 1 Introduction In this chapter, we describe the historical developments that led to the evolution of inductance as a concept in electrical engineering. We introduce the laws of electromagnetism which are used throughout the book. Magnetic materials that are in common use today for inductors and transformers are also discussed. 1.1 Historical Context In 1820, Oersted discovered that electric current ﬂowing in a conductor produces a magnetic ﬁeld. Six years later, Ampere quantiﬁed the relationship between the current and the magnetic ﬁeld. In 1831, Faraday discovered that a changing magnetic ﬁeld causes current to ﬂow in any closed electric circuit linked by the magnetic ﬁeld, and Lenz showed that there is a relationship between the changing magnetic ﬁeld and the induced current. Gauss established that magnetic poles cannot exist in isolation. These phenomena established the relationship between electricity and magnetism and became the basis for the science of electromagnetism. In 1865, Maxwell uniﬁed these laws in the celebrated form of Maxwell’s equations, which established the basis for modern electrical engineering. He also established the link between phenomena in electromagnetics and electrostatics. Father Nicholas Joseph Callan, who was Professor of Natural Philosophy at the National University of Ireland, Maynooth, in the 1830 s, invented the induction coil. Alexander Anderson was Professor of Natural Philosophy at the National University of Ireland, Galway in the early 1900 s and gave his name to the Anderson Bridge for measuring inductance. These individuals provide the inspiration for a textbook on magnetic design that focuses on the issues that arise in power electronics. Power electronics is an enabling technology for modern energy conversion systems and inductors and transformers are at the heart of these systems. Figure 1.1 shows a straight conductor carrying a current, i. The presence of the magnetic ﬁeld is detected by placing a freely-suspended magnet in the vicinity of the conductor. The direction of the magnetic ﬁeld (a vector) is given by the direction in which the north pole of the search magnet points. It turns out that the magnitude of the magnetic ﬁeld is constant on any circle concentric with the conductor, and its direction is tangential to that circle, given by Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 2 Transformers and Inductors for Power Electronics search magnet S magnetic field N i conductor Figure 1.1 Magnetic ﬁeld created by a current. the right hand rule – that is, a conventional (right-handed) cork screw will rotate in the direction of the magnetic ﬁeld if it is driven in the direction of the current ﬂow. It also turns out that the magnitude of the magnetic ﬁeld is proportional to the current in the conductor and is inversely proportional to the radial distance from the conductor axis. The magnetic ﬁeld around a straight conductor is illustrated in Figure 1.2. The direction of the magnetic ﬁeld as shown complies with the right hand screw rule. An alternative to the right hand screw rule for establishing the direction of the magnetic ﬁeld created by the current is to point the thumb of your right hand along the conductor in the direction of the current ﬂow, and your ﬁngers will wrap themselves around the conductor in the direction of the magnetic ﬁeld. The higher density of the lines near the conductor indicates a stronger magnetic ﬁeld in this area. The magnetic ﬁeld around the current carrying conductor is described by two vector quantities: the magnetic ﬂux density B and the magnetic ﬁeld intensity H. The magnetic ﬁeld intensity H is best explained by Ampere’s law, which expresses these observations about the current-carrying conductor in their most general form: þ ð H Á dl ¼ Jf Á nda C S ð1:1Þ Current into paper Current out of paper Figure 1.2 Magnetic ﬁeld around a current-carrying conductor. Introduction 3 φ (t) + N e turns _ Figure 1.3 Conductor in a changing magnetic ﬁeld. The closed contour C, the surface S and the normal vector are deﬁned by convention: S is the surface enclosed by C and n is the unit vector normal to S. H is the magnetic ﬁeld intensity in A/m and Jf is the current density in A/m2. The quantity on the right hand side of Equation 1.1 is the current enclosed by the contour. Figure 1.3 shows a coil with N turns in a magnetic ﬁeld. The magnetic ﬂux that links each turn of the coil is f and the electromotive force (emf) induced in the coil is given by: e ¼ ÀN df dt ð1:2Þ This states that the induced electromotive force (emf) in a coil of N turns is proportional to the rate of change of the magnetic ﬂux that links the coil. The negative sign indicates that current ﬂow in the external circuit will create an opposing magnetic ﬁeld. In a more general form, Equation 1.2 may be stated as: ð e ¼ À d B Á n da dt s ð1:3Þ The integral in Equation 1.3 represents the ﬂux linking the coil. The surface S and the normal vector are deﬁned as before. The ﬂux density B in Wb/m2 or tesla is the ﬂux per unit area inside the coil. The magnetic ﬁeld intensity H gives rise to a magnetic ﬂux density B in a medium of permeability m, so that: B ¼ mH ð1:4Þ The units for permeability are H/m and for free space m0 ¼ 4p Â 10–7 H/m. For magnetic media, m could be up to 10 000 times greater than m0. The permeability is usually presented as the product of m0 and the relative permeability mr. m ¼ mrm0 ð1:5Þ 4 Transformers and Inductors for Power Electronics Typically, relative permeability ranges from about 400 for ferrites used for power electronics applications to 10 000 for silicon steel that is used in power transformers at 50 Hz or 60 Hz. mr is taken as 1 for air. Permeability is treated in Section 1.5. 1.2 The Laws of Electromagnetism In Maxwell’s equations, the following partial differential equation relates the magnetic ﬁeld intensity H to the current density Jf and the electric displacement D: r Â H ¼ Jf þ @D @t ð1:6Þ In general, the laws of electricity and magnetism are broadly divided into quasi-static magnetic ﬁeld systems and quasi-static electric ﬁeld systems. In this book, we concern ourselves with quasi-static magnetic ﬁeld systems and the contribution of the displacement current is considered negligible. The electric ﬁeld intensity is then: r Â H ¼ Jf ð1:7Þ This is Ampere’s law in differential form. 1.2.1 Ampere’s Magnetic Circuit Law This law states that the line integral of H around any closed contour is equal to the total current enclosed by that contour, and it may be stated in the integral form of Equation 1.4: þ H Á dl ¼ i C ð1:8Þ The right hand side of Equation 1.1 is simply the current i enclosed by the contour, and corresponds to the right hand side of Equation 1.8. We have already discovered that magnitude of H is constant around a circle concentric with the axis of the conductor of Figure 1.2. Evaluation of the closed integral of Equation 1.8 for the straight conductor of Figure 1.2 gives the magnitude of the magnetic ﬁeld intensity H(r) at a radius r from the conductor: ð2p HðrÞdr ¼ i 0 ð1:9Þ 2pHðrÞ ¼ i ð1:10Þ HðrÞ ¼ i 2pr ð1:11Þ Introduction 5 and the corresponding magnetic ﬂux density in air is from Equation 1.4: BðrÞ ¼ m0 i 2pr ð1:12Þ We will meet further examples of this law later in our study of inductors. We have seen Ampere’s law in the form of a differential equation (Equation 1.7) and in the form of an integral equation (Equation 1.8). In many practical applications, it makes more sense to state the law in discrete form or in the form of a difference equation. Speciﬁcally, if there are a limited number of discrete sections with a constant value of H over a length l, then: X H Á l ¼ Ni ð1:13Þ In this form, H is summed around the loop for discrete lengths, as in the case of the closed core of an inductor or transformer, and the loop encloses a total current corresponding to N turns, each carrying a current i. We will return this topic in Chapter 2. 1.2.2 Faraday’s Law of Electromagnetic Induction In Maxwell’s equations, Faraday’s law of Electromagnetic Induction takes the form: In its integral form, this is: r Â E ¼ À @B @t þ ð E Á dl ¼ À d B Á nda C dt s ð1:14Þ ð1:15Þ This states that the integral of the electric ﬁeld intensity E around a closed loop C is equal to the rate of change of the magnetic ﬂux that crosses the surface S enclosed by C. E normally includes a velocity term in the form of v Â B, which takes into account the movement of a conductor in a magnetic ﬁeld, such as an electric motor or generator. However, for inductors and transformers, this does not arise. The differential form of Equation 1.15 is described by Equation 1.2. Nf is called the ﬂux linkage, which is the total ﬂux linking the circuit. A coil with N turns may have a ﬂux f linking each turn, so that the ﬂux linkage is l ¼ Nf ð1:16Þ The polarity of the induced electromagnetic ﬁeld (emf) is established by noting that the effect of the current caused by the emf is to oppose the ﬂux creating it; this is Lenz’s law. The induced emf opposes the creating ﬂux by generating secondary currents called eddy 6 Transformers and Inductors for Power Electronics currents. Eddy currents ﬂow in magnetic materials and in conductors operating at high frequency. These topics arise in later chapters. We will see examples of Faraday’s law and Lenz’s law in our study of transformers. In a simple magnet consisting of a north pole and a south pole, ﬂux emanating from the north pole returns to the south pole and through the magnet back to the north pole. This means that the total ﬂux emanating from a closed surface surrounding the magnet is zero. This is Gauss’ law and, in the form of Maxwell’s equations, it states that the divergence of the magnetic ﬁeld is zero: rÁB¼0 ð1:17Þ So for a closed surface S: þ B Á nda ¼ 0 S ð1:18Þ In other words the lines of magnetic ﬂux are continuous and always form closed loops as illustrated in Figure 1.2. Kirchhoff’s current law is another example of this and in Maxwell’s equations it is expressed as: r Â Jf ¼ 0 ð1:19Þ with the more recognized integral form of þ Jf Á nda ¼ 0 S ð1:20Þ This means that when a node in an electrical circuit is surrounded by a closed surface, the current into the surface is equal to the current leaving the surface. Example 1.1 Derive an expression for the magnetic ﬂux density inside a conductor of radius ro carrying current I that is uniformly distributed over the cross-section. We have already established the magnetic ﬁeld outside the conductor in Equation 1.12. The magnetic ﬁeld inside of the conductor is shown in Figure 1.4 and observes the right hand rule. Assuming that uniform current density in the conductor, the current inside a loop of radius r is: iðrÞ ¼ pr2 pr2o I We can now apply Ampere’s law on a closed contour at radius r, yielding: HðrÞ ¼ r 2pr2o I Introduction 7 B(r ) 0 ro r Figure 1.4 Magnetic ﬁeld inside a current carrying conductor. B(r ) 0 ro r Figure 1.5 Magnetic ﬁeld inside and outside a current carrying conductor. and the ﬂux density for a non-magnetic conductor (mr ¼ 1) is: BðrÞ ¼ m0r 2pr2o I Combining this result with Equation 1.12, the internal ﬂux increases linearly with radius inside the conductor, and outside the conductor the ﬂux density falls off inversely with radius. The result in shown in Figure 1.5. 1.3 Ferromagnetic Materials We have already seen that for a current-carrying coil, H is proportional to the current in the coil (Ampere’s law) and B is proportional to the integral of the voltage across the coil (Faraday’s law). The magnetic ﬂux density is related to the magnetic ﬁeld intensity by the magnetic permeability m. In a ferromagnetic material, the magnetic ﬂux density is enhanced or ampliﬁed compared to a medium such as air; the ampliﬁcation factor is the permeability m. 8 Transformers and Inductors for Power Electronics B Bsat Br (c) (a) (b) –Hc Hc –Br (d) H Figure 1.6 B-H magnetization curve. The permeability in a ferromagnetic material can be very large (unless it is limited by saturation), which means that for the same current, a greater ﬂux density is achieved in a core made of ferromagnetic material compared to that achieved in a coil in air. Taking a completely demagnetized ferromagnetic core and slowly increasing the ﬂux density by increasing the magnetic ﬁeld intensity, the B-H curve will follow the curve (a) in Figure 1.6; the details of this experiment will be described in Section 1.4.2. As the H ﬁeld is increased, the ﬂux density saturates at Bsat. If we now decrease H, the ﬂux density B will follow curve (c) in Figure 1.6. This phenomenon is called hysteresis. When the magnetizing force is returned to zero, a residual ﬂux density exists in the magnetic material. The magnitude of the ﬂux density at this point is called the remanent magnetization Br. In order to return the material to a level of zero ﬂux density, a negative value of H is required; ÀHc is called the coercive force of the material. Further decreases in H will eventually cause the material to saturate at ÀBsat and a positive coercive force þHc will again return the material to a state of zero ﬂux. Increasing H further causes B to follow curve (b). If we continue to vary H in a periodic manner, the B-H loop will settle into a ﬁxed loop as illustrated, and the closed loop is called the hysteresis loop. In its most simpliﬁed form, the hysteresis loop is characterized by the saturation ﬂux density Bsat, the coercive force Hc and the slope of the B-H curve m. If the core material were non-magnetic such as air, then the B-H magnetization curve would be linear, as shown by (d) in Figure 1.6. Clearly, the magnetic medium has a much higher ﬂux density for the same magnetizing force. The relationship in a ferromagnetic medium is not linear, although it is reasonably linear up to a value labelled Bsat in Figure 1.6. Beyond Bsat, the medium assumes the characteristic of a non-magnetic medium such as air, and the relative permeability mr approaches 1; this effect is called saturation. In a practical design, it is customary to set the maximum ﬂux density Bmax at a value below the saturation ﬂux density Bsat. The explanation of the above phenomena is rooted in the complex area of atomic physics. However, we can explain the macro effects by magnetic domains. An electron spinning around an atomic nucleus produces a magnetic ﬁeld at right angles to its orbital plane. An electron can also spin about its own axis, giving rise to a magnetic ﬁeld. These effects combine to form a magnetic moment or dipole. The atoms in ferromagnetic material form Introduction 9 B B Br Br –Hc Hc H –Hc Hc H –Br –Br (a) (b) Figure 1.7 B-H magnetization curve for: (a) hard magnetic materials; (b) soft magnetic materials. molecules, and the magnetic moments form a magnetic domain that may be thought of as a microscopic magnet. Returning to the hysteresis loop in Figure 1.6, when the material is exposed to an external magnetic ﬁeld, the magnetic domains line up with the direction of the applied ﬁeld, thus reinforcing the ﬁeld inside the material. As the ﬁeld is further increased, there is less and less opportunity for orientation of the domains and the material becomes saturated; this is part (a) of the loop. If the ﬁeld is now reduced or removed along path (c), the domains will resist and will retain at least some alignment; this is the residual or remanent magnetism, labelled Br. The resistance to realignment following saturation is the hysteresis effect. The coercive force describes the effort involved in rotating the domains to the point where the net ﬁeld is zero. This is the distinction between soft, easily realigned magnetic materials and hard, difﬁcult to realign, magnetic materials. The materials that retain most of their magnetization and are the most difﬁcult to realign are called permanent magnets. These materials are easily identiﬁed by their B-H loops, as illustrated in Figure 1.7. Hard magnetic materials have higher remanent magnetism and higher coercive force than their soft counterparts. It has been observed that when the external ﬁeld is increased at a uniform rate, the magnetization displays ﬁnite step jumps known as the Barkhausen effect, which may be explained by the sudden reorientation of the domains. This observation is often taken to validate the domain theory of ferromagnetism because, in the past, the effect was audible in speakers. The permeability of a ferromagnetic material is temperature-dependent, as is the saturation ﬂux density. This is illustrated for a Mn-Zn ferrite in Figure 1.8. The saturation ﬂux density falls from approximately 450 mT at 25 C to about 360 mT at 100 C. As with any atomic level activity, the domains are inﬂuenced by temperature. At high temperatures (above 760 C for iron), the thermal motion of the molecules is sufﬁciently agitated to block the alignment of the domains with the external ﬁeld. The relative permeability returns to approximately 1; materials with mr % 1 are called paramagnetic materials, and examples include metals such as aluminium and titanium. The temperature above which a ferromagnetic material becomes paramagnetic is called the Curie temperature; it is a property of the ferromagnetic material and is normally given by the manufacturer. For ferrites, the Curie temperature may be as low as 200 C and, for this reason, the control of the temperature in an inductor or transformer core of ferrite material is important. 10 Transformers and Inductors for Power Electronics Figure 1.8 Temperature dependence of saturation ﬂux density [Reproduced with permission of Ferroxcube]. In addition to paramagnetic and ferromagnetic materials, there is another broad class called diamagnetic materials. In these materials, the dipoles formed by the orbiting electrons oppose the applied ﬁeld in accordance with Lenz’s law. The effect is very weak; examples include copper, silver, and gold. In silver, the ﬁeld is reduced by one part in 40 000. It may be argued that the diamagnetic effect exists in all materials, but that the paramagnetic effect and the ferromagnetic effect dominate in some materials. Under AC operating conditions, the domains are constantly rotating and this requires an input of energy to overcome the molecular resistance or friction created by the changing domains. Over a complete cycle, the net energy expended appears in the form of heat, and this is known at the hysteresis loss. We will take a closer look at hysteresis loss in Section 1.4. 1.4 Losses in Magnetic Components Losses in inductors and transformers can be classiﬁed as core loss and winding or copper loss. In the core there are hysteresis loss and eddy current loss. 1.4.1 Copper Loss The resistance R of the wire used to build a winding in an inductor or transformer causes heat generation in the form of I2R loss, where I is the DC or RMS current. At high frequencies, the copper loss is aggravated by a phenomenon known as ‘skin depth’. At high frequencies, the current in a conductor bunches towards the surface of the conductor, due to the AC magnetic ﬁeld created by the conductor current. This is a direct result of Faraday’s law, whereby current will ﬂow inside a conductor to oppose the AC ﬂux, Introduction 11 in the form of eddy currents. This increases the effective resistance of the conductor, called Rac, by reducing the net area available for current ﬂow. Skin depth d can be thought of as the thickness of a hollow conductor which has the same resistance as the solid conductor with skin effect. This topic is covered in Chapter 6. We will see that the skin depth is given by: d ¼ pﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃ pf ms ð1:21Þ where f is the frequency and s is the conductivity of the conductor material. For copper at 50 Hz, d ¼ 9.5 mm and at 10 kHz, d ¼ 0.56 mm. Most of the current is con- tained within one skin depth of the surface, so that the conduction area decreases as the fre- quency increases. A useful approximation for the AC resistance of a round conductor of radius ro and DC resistance Rdc is: " # Rac ¼ Rdc 1 þ 48 ðro=dÞ4 þ 0:8ðro =dÞ4 ð1:22Þ We will take a closer look at this approximation in Chapter 6. Another effect at high frequencies is the ‘proximity effect’, where the magnetic ﬁeld of the current in one conductor interferes with that of another conductor nearby, increasing the resistance further. The limitations of skin depth and proximity effects can be avoided by using stranded wire, with each strand insulated. Each strand is transposed over the length of the wire, so that it occupies the positions of all the other strands over the cross-section; in this way, all strands are equally exposed to the prevailing magnetic ﬁeld. Transposition ensures that each strand has equal inductance over the length of the wire. Litz wire is commercially available for this purpose. In cases where only a few turns are required in a coil, thin foil may be used. 1.4.2 Hysteresis Loss At this point, we can take a closer look at the hysteresis loss that occurs as a result of the application of an AC ﬁeld to the material. Hysteresis may be measured by uniformly winding insulated wire on a toroidal core, as shown in Figure 1.9. Internal molecular friction resists lc Ac i _e + ϕ = BAc Figure 1.9 Circuit setup to measure hysteresis loss in a ferromagnetic material. 12 Transformers and Inductors for Power Electronics the continuous re-orientation of the microscopic magnets or domains in ferromagnetic materials, and energy is expended in the form of heat as the material undergoes its cyclic magnetization. The measurement of the B-H loop and hysteresis loss will be described in detail in Chapter 8. For present purposes, we can establish that the integral of the applied voltage is related to the ﬂux by Faraday’s law and the magnetic ﬁeld intensity is related to the current by Ampere’s law. The applied voltage e in Figure 1.9 is: e ¼ Ri þ N df dt ð1:23Þ The instantaneous power supplied to the circuit is: ei ¼ Ri2 þ Ni df dt ð1:24Þ Taking A as the cross-sectional area, the ﬂux f is BA. Taking l as the mean length of the path around the toroid, from Ampere’s law, H.l ¼ Ni and the term representing hysteresis in Equation 1.24 may be rewritten: ei ¼ Ri2 þ Ni ‘ A‘ dB dt ð1:25Þ or ei ¼ Ri2 þ H dB V dt ð1:26Þ where V ¼ Al is the volume of the core. The ﬁrst term on the right hand side of Equation 1.26 represents the copper loss in the windings and the second term represents the hysteresis loss. The hysteresis loss per unit volume is found by integrating the hysteresis term in Equation 1.26, yielding: ð ð H dB dt ¼ H dB dt ð1:27Þ In terms of Figure 1.10, the integral in Equation 1.27 represents a strip of width H and height dB, and for the limits a and b it represents the area between the H curve and the B axis. When the H ﬁeld returns to zero between b and c, some of this energy is returned to the circuit, the returned energy being given by the area bcd. The remaining unrecoverable energy is stored in the spinning electrons within the magnetic material, which produce the residual ﬂux. Completing the loop, the total area inside the B-H loop represents the hysteresis loss over a complete cycle. The total hysteresis loss is a product of the area of the hysteresis loop, the frequency of the applied signal and the core volume. Manufacturers normally specify loss in the form of watts/m3 as a function of B for different frequencies. Soft magnetic materials normally have Introduction 13 B d b c dB 0 H a Figure 1.10 Hysteresis loss in a ferromagnetic material. smaller hysteresis loops than hard magnetic materials, as illustrated in Figure 1.7, and consequently they have lower hysteresis loss at a given frequency. 1.4.3 Eddy Current Loss Many magnetic cores are made of materials that are, themselves, conductors of electricity; therefore, under AC conditions, currents are induced in the core, giving rise to an I2R type loss. In particular, silicon steel, used in power transformers, falls into this category. Many ferrites are classiﬁed as non-conductors but, at a high enough frequency, they are subjected to eddy currents as a direct consequence of Faraday’s law of electromagnetic induction. Cores are laminated to reduce eddy current loss. Essentially, the laminations consist of insulated sheets of magnetic material such as grain-orientated steel. A magnetic ﬁeld along the lamination induces an emf, which drives a current through a resistance path as shown in Figure 1.11. The resistance is proportional to the length and thickness of the lamination, while the induced voltage is proportional to the cross-sectional area of the lamination. Consider two equal areas, one solid and the other with n laminations, as shown in Figure 1.11. ie ie /n t Figure 1.11 Eddy current loss in a ferromagnetic material. 14 Transformers and Inductors for Power Electronics Taking ec as the induced voltage in the solid core and Rc as the resistance in the solid core, then the voltage induced per lamination is ec/n. The resistance of one lamination will be n times that of the solid area, i.e. nRc. The power loss/lamination is: ec2 n nRc ¼ 1 e2c n3 Rc ð1:28Þ For n laminations, the total loss is (1/n2)[e2c =Rc] or (1/n2) times the loss for the solid core. Clearly, laminations are very effective at reducing eddy current loss. Grain-orientated silicon steel at 50 Hz has a skin depth of 0.5 mm. A solid core of this material would contain all its ﬂux in an outer shell of 0.5 mm thickness, while a core of greater thickness would contain little ﬂux at its centre. 1.4.4 Steinmetz Equation for Core Loss The celebrated general Steinmetz equation [1] for core loss is commonly used to describe the total core loss under sinusoidal excitation: Pfe ¼ Kcf aBbmax ð1:29Þ where: Pfe is the time-average core loss per unit volume; Bmax is the peak value of the ﬂux density with sinusoidal excitation at the frequency f; Kc, a and b are constants that may be found from manufacturers’ data (examples are given in Table 1.1). For power electronics applications, non-sinusoidal excitation is common, and also AC excitation under DC bias conditions. These effects are discussed in Chapter 7. 1.5 Magnetic Permeability The magnetic ﬂux density is related to the magnetic ﬁeld intensity by the magnetic permeability in Equation 1.1. We have seen that the relationship is non-linear, as depicted in the B-H loop shown in Figure 1.6. At this point, it is worthwhile to revisit permeability and take a closer look. The magnetization density M describes the manifestation of the effects of the magnetic dipoles in the magnetic material: M ¼ XmH ð1:30Þ where Xm is called the magnetic susceptibility and is dimensionless. The permeability m may be deﬁned as: m ¼ m0ð1 þ XmÞ ð1:31Þ where m0 is the permeability of free space and, from Equation 1.4: B ¼ mH Introduction 15 B B H (a) H (b) Figure 1.12 Normal magnetization curve of a ferromagnetic material. The relative permeability mr is deﬁned in terms of Equation 1.5: m ¼ mrm0 and therefore mr ¼ 1 þ Xm ð1:32Þ Taking this approach, Xm is slightly greater than 1.0 for paramagnetic materials, slightly less than 1.0 for diamagnetic materials and much greater than 1.0 for ferromagnetic materials operating below their Curie temperature. On a macro scale, we can think of the magnetic ﬁeld intensity H being produced by electric current as the cause or driving force, and the magnetic ﬁeld intensity B is the result or effect of H. Permeability quantiﬁes the ease with which H give rise to B. One of the features of the B-H hysteresis loop shown in Figure 1.6 is that the tips of the loop are a function of the maximum value of H. For any sample of material, we can generate a whole series of loops with different values of Hmax, as shown in Figure 1.12(a). If we now plot the value of Bmax and Hmax corresponding to the tips, we have a plot of the normal magnetization curve as shown in Figure 1.12(b). The single value of permeability as deﬁned in Equation 1.4 is obtained by taking the ratio of B/H at any point on the magnetization curve. This is sometimes referred to as the static permeability or absolute permeability. When the material is saturated, this value approaches m0, the permeability of free space. Figure 1.13 shows the static permeability for the normal magnetization curve. The value of permeability at very low values of B is called the initial permeability, mi. The permeability continues to increase from mi until it reaches a maximum value mmax, and then decreases in the saturation region with a limiting value of m0. In many applications involving inductors, there may be a DC bias with an AC signal. The AC components of B-H give rise to minor B-H loops that are superimposed on the normal magnetization curve, as shown in Figure 1.14, the DC component of H is different in each loop and the amplitude of the AC component of ﬂux density is the same. The slopes of these minor loops are called incremental permeability minc. 16 Transformers and Inductors for Power Electronics μ μmax μi H Figure 1.13 Permeability versus H ﬁeld. B ΔB ΔH H Figure 1.14 Minor B-H loops in a ferromagnetic material. Complex permeability is often used to describe both the ferromagnetic effect and the attendant core loss in inductor design. It is particularly useful to describe high-frequency effects in magnetic cores, and we will return to this topic in Chapter 7. 1.6 Magnetic Materials for Power Electronics The magnetic materials used in power electronics applications can be classiﬁed into soft magnetic materials and hard magnetic materials, the main criteria for classiﬁcation being the width and slope of the hysteresis loop. Hard magnetic materials have a wide hysteresis loop, as shown in Figure 1.7(a). The coercive force Hc of hard material is higher than the corresponding value for a soft material illustrated in Figure 1.7(b). Comparing hard and soft materials, a strong ﬁeld is required to rotate the atomic level domains in a hard material so that when the material is fully magnetized a strong reverse magnetic ﬁeld is needed to decrease the magnetic ﬂux density in the material to zero. Hard magnetic materials are used in permanent magnets and mainly include an iron alloy of aluminium (Al), nickel (Ni) and cobalt (Co), sometimes called alnico; an alloy of samarium (Sm) and cobalt (SmCo) and an alloy of neodymium (Nd), iron (Fe), and boron Introduction 17 (B) designated NdFeB. Hard magnetic materials are commonly utilized for generating the magnetic ﬁeld in electrical motors and generators. Soft magnetic materials, on the other hand, can achieve a high value of ﬂux density in the presence of a relatively low value of magnetic ﬁeld intensity, as shown in Figure 1.7(b). This means that soft materials can be easily magnetized and demagnetized. The coercive force is low and the B-H loop is narrow. Soft magnetic materials include ferrites, silicon steel, nickel iron (Fe-Ni) alloy, iron-cobalt-vanadium (Fe-Co-V) alloy and amorphous alloy. 1.6.1 Soft Magnetic Materials Soft magnetic materials ﬁnd many applications in power electronics. They are widely used in high frequency transformers, in isolation transformers for driving and triggering switching components, in ﬁlter inductors for rectiﬁers, power factor correction and EMI control, and in resonance inductors for soft switching, as well as current transformers. Soft magnetic materials are classiﬁed as: Ferrites Laminated iron alloys Powder iron Amorphous alloys Nanocrystalline materials. The main features of each material may be summarized as follows: Ferrites In power electronic devices, ferrites are the most commonly applied magnetic materials. Ferrites are deep grey or black ceramic materials and are typically brittle. They are made up of iron oxide (Fe2O3), mixed with other metals such as cobalt (Co), copper (Cu), magnesium (Mg), manganese (Mn), nickel (Ni), silicon (Si) and zinc (Zn). The two most common types of ferrite are Mn-Zn and Ni-Zn. The magnetic and electrical properties vary with each alloy. For example, Ni-Zn ferrite has high electrical resistivity around 10 000 Vm, which makes it more suitable for high-frequency operation above 1 MHz. On the other hand, the lower resistivity of Mn-Zn ferrite, around 1 Vm, is offset by higher permeability and saturation ﬂux density, making it a good candidate for applications below 1 MHz. Ferrites in general have low Curie temperature, and this must be taken into account in the design. Ferrites come in many shapes and are found in inductors, transformers and ﬁlters. The saturation ﬂux density of ferrites is considerably lower than that of laminated or powdered iron cores and this restricts their use in high current applications. Laminated Iron Alloys Laminated iron is used in magnetic cores for low- to medium-frequency applications. Laminations reduce eddy currents when they are electrically insulated from each other in the core 18 Transformers and Inductors for Power Electronics of a transformer or inductor under AC operating conditions. The laminations may be stamped into any shape, with E or C shapes the most common. They are often used in cut cores, and the small gap that inadvertently arises when the cut cores are assembled reduces the likelihood of saturation. Tape-wound toroidal cores are available for higher frequencies up to 20 kHz. The iron alloys may be divided into two broad categories: silicon-iron and nickel-iron alloys. In silicon-iron alloys, silicon is added to iron in order to reduce the overall conductivity compared with iron and hence to reduce the eddy current loss in the alloy. Additionally, the effects of magnetostriction due to domain wall rotation in AC applications are reduced and this effect is manifested in reduced acoustic noise. However, silicon-iron alloys exhibit reduced saturation ﬂux density, and they are more brittle than iron. The silicon content is normally around 3% due to manufacturing considerations, although it can be as high as 6.5%. Silicon steel has been the workhorse of laminated core materials for power transformers and inductors for over 100 years. The steel is annealed in laminations and grainorientated to give maximum ﬂux density along the main axis. Silicon steel is also found in generators and motors. Nickel-iron alloy is usually made up of 80% nickel and 20% iron in laminated and tape-wound cores The alloy is characterized by low coercive force, high saturation ﬂux density, high permeability (up to 100 000) and high Curie temperature. These alloys are mainly found in current transformers, audio transformers and magnetic ampliﬁers. Increasingly, they are being used in power electronics applications up to 20 kHz. They also exhibit very low levels of magnetostriction, making them suitable for thin ﬁlm applications. The cores are often encased in non-metallic enclosures to protect the material against winding stresses. Powder Core A magnetic powder core is manufactured by having iron or iron alloy powder mixed or glued with an insulation material and compressed into a ring or toroidal form. The combination of the magnetic powder and insulating resin results in a distributed gap, which gives the material its characteristic low value of effective relative permeability. The effective permeability is a function of the size and spacing of the iron particles, their composition and the thickness of the insulation binder. The bonding material has the same effect as that of an air gap distributed along the core. The distributed gap means that high DC current can be tolerated before the iron saturates. The linear dimension of the iron particles is less than the skin depth at the desired operating frequency, resulting in low eddy current loss. The effective permeability usually ranges from 15 to 550 and the core electrical resistivity is around 1 Vm. The maximum ﬂux density may be as high as 1.5 T. The resulting inductance values tend to be very stable over a wide temperature range. Molybdenum permalloy or MPP is one of the most popular materials used in the manufacture of powdered cores, while those made of carbonyl iron are very stable up to 200 MHz. Powder cores are suited to applications where the advantages of an air gap are desired, such as energy storage inductors. They are commonly used in switched mode power supplies, high Q inductors, ﬁlters and chokes. Introduction 19 Amorphous Alloys The chemical composition of amorphous alloys contains two types of elements: ferromagnetic elements (Fe, Ni, Co and their combinations) generate the magnetic properties, while metallic elements Si, B and carbon (C) are introduced to decrease the melting point of the alloy to aid the manufacturing process. The resulting structure is similar to that of glass, and these alloys are often called metallic glass. In general, the resistivity of the amorphous alloy can reach 1.6 mVm, which is three times of silicon steel, but several orders of magnitude lower than ferrites. Their Curie temperature is around 350 C, and the saturation magnetic ﬂux density is typically up to 1.6 T, which is much higher than the corresponding values for ferrites. Relative permeability values up to 100 000 are not unusual. Amorphous alloys are also low in coercive force. The core loss is reduced by the lamination effect of the tape-wound (thin ribbon) cores. However, amorphous alloys do not share the temperature stability of nanocrystalline materials. When the temperature goes from 25 C to 250 C, the saturation ﬂux density may be reduced by as much as 30%. Iron-based amorphous alloys have found application in low-frequency transformers and high-power inductors due to their low loss compared with grain-orientated steel, but with comparable saturation ﬂux density levels. They may be found in pulse transformers, current transducers and magnetic ampliﬁers. Nickel-iron-based amorphous alloys can achieve very high relative permeability, with saturation ﬂux densities around 1 T. These are used in low- to medium-frequency transformers to replace iron cores. Cobalt-based amorphous alloys tend to be expensive, with very high relative permeability, but the maximum value of the saturation ﬂux density is below 1 T. They tend to be used in specialist applications. Nanocrystalline Materials Nanocrystalline materials contain ultra-ﬁne crystals, typically 7–20 nm in size, that are iron (Fe) based. In addition to Fe, there are traces of Si, B, Cu, molybdenum (Mo) and niobium (Nb). Among these, Cu, B and Nb are nearly always present. These materials combine the high saturation magnetic ﬂux density of silicon steels with the low loss of ferrites at high frequencies. The relative permeability is typically 20 000, and the saturation ﬂux density could be as high as 1.5 T. Core loss due to eddy currents is low, because these materials are supplied in nano-ribbon form with a thickness of 15–25 mm and an electrical resistivity of 0.012 mVm. The thin ribbon is a form of lamination and reduces eddy current loss. The nanocrystalline material is very stable over a wide temperature range, and the Curie temperature at 600 C is much higher than that for ferrites. Nanocrystalline tape-wound cores are used in applications up to 150 kHz. Their high relative permeability makes them suitable for applications in current transformers, pulse transformers and common-mode EMI ﬁlters. In some cases, nanocrystalline materials are favoured over ferrites in military applications. 1.6.2 The Properties of some Magnetic Materials Table 1.1 shows the magnetic and operating properties of some magnetic materials. Introduction 21 1.7 Problems 1.1 Derive the B-H relationship in the toroidal setup of Figure 1.9. 1.2 Derive the B-H relationship in a long solenoid with a uniformly wound coil of N turns per metre, neglect fringing at the ends of the solenoid. 1.3 Calculate the H ﬁeld in the dielectric of a coaxial cable with the following dimensions: the radius of the inner conductor is ri and the inner and outer radii of the outer conductor are roi and roo respectively. 1.4 Describe the three types of power loss in a magnetic component. Reference 1. Steinmetz, C.P. (1984) On the law of hysteresis. Proceedings of the IEEE 72 (2), 197–221. Further Reading 1. Blume, L.F. (1982) Transformer Engineering, John Wiley & Sons, New York. 2. Bueno, M.D.A. (2001) Inductance and Force Calculations in Electrical Circuits, Nova Science Publishers, Huntington. 3. Del Vecchio, R.M., Poulin, B., Feghali, P.T. et al. (2001) Transformer Design Principles: With Applications to Core-Form Power Transformers, 1st edn, CRC Press, Boca Raton, FL. 4. Erickson, R.W. (2001) Fundamentals of Power Electronics, 2nd edn, Springer, Norwell, MA. 5. Flanagan, W.M. (1992) Handbook of Transformer Design and Application, 2nd edn, McGraw-Hill, New York. 6. Georgilakis, P.S. (2009) Spotlight on Modern Transformer Design (Power Systems), 1st edn, Springer, New York. 7. Hui, S.Y.R. and Zhu, J. (1995) Numerical modelling and simulation of hysteresis effects in magnetic cores using transmission-line modelling and the Preisach theory. IEE Proceedings – Electric Power Applications B, 142 (1), 57–62. 8. Jiles, D.C. and Atherton, D.L. (1984) Theory of ferromagnetic hysteresis (invited). Journal of Applied Physics 55 (6), 2115–2120. 9. Kaye, G.W.C. and Laby, T.H. (2008) Tables of Physical and Chemical Constants, vol. Section 2.6.6, National Physical Laboratory, Teddington, Middlesex, UK. 10. Kazimierczuk, M.K. (2009) High-Frequency Magnetic Components, John Wiley & Sons, Chichester. 11. Krein, P.T. (1997) Elements of Power Electronics (Oxford Series in Electrical and Computer Engineering), Oxford University Press, Oxford. 12. Kulkarni, S.V. (2004) Transformer Engineering: Design and Practice, 1st edn, CRC Press, New York. Section One Inductors 2 Inductance1 Inductors and transformers are present in almost every power electronics circuit. Broadly speaking, inductors are dynamic energy storage devices and, as such, they are employed to provide stored energy between different operating modes in a circuit. They also act as ﬁlters for switched current waveforms. In snubber circuits, they are used to limit the rate of change of current and to provide transient current limiting. Transformers, on the other hand, are energy transfer devices, for example converting power at a high voltage and a low current to power at a lower voltage and a higher current. In a mechanical analogy, an inductor is like a ﬂywheel which stores energy, while a transformer is like a gearbox which trades speed for torque. Besides transforming voltage levels, transformers are also used to provide electrical isolation between two parts of a circuit; to provide impedance matching between circuits for maximum power transfer; and to sense voltage and currents (potential and current transformers). 2.1 Magnetic Circuits We have seen in Chapter 1 that Ampere’s law describes the relationship between the electric current and magnetic ﬁeld in a magnetic circuit. Figure 2.1 shows a coil with N turns wound on a closed magnetic core of mean length lc and cross-sectional area Ac, from Equation 2.1. þ ¼ H Á dl ¼ Ni C ð2:1Þ Recall that the direction of H around the loop is related to i by the right hand screw rule. At this point we need not concern ourselves with the shape of the core or the shape of the cross-section. For simplicity, we assume that the cross-section is constant over the length of 1 Part of this chapter is reproduced with permission from [1] Hurley, W.G. and Duffy, M.C. (1995) Calculation of self and mutual impedances in planar magnetic structures. IEEE Transactions on Magnetics 31 (4), 2416–2422. Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 26 Transformers and Inductors for Power Electronics φ i + e _ N turns Ac lc Figure 2.1 Magnetic circuit. the core. We also assume that the ﬂux density is uniform over the cross-sectional area and that the value of the magnetic ﬁeld intensity Hc is constant around the loop C. This assumption is reasonable as long as there is no ﬂux ‘leaking’ from the core; we can take leakage effects into account later. Thus, applying Equation 2.1 yields the expression for Hc in terms of the product Ni: Hclc ¼ Ni ð2:2Þ The magnetic ﬁeld intensity Hc produces a magnetic ﬂux density Bc: Bc ¼ mrm0Hc ð2:3Þ where mr is the relative permeability and m0 is the permeability of free space (m0 ¼ 4 p Â 10À7 H/m), as before. The ampere turns product Ni drives the magnetic ﬁeld around the core in a manner analogous to the emf in an electrical circuit driving current through the conductors. Ni is called the magnetomotive force or mmf, and it is normally denoted by the symbol F. The ﬂux in the core in Figure 2.1 is: f ¼ BcAc ¼ mrm0HcAc ð2:4Þ Hc may found from Equation 2.4: Hc ¼ f mrm0Ac ð2:5Þ Inductance 27 We can rewrite Ampere’s law in Equation 2.2 with this value of Hc: Ni ¼ Hclc ¼ f lc mrm0Ac ¼ fR c ð2:6Þ R is called the reluctance of the magnetic circuit. This form of the magnetic circuit equation is analogous to the Ohm’s law for electrical circuits. The mmf F is related to the reluctance R by: F ¼ fRc ð2:7Þ and the reluctance is deﬁned as: Rc ¼ lc mrm0Ac ð2:8Þ Reluctance is a measure of the effect of the magnetic circuit in impeding the ﬂow of the magnetic ﬁeld; its units are At/Wb. The inverse of reluctance is permeance. There is an analogy here between reluctance in a magnetic circuit and resistance in an electrical circuit. Just like resistance, the reluctance of the magnetic circuit is proportional to the length of the circuit and inversely proportional to the cross-sectional area, as shown by Equation 2.8. However, it is important to note that while resistance represents power dissipation in an electric circuit, reluctance is closer to electrical impedance, in the sense that it does not involve power loss. We shall see later that it is inherently connected to inductance and energy storage. The shape of a wire in an electrical circuit does not affect Ohm’s law; likewise, the shape of the magnetic circuit does not affect the magnetic circuit law in Equation 2.7. The analogy with the electrical circuit can be used to analyze more complex magnetic circuits. In the same way that individual parts of a circuit are represented by electrical resistance, a magnetic circuit can be made up of different reluctances. The total mmf is then: F ¼ Ni ¼ f1R1 þ f2R2 þ f3R3 þ . . . ð2:9Þ In many cases, the ﬂux is common to all parts of the magnetic circuit in the same way that current could be common to a set of series resistances. Equation 2.9 is then: Ni ¼ fðR1 þ R2 þ R3 þ . . .Þ ð2:10Þ We can extend the analogy to both series and parallel reluctances. One drawback with reluctance is that it is a function of the relative permeability of the material, and we saw in Chapter 1 that this is a non-linear function of the applied magnetic ﬁeld intensity in ferromagnetic materials. In the magnetic core of Figure 2.1, the ﬂux is limited by the core material, and a small current with sufﬁcient turns can easily drive the magnetic core material into saturation. However, if a gap is introduced into the core, as shown in Figure 2.2, a larger current will be required to achieve saturation of the magnetic material because of the increased reluctance 28 Transformers and Inductors for Power Electronics φ i + e _ N turns g lc Ag Figure 2.2 Magnetic circuit with an air gap. introduced by the air gap. The analysis to follow also shows that the air gap gives rise to more energy storage. This arrangement is the basis for most inductors encountered in power electronics. In the structure of Figure 2.2, there are two reluctances in series, so: F ¼ fðRc þ RgÞ ð2:11Þ The individual reluctances are given by their respective physical properties of length and cross-section; subscript c refers to the core and subscript g refers to the air gap. For the core: and for the air gap mr ¼ 1: Rc ¼ lc mrm0Ac Rg ¼ g m0Ag ð2:12Þ ð2:13Þ Where g is the length of the air gap. The equivalent reluctance of the gapped core is: R eq ¼ Rc þ Rg ¼ lc mrm0Ac þ g m0Ag ð2:14Þ At this point, we will set the cross-sectional area of the core equal to the cross-sectional area of the gap. This is a reasonable assumption when g is small compared to the dimensions of Inductance 29 the cross-section, and we will return to it later when we discuss fringing effects. Equation 2.14 becomes: ! R eq ¼ lc m0Ac 1 mr þ 1 lc=g ¼ lc meff m0Ac ð2:15Þ This means that the reluctance of the gapped core is equivalent to the reluctance of a core of length lc and relative permeability meff, with: meff ¼ 1 mr 1 þ l 1 c=g ð2:16Þ If mr ) 1, then the effective relative permeability is: meff % lc g ð2:17Þ One way to interpret this result is to say that if the air gap (length g) is distributed over the whole core of length lc, the effective permeability of the core is reduced from mr to meff. Example 2.1 The magnetic circuit of Figure 2.2 has the following dimensions: Ac ¼ Ag ¼ 12 mm Â 15 mm, g ¼ 0.5 mm, lc ¼ 10.3 cm, N ¼ 5 turns. The core material is ferrite, with relative permeability mr ¼ 2000. Find: (a) the reluctance of the core; (b) the reluctance of the air gap; (c) the ﬂux for a magnetic ﬂux density of Bc ¼ 0.2 T; (d) the magnetic ﬁeld intensity inside the core and inside the air gap. The area of the gap is 12 Â 15 Â 10À2 ¼ 1.80 cm2. The reluctances are given by Equation 2.12 and Equation 2.13 for the core and air gap respectively: (a) Rc ¼ lc mrm0Ac ¼ 10:3 Â 10À2 ð2000Þð4p Â 10À7Þð1:8 Â 10À4Þ ¼ 0:228 Â 106 At=Wb (b) Rg ¼ g m0Ag ¼ 0:5 Â 10À3 ð4p Â 10À7Þð1:8 Â 10À4Þ ¼ 2:21 Â 106 At=Wb 30 Transformers and Inductors for Power Electronics The ﬂux is given by Equation 2.4: (c) f ¼ BcAc ¼ ð0:2Þð1:8 Â 10À4Þ ¼ 3:6 Â 10À5 Wb (d) Hc ¼ Bc mrm0 ¼ 0:2 ð2000Þð4p Â 10À7Þ ¼ 79:6 A=m Hg ¼ Bc m0 ¼ 4p 0:2 Â 10À7 ¼ 159:2 Â 103 A=m As expected, the H ﬁeld is much smaller inside the core than in the gap. The discussion so far has concerned itself with series reluctance. Many practical designs have parallel reluctance paths and these can be handled by noting the analogy with the electrical circuit. 2.2 Self and Mutual Inductance The quasi-static magnetic ﬁeld system depicted in Figure 2.3 consists of a coil with N turns in a magnetic ﬁeld. If the magnetic ﬂux is allowed to vary with time, then a voltage is produced in accordance with Faraday’s law: e ¼ ÀN df dt ð2:18Þ In other words, an emf is induced in the circuit as long as the ﬂux is changing. Furthermore, the magnitude of the induced emf is equal to the time rate of change of ﬂux linkage in the circuit. Lenz’s law tells us that the induced emf due to the changing ﬂux is always in such a direction that the current it produces tends to oppose the changing ﬂux that is causing it. Figure 2.4 shows a current carrying coil near a solenoid. The solenoid has N turns. The current in the coil creates ﬂux as a direct consequence of Ampere’s law. The total ﬂux φ(t) + N e turns _ Figure 2.3 Coil of N turns in a changing magnetic ﬁeld. Inductance 31 Flux Solenoid N turns Currentcarrying coil Figure 2.4 Current carrying coil and solenoid. linking the solenoid is the sum of the ﬂuxes linking each turn of the solenoid, in this case the ﬂux linking each turn is not the same, but the total ﬂux linkage for the N turn solenoid is l ¼ f1 þ f2 þ . . . þ fn þ . . . þ fN ð2:19Þ where fn is the ﬂux linking the nth turn. If f is the average ﬂux linking each turn, then the total ﬂux linkage is: l ¼ Nf ð2:20Þ Let us now take two coils: coil 1 has a current i1, and the ﬂux linkage in coil 2 is l2, as shown in Figure 2.5. The ﬂux linkage of coil 2 due to the current i1 in coil 1 gives rise to an induced emf in coil 2, given by: e2 ¼ À d l2 dt ¼ À d l2 di1 Á di1 dt ¼ ÀM21 di1 dt ð2:21Þ M21 is called the coefﬁcient of mutual inductance of coil 2 w.r.t. coil 1. If the ﬂux linkage is proportional to the current, which is the case unless there are saturation effects at play, then d l2 di1 is constant and M21 is constant. Flux λ2 i1 Figure 2.5 Coils with mutual ﬂux. 32 Transformers and Inductors for Power Electronics A current i2 in coil 2 will establish a ﬂux linkage l1 in coil 1 and an associated emf e1. The coefﬁcient of mutual inductance is M12: e1 ¼ À d l1 dt ¼ À d l1 di2 Á di2 dt ¼ ÀM12 di2 dt ð2:22Þ Energy considerations in the combined circuits show that M12 and M21 are equal so that M12 ¼ M21 ¼ M in a linear system. In general: M ¼ l2 ¼ l1 i1 i2 ð2:23Þ Two coils have a mutual inductance of one Henry (H) if a current of one ampere (A) ﬂowing in either coil sets up a ﬂux linkage of one Weber (Wb) in the other coil. Alternatively, two coils have a mutual inductance of one Henry if a uniformly changing current at the rate of one ampere per second (A/s) in one coil induces an emf of one volt (V) in the other. The mutual inductance is dependent on the shapes of the coils involved, their proximity to each other and the number of turns in each coil. Self inductance is a special case of mutual inductance. A current carrying coil has a ﬂux linkage due to its own current i, apart from any outside sources of ﬂux. The self inductance L is deﬁned as: L¼l i ð2:24Þ A coil carrying 1 A that sets up a ﬂux linkage of 1 Wb has a self inductance of 1 H. If the current changes at a steady rate of 1 A/s and induces an emf of 1 V, then the self inductance is 1 H. The circuit symbol for inductance is shown in Figure 2.6. The question of the polarity of the voltage across the inductor can be resolved by adopting the convention shown in Figure 2.6 for the polarity of e: e ¼ dl ¼ L di dt dt ð2:25Þ L i L i + e _ + e _ Figure 2.6 Polarity convention and symbol for inductance. Inductance 33 If the inductor represented by the inductance in Figure 2.6 were connected to an external circuit, with i increasing with time, then e would generate an induced current out of the positive terminal through the external circuit and into the negative terminal, correctly opposing i as expected from Lenz’s law. Let us take a closer look at Equation 2.24, using Equations 2.20, 2.4 and 2.2: L ¼ l i ¼ Nf i ¼ N2BA Ni ¼ N2BA Hl ¼ N2mrm0A l ¼ N2 R ð2:26Þ This result shows that the self inductance may be computed from the total number of turns and the reluctance of the magnetic circuit. This approach may be applied to the inductance of a gapped core: L ¼ N2 Rc þ Rg ¼ N2 Rg 1 1 þ meff mr ð2:27Þ Thus, the value of the inductance can be made independent of the magnetic properties of the core material if mr ) meff, as long as the core material has not saturated. In saturation, there is no guarantee that the inequality mr ) meff holds. Example 2.2 For the magnetic circuit in Example 2.1, ﬁnd: (a) the self inductance and (b) the induced emf for a sinusoidal ﬂux density of magnitude 0.2 T at 50 kHz. (a) The self inductance is found from Equation 2.27 The reluctance of the core and the reluctance of the air gap were found in Example 2.1. The reluctance of the series combination of the core and air gap is: R ¼ Rc þ Rg ¼ ð0:228 Â 106Þ þ ð2:21 Â 106Þ ¼ 2:438 Â 106 At=Wb And the self inductance for 5 turns is: L ¼ 52 2:438 Â 106 Â 103 ¼ 0:01 mH In manufacturers’ data books, the quantity 1/R (inverse of reluctance) is sometimes called the AL value or speciﬁc inductance; this is the inductance for a single turn, and the inductance for N turns is L ¼ ALN2. 34 Transformers and Inductors for Power Electronics (b) The core is operating at 50 kHz and the induced emf is: e ¼ N df dt ¼ NAc dB dt ¼ NAcvBmaxcosðvtÞ ¼ ð5Þð1:8 Â 10À4Þðð2pÞð50 Â 103ÞÞð0:2Þcosð2p50 Â 103tÞ ¼ ð56:55Þcosð314:159 Â 103tÞ V 2.3 Energy Stored in the Magnetic Field of an Inductor One way to interpret Equation 2.25 is that a voltage is required to change the current when inductance is present – or, in other words, the current in the inductor has inertia. A consequence of this state of affairs is that electrical energy is stored in the magnetic ﬁeld of the inductor. Consider the inductor represented in Figure 2.7, with a current i and voltage e. The power or rate of energy ﬂow into the winding is: p ¼ ie ¼ i dl dt ð2:28Þ The change in magnetic ﬁeld energy between t1 and t2 is: ð t2 ð l2 Wm ¼ pdt ¼ i dl t1 l1 Noting that l ¼ Li Wm ¼ ði 0 Li di ¼ 1 2 Li2 ð2:29Þ ð2:30Þ i HC + e N _ Figure 2.7 A closed magnetic circuit. Inductance 35 Furthermore, l ¼ Nf and idl ¼ Nidf, the mmf Ni ¼ Hclc (Ampere’s law), f ¼ BcAc. Substituting these relationships into Equation 2.30 allows us to express the stored energy in terms of B and H: ðB2 Wm ¼ Aclc Hc dBc B1 ð2:31Þ Aclc ¼ Vc is the volume of the core, so that the stored energy per unit volume is the integral of the magnetic ﬁeld intensity between the limits of the ﬂux density. This is similar to the expression we had in Chapter 1 for hysteresis loss. For an inductor with ﬂux B, the integration of Equation 2.31 yields: Wm ¼ B2V c 2mrm0 ð2:32Þ where Vc is the volume of the core. The result in Equation 2.32 may be applied to a gapped core: Wm ¼ B2 V c 2mrm0 þ B2 V g 2m0 ð2:33Þ where Vg ¼ Acg is the volume of the gap. Equation 2.33 may be rewritten: Wm ¼ B2V g 2m0 ! 1 þ meff mr ð2:34Þ with meff ¼ lc/g as before. The result in Equation 2.34 shows that, in a gapped inductor, most of the energy is stored in the gap itself. This arises because the H ﬁeld is diminished inside the core (see Example 2.1), and it raises the obvious question: why do we need a core if there is so little energy stored in it? 2.3.1 Why Use a Core? Consider a core made of ferrite material which saturates at about 0.3 T. The B-H magnetization curve of the ferrite has the non-linear characteristic shown in Figure 2.8, with saturation occurring at about H ¼ 2 A/m. The B-H characteristic of the air gap is linear, with mr ¼ 1. If the air gap were spread out over the whole core, then for the same reluctance: R ¼ g ¼ lc m0Ag meff m0Ag ð2:35Þ 36 Transformers and Inductors for Power Electronics B(T) Ferrite 0.3 μ0 μr μ0 μeff μ0 gapped core air μ0 0 10 20 30 H(A/m) Figure 2.8 B-H magnetization curve of a ferrite material. For a typical gapped core with lc ¼ 100g, the effective relative permeability of the core is 100. The gap dominates the core characteristics if mr ) meff, as shown by Equation 2.27. The B-H characteristic of the combination is shown in Figure 2.8, which is dominated by the air gap but eventually reaches saturation. It is clear from the combination that saturation occurs at a much higher value of H, in this case about 30 A/m. This means that much higher currents can be tolerated before the onset of saturation at the expense of a lower value of inductance. From Equation 2.17, the slope of the B-H curve in the combination is meffm0. Recall that Equation 2.26 shows that higher reluctance leads to lower overall inductance. Remembering that meff ¼ lc/g, it is worth noting that increasing g reduces the slope of the B-H characteristic and allows more stored energy even though the inductance is reduced (see Equation 2.27). The energy stored in the magnetic ﬁeld is the area to the left of the B-H curve from the origin up to the operating point. Evidently, the area in question is increased by the presence of the air gap and more energy can be stored in the gapped core. The properties of most core materials vary with temperature, ﬂux level, sample and manufacture. The gap reduces the dependence of the overall inductance on these parameters to make the value of inductance more predictable and stable. The issues associated with saturation, hysteresis and remanance are mitigated by the introduction of an air gap as long as meff ) mr. The advantages of the air gap can thus be summarized: Saturation occurs at higher values of current. More energy can be stored, which is the basic function of an inductor. The inductor is less susceptible to variations in the magnetic properties of the core. In the case of a transformer, it is energy transfer and not energy storage that counts, so that although a gap is not normally used, a small gap may store some energy to feed parasitic capacitances. With problems of saturation, hysteresis and eddy current losses, and with the expense and weight of magnetic materials, one might ask why not build an air-core inductor with sufﬁcient turns? First, the overall size of an air-core inductor is much larger. Also, the lower reluctance of a ferromagnetic core ensures that most of the ﬂux is inside the core and in an Inductance 37 air-core coil there is more leakage ﬂux, which may give rise to electromagnetic interference (EMI). For a given value of inductance and a given overall volume, a lot fewer turns are required when a core is used and, therefore, there is less winding resistance. Example 2.3 For the magnetic circuit in Example 2.1, ﬁnd the energy stored (a) in the core; and (b) in the air gap for B ¼ 0.2 T. (a) The energy stored in the core is: Wc ¼ B2 V c 2mrm0 ¼ ð0:2Þ2ð1:8 Â 10À4Þð10:3 Â 10À2Þ ð2Þð2000Þð4p Â 10À7Þ Â 103 ¼ 0:148 mJ (b) The energy stored in the gap is: Wg ¼ B2 V g 2m0 ¼ ð0:2Þ2ð1:8 Â 10À4Þð0:5 Â ð2Þð4p Â 10À7Þ 10À3Þ Â 103 ¼ 1:43 mJ As expected, most of the energy in this inductor is stored in the air gap. The total energy stored is 1.578 mJ. Example 2.4 In this example, we take a look at a more practical structure that is often encountered in power electronics. The inductor shown in Figure 2.9 consists of two E-shaped cores with a gap in the centre leg. Calculate the equivalent reluctance of the structure. μr =2000 44 2 2 1 44 0.5 32 1 32 6 10 12 10 6 15 Figure 2.9 Core dimensions given in mm. 38 Transformers and Inductors for Power Electronics R1 Rg R2 + R2 Ni _ R1 Figure 2.10 Equivalent magnetic circuit. The equivalent magnetic circuit is shown in Figure 2.10. R1 represents the reluctance of the centre leg on either side of the air gap. R2 represents the reluctance of the outer leg and Rg represents the reluctance of the air gap. R1 ¼ l1 mrm0Ac ¼ ð2000Þð4p 19 Â 10À3 Â 10À7Þð12Þð15 Â 10À6Þ ¼ 0:42 Â 105 At=Wb R2 ¼ l2 mrm0Ac ¼ ð2000Þð4p 76:5 Â 10À3 Â 10À7Þð6Þð15 Â 10À6Þ ¼ 3:38 Â 105 At=Wb Rg ¼ g m0Ac ¼ ð4p Â 0:5 Â 10À3 10À7Þð12Þð15 Â 10À6Þ ¼ 22:1 Â 105 At=Wb The equivalent reluctance is found by analogy with an electrical circuit with resistances. R eq ¼ À Rg þ Á 2R1 jj R2 2 ¼ ½ð22:1 þ ð2Þð0:42ÞÞjjð3:38=2Þ Â 105 ¼ 1:57 Â 105 At=Wb 2.3.2 Distributed Gap The discussion so far has been concerned with cores with discrete gaps. Powder iron cores consist of high-permeability magnetic alloy beads glued together with non-ferromagnetic material. The non-ferromagnetic material plays the role of the gap except that, in this case, it is distributed over the whole core. The deﬁnition of effective relative permeability remains valid and the value for powder iron cores is typically in the range of 15–550. Distributing the gap in this manner has a number of beneﬁts: There is less ﬂux fringing compared to a discrete gap. EMI effects are mitigated. Winding loss due to fringing effects is reduced. Core assembly is mechanically more stable. Inductance 39 However, ferrites with discrete gaps tend to be less expensive and there is a greater variety of shapes and sizes. There is another important distinction. In a gapped ferrite, when the onset of saturation is reached (meff % mr), the permeability of the ferrite rolls off quite sharply, resulting in a dramatic lowering of inductance. With the distributed gap, this effect is less severe, because the intrinsic permeability of the beads is much higher than that of ferrite to begin with. The ferrite is also more dependent on temperature than the beads, and saturation in the ferrite occurs at a lower value of H at higher temperatures. 2.4 Self and Mutual Inductance of Circular Coils 2.4.1 Circular Filaments The fundamental building block for inductance calculations in coils is the mutual inductance between two ﬁlaments, as shown in Figure 2.11 [1]. A ﬁlamentary energizing turn of radius a at z ¼ 0 carries a sinusoidal current represented by if(t) ¼ Ifejvt. A second ﬁlamentary turn, of radius r, is located an axial distance z from the energizing turn. For a magnetoquasistatic system, the following forms of Maxwell’s equations hold in a linear homogeneous isotropic medium: r Â H ¼ Jf r Â E ¼ À @B @t ð2:36Þ ð2:37Þ On the basis of cylindrical symmetry, the following identities apply to the electric ﬁeld intensity E and the magnetic ﬁeld intensity H: Er ¼ 0; Ez ¼ 0; @Ef @f ¼ 0 Hf ¼ 0; @Hr @f ¼ 0; @Hz @f ¼ 0 ð2:38Þ ð2:39Þ z z= 0 a b r EΦ iΦ Figure 2.11 Circular ﬁlaments. 40 Transformers and Inductors for Power Electronics Maxwell’s equations reduce to: @Hr @z À @Hz @r ¼ Ifdðr À aÞdðzÞ @Ef @z ¼ jvm0Hr 1 r @ðrEfÞ @r ¼ Àjvm0Hz ð2:40Þ ð2:41Þ ð2:42Þ Eliminating H gives the following result for Ef: @2Ef @z2 þ @2Ef @r2 þ 1 r @Ef @r À Ef r2 ¼ jvm0Ifdðr À aÞdðzÞ ð2:43Þ where d(r-a) and d(z) are Dirac unit impulse functions which locate the energizing current at the co-ordinates (a, 0). The mutual inductance between the two ﬁlaments is found from the solution to the Bessel equation in Equation 2.43 and is well known; it was ﬁrst proposed by Maxwell [2]: ð1 M ¼ m0par J1ðkrÞJ1ðkaÞeÀkjzj dk 0 ð2:44Þ where J1 is a Bessel function of the ﬁrst kind. The solution of Equation 2.43 may also be written in the form of elliptic integrals given by Gray [3]: M ¼ pﬃﬃﬃﬃ m0 ar 2 f 1 À f 2 Kðf Þ À Eðf ! Þ 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f¼ 4ar z2 þ ða þ rÞ2 ð2:45Þ ð2:46Þ where K( f ) and E( f ) are complete elliptic integrals of the ﬁrst and second kind, respectively. 2.4.2 Circular Coils The mutual inductance formula Equation 2.44 relates to ﬁlaments. As designers, however, in practice, we need formulae relating to coils, i.e. groups of turns. Figure 2.12 indicates two such coils. In this ﬁgure, the coils are shown to be of rectangular cross-section (the shape of cross-sections normally of interest in power electronics, and it is understood that coil 1 has N1 turns and coil 2 has N2 turns. The formula for mutual inductance between ﬁlaments in air can be formally modiﬁed to apply to coils of rectangular cross-section. This is achieved by integrating the ﬁlament Inductance 41 z z =0 N1 turns a1 a2 h1 r2 r1 h2 N2 turns w2 w1 Figure 2.12 Circular coils. formula over the coil cross-section, assuming uniform current density across the crosssection. We shall see later, in Chapter 9, that this is not necessarily true when the widthto-height ratio of the coil cross-section is very large and must be taken into account. Figure 2.13 shows the arrangement and dimensions of two illustrative coils, which are assumed to be circular and concentric. The impedance formula for this arrangement is derived from the ﬁlamentary formula by an averaging process. Consider a representative point (t1,a) within the cross-section of coil 1 and a representative point (z þ t2, r) within the cross-section of coil 2, as indicated in Figure 2.13. If these two points are taken to deﬁne two ﬁlaments, Equation 2.44 gives the following formula for the mutual inductance between them: ð1 M ¼ m0par J1ðkrÞJ1ðkaÞeÀkjzþt2Àt1j dk 0 ð2:47Þ The basic mutual inductance formula for coils, rather than ﬁlamentary turns, may then be obtained from Equation 2.46 by taking this mutual inductance as the average effect of all τ1 N1 turns τ2 N2 turns a1 a a2 w1 h1 r1 r r2 w2 h2 z Figure 2.13 Filaments in circular coils. 42 Transformers and Inductors for Power Electronics such ﬁlamentary turns within the structure of Figure 2.13. This assumes uniform current density over either coil cross-section, an assumption which is justiﬁed in most practical cases. However, an exception is planar magnetic structures, which will be dealt with in Chapter 9. The proposed formula is: M0 ¼ N1N2 h1w1h2w2 ð a2 a1 ðr2 r1 ð w1 2 Àw1 2 ð w2 2 Àw2 2 M da dr dt1 dt2 ð2:48Þ The integration of Equation 2.48 has been carried out in the past, but the exact integral solution converges slowly and an alternative solution, such as the elliptic integral formula, does not exist. Approximations to the integral formula for coils in air have been developed by Gray [3] and Lyle [4]. Dwight [5] and Grover [6] have published extensive tables to calculate the inductance between coil sections. Maxwell [2] proposed that the elliptic integral formula for ﬁlaments could be used if the coil separation is replaced by the Geometric Mean Distance (GMD) between the coil sections. Basically, if conductor 1 is divided into n identical parallel ﬁlaments, each carrying an equal share of the current, and if conductor 2 is similarly divided into m identical parallel ﬁlaments, then the GMD between the two conductors is the mnth root of mn terms. These are the products of the distances from all the n ﬁlaments in conductor 1 to all the m ﬁlaments in conductor 2. Using GMD to replace coil separation z in Equation 2.45 gives: M ¼ pﬃﬃﬃﬃﬃﬃ m0 AR 2 f 1 À f 2 Kðf Þ À Eðf ! Þ 2 ð2:49Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f¼ 4AR GMD2 þ ðA þ RÞ2 ð2:50Þ The and accuracy R are the of Equation 2.49 geometric means doef ptheendcsoiolnrathdeii,cworirtehcAt c¼hopicﬃaeﬃﬃ1ﬃoﬃaﬃfﬃ2ﬃﬃvaanludeRs f¼orpAﬃr,ﬃ1ﬃRﬃrﬃﬃ2ﬃﬃa. nInd GMD. A practice, there is negligible error in taking A and R as the mean radii of the respective coils. Self inductance of a circular coil of rectangular cross-section is a special case of Equation 2.49. In this case, GMD is interpreted as the geometric mean distance of the coil from itself and is given by Lyle [4]: lnðGMDÞ ¼ lnðdÞ À u ð2:51Þ where: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d ¼ h2 þ w2 ð2:52Þ Inductance 43 1 A h 2 B c 3 A h w Figure 2.14 GMD between coils of equal area. and: with: u ¼ u þ v þ 25 À 2 ðx þ yÞ 12 3 u ¼ w2 h2 d2 ln w2 ð2:53Þ ð2:54Þ and: v ¼ h2 w2 d 2 ln h2 ð2:55Þ and: x ¼ w tanÀ1 h h w ð2:56Þ and: y ¼ h tanÀ1w w h ð2:57Þ A simple approximation for the GMD of a coil from itself is GMD % 0.2235 (h þ w). For coils of equal section, as shown in Figure 2.14, the GMD between the coils may be obtained using the formula for the GMD of a coil from itself. We want to calculate the GMD between areas 1 and 3, each having an area A ¼ w Â h. Area 2, between Sections 2.1 and 2.3, has an area B ¼ w Â c. Rs is the GMD of the total area (1 þ 2 þ 3) from itself. R1 is the GMD of area 1 from itself (the same as R2) and R12 is the 44 Transformers and Inductors for Power Electronics w1 2 h1 1 r2 R r1 w2 3 4 h2 a2 A a1 Figure 2.15 Lyle’s method of equivalent ﬁlaments. GMD of the combined area 1 and 2 from itself. It may be shown that R13, the GMD between Sections 2.1 and 2.3, is given by Grover [6]: ln ðR13Þ ¼ ð2w þ 2w2 cÞ ln ðRsÞ À ln ðR1Þ À c2 2w2 ln ðR2Þ À 2c w ln ðR12Þ ð2:58Þ where: R1 ¼ 0:2235ðw þ hÞ R2 ¼ 0:2235ðw þ cÞ R12 ¼ 0:2235ðw þ h þ cÞ Rs ¼ 0:2235ðw þ 2h þ cÞ ð2:59Þ The elliptic integral formula can also be used for coils of unequal sections with Lyle’s method [4], where each coil is replaced by two equivalent ﬁlaments, each carrying half the ampere-turns of the actual coil, as shown in Figure 2.15. For w < h, the equivalent ﬁlaments are co-planar, with radii r þ a and r À a: r ¼ R1 þ w21 24R2 ð2:60Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a ¼ h21 À w21 12 ð2:61Þ Inductance 45 For w > h the equivalent ﬁlaments have the same radius at a distance b on either side of the mid-plane of the coil. r ¼ A1 þ h22 24A2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b ¼ w22 À h22 12 ð2:62Þ ð2:63Þ The mutual inductance is then given by: M ¼ N1N2 M13 þ M14 þ 4 M23 þ M24 ð2:64Þ where the Mij terms are given by the Elliptic Integral Formula (2.45). Example 2.5 For the sections shown in Figure 2.16, calculate the self inductance of Section 2.0 and the mutual inductance between Sections 2.1 and 2.2 and between Sections 2.1 and 2.4, with the dimensions shown. The required dimensions for the mutual inductance calculations are: Coils 1 and 2: Coils 1 and 4: r1 ¼ a1 ¼ 1:15 mm; r2 ¼ a2 ¼ 1:75 mm; h1 ¼ h2 ¼ 15 mm z ¼ 0: r1 ¼ 2:00 mm; r2 ¼ 2:60 mm; h1 ¼ 15 mm; a1 ¼ 2:00 mm; a2 ¼ 2:60 mm; h2 ¼ 15 mm z ¼ 55 mm: Formulae 2.49 and 2.58 may be used to ﬁnd L1 and M12. Formula 2.64 may be used to ﬁnd M14. 5.2 mm 2.3 mm 2 3 1 4 600 µ 250 µ 40 µ t 15 µ Figure 2.16 Planar coils. 46 Transformers and Inductors for Power Electronics The MATLAB program is listed at the end of this chapter, yielding the follow results: L1 ¼ 4.316 nH M12 ¼ 4.009 nH M14 ¼ 2.195 nH Example 2.6 Calculate the inductance of a two-wire line consisting of a go-and-return circuit, as shown in Figure 2.17. The straight conductor is round, with a radius ro, and the separation is D. Assume the length of the conductor is inﬁnite in both directions. In Example 1.1, we found expressions for the ﬂux density inside and outside a straight round conductor. Consider an annular ring of inside radius r1 and outside radius r2 concentric with conductor 1 as shown. The ﬂux linking the annular element between r1 and r2 of length one metre is found from B(r) in Equation 1.12: df ¼ m0I dr 2pr Integrating from r1 to r2 yields the ﬂux linkage (in effect, the number of turns N ¼ 1) due to the current I as: l12 ¼ ð r2 r1 m0I 2pr dr ¼ m0I 2p ln r2 r1 The corresponding inductance per metre is then: L12 ¼ m0 ln 2p r2 r1 r2 r1 ro ro D Figure 2.17 Two-wire transmission line. Inductance 47 This only accounts for the ﬂux linkage outside the conductor. We need to add the contribution from the internal ﬂux, based on the ﬂux density derived in Example 1.1. Taking an annular section inside the conductor, the ﬂux density due to the current inside the radius r is, by Equation 1.23: The ﬂux f is: BðrÞ ¼ m0r 2pr2o I df ¼ m0rI 2pr2o dr In calculating the ﬂux linkage, the number of turns is interpreted as the proportion of the total current that is linked (N ¼ 1 represents the total current), i.e. ðpr2=pr2oÞ. This yields the ﬂux linkage as: dl ¼ m0r3I 2pr4o dr Performing the integration from r ¼ 0 to r ¼ ro results in the internal inductance per metre in terms of the exponential function: Lint ¼ m0 8p ¼ m0 ln e1=4 2p Combining the internal and external contributions gives a compact expression for the inductance at a radius r2 outside a conductor of radius ro: L12 ¼ m0 2p ln r2 roeÀ1=4 ¼ m0 ln 2p r2 r0o Taking the return conductor at r2 ¼ D, the inductance resulting from the ﬂux linking conductor 2 due to the current is 1 is given by this expression, with r2 ¼ D. The inductance resulting from the ﬂux linking conductor 1 due to the current in conductor 2 is given by the same expression. The two components of ﬂux linkage add together and the total inductance per unit length becomes: L12 ¼ m0 p ln D r0o r0o is called the geometric mean radius (GMR) or self Geometric Mean Distance (GMD) of the round conductor cross-section from itself. We have already used GMD to calculate self and mutual inductance of circular coils. Strictly speaking, the concept only applies to straight conductors; it works for circular coils with rectangular cross-sections when the radius of curvature is much greater than the dimensions of the coil cross-section. 48 Transformers and Inductors for Power Electronics Ic Ac i _ e+ φ = BAc Figure 2.18 Circuit setup to measure hysteresis loss in a ferromagnetic material. Example 2.7 Calculate the inductance of the coil on a toroidal core shown in Figure 2.18 of mean length lc and with a cross-sectional area Ac. The relative permeability of the core material is mr. The toroid is shown with N turns tightly wound over the whole core to ensure that there is no leakage of ﬂux from the core. We begin by applying Ampere’s law along the contour of the mean path of the core: Hc ¼ NI lc The corresponding ﬂux linkage is: and the self inductance of the toroid is: l ¼ mrm0N2AcI lc L ¼ mrm0N2Ac lc 2.5 Fringing Effects around the Air Gap The expression for self inductance of a gapped core in Equation 2.27 assumes that there are no fringing effects around the air gap. In practice, the ﬂux lines do not cross the gap in straight lines, but rather bulge out around the gap as shown in Figure 2.19. Fringing increases loss in neighbouring conductors, and the overall inductance is increased because the reluctance of the air gap is reduced. This fringing of the ﬂux ﬁeld is a function of the geometry of the gap and can be evaluated by resorting to magnetic ﬁeld plotting using ﬁnite element analysis tools. In practice, a useful rule of thumb is to add the gap length to each dimension in the crosssection. For a rectangular cross-section with dimensions a by b, the cross-section would Inductance i + N e turns _ 49 g Figure 2.19 Fringing effect in an air gap. become (a þ g) by (b þ g), where g is the length of the air gap and does not change. The overall inductance is then increased because the reluctance of the air gap is reduced: L0 ¼ N2 R c þ R 0 g % N2 R 0 g ¼ L A0g Ag ¼ ða L þ gÞðb ab þ gÞ ð2:65Þ We have inherently assumed that g ( a or b in a gapped core, so we can neglect the g2/ab term and the expression is reduced to: L0 % L 1 þ a þ b g ab ð2:66Þ Finally, in a square cross-section, the self inductance is increased by fringing by the factor (1 þ 2g/a), so if g is 5% of a, then the overall inductance is estimated to increase by 10%. In the case of a round core, the diameter should be increased by g. Referring to Figure 2.20, the reluctance term due to the fringing (Rfringing) is parallel to the reluctance of the gap without fringing, and the parallel combination reduces the overall reluctance of the gap. Fringing ﬂux can introduce eddy currents into neighbouring conductors, so winding placement must be considered to reduce associated loss. + Ni _ Rgap(ideal) Rcore Rfringing Figure 2.20 Equivalent magnetic circuit for fringing effects in an air gap. 50 Transformers and Inductors for Power Electronics Example 2.8 Estimate the increase in self inductance of the structure in Example 2.2 due to fringing effects around the air gap. In Example 2.2, the cross-sectional dimensions of the gap are 12 mm Â 15 mm and the gap length is 0.5 mm. The new value of inductance, taking fringing into account, is from Equation 2.66: L0 % L 1 þ a þ b g ab ¼L 1 þ 12 þ 15 ð12Þð15Þ ð0:5Þ ¼ 1:075L This represents an increase of 7.5% in the self inductance. Let us look at this in more detail by considering four options: 1. Base inductance on the equivalent reluctance without fringing at the gap: L ¼ 52 2:438 Â 106 Â 106 ¼ 10:3 mH 2. Base inductance on the equivalent reluctance with fringing at the air gap. Taking the fringing into account, the reluctance of the air gap is: R 0 g ¼ 0:5 Â 10À3 ð4p Â 10À7Þð12:5Þð15:5 Â 10À6Þ ¼ 2:05 Â 106 At=Wb Req ¼ ð0:228 þ 2:05Þ Â 106 ¼ 2:278 Â 106 At=Wb L ¼ N2 R 0 eq ¼ 52 2:278 Â 106 Â 106 ¼ 11:0 mH 3. Base inductance on the air gap approximation without fringing: L ¼ N2 Rg ¼ 52 2:21 Â 106 Â 106 ¼ 11:3 mH 4. Base inductance on the air gap approximation but include fringing: L ¼ N2 R 0 g ¼ 52 2:05 Â 106 Â 106 ¼ 12:2 mH This example shows that there is little merit in making allowance for the fringing effect around the air gap while at the same time neglecting the reluctance of the core. Or, if we look at it in another way, the effect of the core on the inductance may be just as important at the fringing effect. Inductance 51 3 cm 8cm 12 cm B 1.2 1.0 100 400 H At/m (a) (b) Figure 2.21 Problem 2.2 toroidal core. 2.6 Problems 2.1 List ﬁve factors that affect the inductance of a coil. 2.2 A toroidal core has the dimensions shown in Figure 2.21. It is wound with a coil hav- ing 100 turns. The B-H characteristic of the core may be represented by the linearized magnetization curve shown (B is in T, H in At/m). (a) Determine the inductance of the coil if the ﬂux density in any part of the core is below 1.0T. (b) Determine the maximum value of current for the condition of part (a). (c) Calculate the stored energy for the conditions in part (b). 2.3 The centre limb of the magnetic structure in Figure 2.22 is wound with 400 turns and has a cross-sectional area of 750 mm2. Each of the outer limbs has a cross-sectional area of 450 mm2. The air gap has a length of 1 mm. The mean lengths of the various magnetic paths are shown in Figure 2.22. The relative permeability of the magnetic material is 1500. Calculate: (a) The reluctances of the centre limb, the air gap and the two outer limbs; (b) The current required to establish a ﬂux of 1.4 mWb in the centre limb, assume there is no magnetic leakage or fringing; (c) The energy stored in the air gap in (b). 1mm 350 mm 150 mm 350 mm Figure 2.22 Problem 2.3 gapped core. 52 Transformers and Inductors for Power Electronics 2.4 Calculate the self inductance of a solenoid of length 5 cm with 500 turns. Neglect fringing. 2.5 Calculate the self inductance of the gapped core in Example 2.4. Add 10% to the linear dimensions of the gap and ﬁnd the increase in self inductance. The relative permeability of the core material is 2000. 2.6 Calculate the value of the current to give B ¼ 0.2 T in Example 2.3. 2.7 Calculate the inductance of the structure in Figure 2.9 and determine the stored energy for a coil with 25 turns and carrying 2A. 2.8 When the core material in Example 2.8 enters saturation, the relative permeability is reduced, which will counter the effect of increase inductance due to fringing. Calculate the value of the relative permeability to offset the increase the inductance. 2.9 Use Lyle’s method to calculate M12 in Example 2.5. 2.10 Use the GMD method to calculate M14 in Example 2.5. MATLAB Program for Example 2.5 %This MATLAB program is used to calculate the self and mutual inductances %in Example 2.5 %The parameters are shown in Figure 2.16 a1_in = 1150e-6; a1_out = 1750e-6; a2_in = 1150e-6; a2_out = 1750e-6; a4_in = 2000e-6; a4_out = 2600e-6; h1 = 15e-6; h2 = 15e-6; h4 = 15e-6; w = 600e-6; z12 = 55e-6; c12 = z12-(h1+h2)/2; z14 = 0; %This section the self inductance of sections 1 is calculated R1=sqrt (a1_in*a1_out); GMD_1 = 0.2235*(w+h1); L1 = inductance(R1,R1,GMD_1) %This section the mutual inductance between sections 1 and 2 is calculated R2 = sqrt(a2_in*a2_out); GMD_b = 0.2235*(w+c12); GMD_a = 0.2235*(w+h1+c12); Inductance 53 GMD_s = 0.2235*(w+h1+h2+c12); GMD_12 = (2*w+c12)^2/(2*w^2)*log(GMD_s)-log(GMD_1)((c12/w)^2)/2*log(GMD_b)-((2*c12)/w)*log(GMD_a); GMD_12 = exp(GMD_12); M_12 = inductance(R1,R2,GMD_12) %This section the mutual inductance between sections 1 and 4 is calculated %The dimension calculation results with Lyle’s method applied r1_lyle = R1*(1+h1^2/(24*R1^2)); alpha1_lyle = sqrt((w^2-h1^2)/12); R4 = sqrt(a4_in*a4_out); r4_lyle = R4*(1+h4^2/(24*R4^2)); alpha4_lyle = sqrt((w^2-h4^2)/12); r1_ﬁlament = r1_lyle-alpha1_lyle; r2_ﬁlament = r1_lyle+alpha1_lyle; r3_ﬁlament = r4_lyle-alpha4_lyle; r4_ﬁlament = r4_lyle+alpha4_lyle; M13 = inductance(r1_ﬁlament,r3_ﬁlament,z14); M14 = inductance(r1_ﬁlament,r4_ﬁlament,z14); M23 = inductance(r2_ﬁlament,r3_ﬁlament,z14); M24 = inductance(r2_ﬁlament,r4_ﬁlament,z14); M_14 = (M13+M14+M23+M24)/4 %File to deﬁne the function inductance function M = inductance(a,r,GMD) % This function is used to calculate the self and mutual inductance applying the GMD method uo = 4*pi*1e-7; f = sqrt((4*a*r)/(GMD^2+(a+r)^2)); %MATLAB deﬁnition of ellipke integal uses square function [K,E] = ellipke(f^2); M = uo*(2*sqrt(a*r)/f)*((1-(f^2/2))*K-E); end References 1. Hurley, W.G. and Duffy, M.C. (1995) Calculation of self and mutual impedances in planar magnetic structures. IEEE Transactions on Magnetics 31 (4), 2416–2422. 2. Maxwell, J.C. (1881) A Treatise on Electricity and Magnetism, Clarendon Press, Oxford. 3. Gray, A. (1893) Absolute Measurements in Electricity and Magnetism, MacMillan, London. 4. Lyle, T.R. (1914) Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, The Royal Society, London. 54 Transformers and Inductors for Power Electronics 5. Dwight, H.B. (1919) Some new formulas for reactance coils. Transactions of the American Institute of Electrical Engineers XXXVIII (2), 1675–1696. 6. Grover, F.W. (2004) Inductance Calculations: Working Formulas and Tables, Dover Publications Inc., New York. Further Reading 1. Blume, L.F. (1982) Transformer Engineering, John Wiley & Sons, New York. 2. Bueno, M.D.A. (2001) Inductance and Force Calculations in Electrical Circuits, Nova Science Publishers, Huntington. 3. Del Vecchio, R.M., Poulin, B., Feghali, P.T. et al. (2001) Transformer Design Principles: With Applications to Core-Form Power Transformers, 1st edn, CRC Press, Boca Raton, FL. 4. Dowell, P.L. (1966) Effects of eddy currents in transformer windings. Proceedings of the Institute of Electrical and Electronic Engineers 113 (8), 1387–1394. 5. Erickson, R.W. (2001) Fundamentals of Power Electronics, 2nd edn, Springer, Norwell, MA. 6. Flanagan, W.M. (1992) Handbook of Transformer Design and Application, 2nd edn, McGraw-Hill, New York. 7. Georgilakis, P.S. (2009) Spotlight on Modern Transformer Design (Power Systems), 1st edn, Springer, New York. 8. Goldberg, A.F., Kassakian, J.G., and Schlecht, M.F. (1989) Issues related to 1–10-MHz transformer design. IEEE Transactions on Power Electronics 4 (1), 113–123. 9. Kassakian, J.G., Schlecht, M.F., and Verghese, G.C. (1991) Principles of Power Electronics (Addison-Wesley Series in Electrical Engineering), Prentice Hall, Reading, MA. 10. Kazimierczuk, M.K. (2009) High-Frequency Magnetic Components, John Wiley & Sons, Chichester. 11. Krein, P.T. (1997) Elements of Power Electronics (Oxford Series in Electrical and Computer Engineering), Oxford University Press, Oxford. 12. Kulkarni, S.V. (2004) Transformer Engineering: Design and Practice, 1st edn, CRC Press, New York. 13. Kusko, A. (1969) Computer-aided Design of Magnetic Circuits, The MIT Press, Cambridge, MA. 14. B.H.E. Limited (2004) Transformers: Design, Manufacturing, and Materials (Professional Engineering), 1st edn, McGraw-Hill, New York. 15. McLachlan, N.W. (1955) Bessel Functions for Engineers, 2nd edn, Clarendon Press, Oxford. 16. E.S. MIT (1943) Magnetic Circuits and Transformers (MIT Electrical Engineering and Computer Science), The MIT Press, Cambridge, MA. 17. Niemela, V.A., Owen, H.A. Jr, and Wilson, T.G. (1990) Cross-coupled-secondaries model for multiwinding transformers with parameter values calculated from short-circuit impedances. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 822–830. 18. Niemela, V.A., Skutt, G.R., Urling, A.M. et al. (1989) Calculating the short-circuit impedances of a multiwinding transformer from its geometry. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 607–617. 19. Paul, C.R. (2010) Inductance: Loop and Partial, John Wiley & Sons, Hoboken, NY. 20. Pressman, A.I., Bellings, K., and Morey, T. (2009) Switching Power Supply Design, 3rd edn, McGraw-Hill, New York. 21. Ramo, S., Whinnery, J.R., and Van Duzer, T. (1965) Fields and Waves in Communications Electronics, John Wiley & Sons, New York. 22. Smith, B. (2009) Capacitors, Inductors and Transformers in Electronic Circuits (Analog Electronics Series), Wexford College Press, Wexford. 23. Snelling, E.C. (1988) Soft Ferrites: Properties and Applications, 2nd edn, Butterworths, London. 24. Van den Bossche, A. (2005) Inductors and Transformers for Power Electronics, 1st edn, CRC Press, New York. 25. Van den Bossche, A., Valchev, V., and Filchev, T. (2002) Improved approximation for fringing permeances in gapped inductors. Proceedings of the IEEE Industry Applications Conference, IAS, pp. 932–938. 26. Wilcox, D.J., Conlon, M., and Hurley, W.G. (1988) Calculation of self and mutual impedances for coils on ferromagnetic cores. IEE Proceedings A, Physical Science, Measurement and Instrumentation, Management and Education 135 (7), 470–476. 27. Wilcox, D.J., Hurley, W.G., and Conlon, M. (1989) Calculation of self and mutual impedances between sections of transformer windings. Generation, Transmission and Distribution, IEE Proceedings C 136 (5), 308–314. 3 Inductor Design In Chapter 2, we described the fundamentals of inductance, and this will form the basis of the design of practical inductors. Essentially, inductors can have a discrete gap, typically in a laminated or ferrite core, or a distributed gap in an iron powder core. The design outlined in this chapter will apply to both types of cores. The approach is to adopt the concept of effective permeability, as enunciated in Chapter 2. The physical core selection is based on the energy stored in the inductor, the maximum ﬂux density in the core and the temperature rise in the inductor. The core selection is followed by the winding design including turns and wire size. 3.1 The Design Equations 3.1.1 Inductance The inductance of a coil with N turns wound on a core of length lc and cross section Ac with effective relative permeability meff to account for the gap (discrete or distributed) is given by Equation 2.26: L ¼ meff m0N2Ac lc ð3:1Þ 3.1.2 Maximum Flux Density Applying Ampere’s law to the closed loop around a core of length lc with N turns establishes the relationship between the magnetic ﬁeld intensity and the current in the core: H max ¼ N^I lc ð3:2Þ Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 56 Transformers and Inductors for Power Electronics Here, ^I is the peak value of current. For simplicity, we will relate it to the rms value of the current waveform: Irms ¼ Ki^I ð3:3Þ where Ki is the current waveform factor. Hmax is related to the maximum ﬂux density, so: Bmax ¼ meff m0Hmax ¼ meff m0 N^I lc ð3:4Þ The effective relative permeability in Equation 3.4 includes the effect of an air-gap as described in Section 2.1. By rearranging Equation 3.4 we can obtain a relationship for ^I in terms of the maximum ﬂux density in the core: ^I ¼ Bmaxlc meff m0N ð3:5Þ Bmax is limited by the saturation ﬂux density, Bsat for the core material. Typical values for different materials are listed in Table 1.1. 3.1.3 Winding Loss The resistive or I2R loss Pcu in a winding is Pcu ¼ rw lw Aw I 2rms ¼ rw N MLT À K i^IÁ2 Aw ¼ rw N2 MLT À K i^IÁ2 NAw ð3:6Þ The resistivity of the conductor is rw and the length of the conductor in the winding is lw, which is the product of the number of turns N and the mean length of a turn (MLT). The rms value of the coil current is related to the peak current by Ki as before. Extracting ^I from Equation 3.6: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^I ¼ 1 PcuNAw NKi rwMLT ð3:7Þ The maximum dissipation PD in the core is related to the temperature rise and the heat transfer from the surface of the inductor to the surroundings. This relationship is normally described by the thermal resistance: DT ¼ RuPD ð3:8Þ Inductor Design 57 This relationship will be examined in more detail in Section 3.2, but for now PD represents the maximum allowable dissipation in the inductor. In choosing the best core for the application, the choice is ultimately limited by PD, which is the limiting value in Equation 3.7. Equally, Bsat is the limiting value of ﬂux density in Equation 3.5. 3.1.4 Optimum Effective Permeability The maximum energy stored in an inductor is 1/2L^I2. Combining Equations 3.1 and 3.5 gives an expression for the stored energy in terms of Bmax: 1 L^I2 2 ¼ 1 Aclc 2 meff m0 B2max ð3:9Þ Combining Equations 3.1 and 3.7 yields another expression for the stored energy in terms of copper loss: 1 2 L^I2 ¼ 1 2 meff m0AcNAw rw MLT K2i lc Pcu ð3:10Þ Both Equations 3.9 and 3.10 express the stored energy in terms of the effective permeability of the core, the maximum ﬂux density in the core and the winding dissipation. All of the other parameters are derived from the physical dimensions of the core. Figure 3.1 shows the stored The energy design as a function of both Bmax and Pcu. space of the inductor is bounded by L^I2, Pcumax and Bsat. The curves for L^I2, as a function of Pcu and Bmax respectively, are drawn for three values of meff. The solid lines in Figure 3.1 (point 1) show the stored energy for the optimum value of the effective permeability, which is the maximum stored energy for that particular core. The core is operating at its maximum permissible ﬂux density and its maximum permissible dissipa- tion. The application may require less stored energy. Point 2 represents a design at which the LÎ 2 µeff 1 2 3 µeff µopt µmax Pcu PD 0 Bsat B Figure 3.1 Stored energy as a function of ﬂux density and dissipation. 58 Transformers and Inductors for Power Electronics current is set at the maximum dissipation. The core is operating at a lower value of ﬂux density at point 2, and meff is smaller than the optimum value. On the other hand, at point 3, the core is operating at the maximum allowed value of ﬂux density but the dissipation is less than the maximum permissible value. This means that meff is greater than the optimum value. The design space in Figure 3.1 shows that increasing meff above the optimum value while maintaining maximum dissipation will cause the core to go into saturation. Reducing meff below the optimum value while maintaining the maximum ﬂux density will cause the core to overheat. Equating the expressions for ^I in Equations 3.5 and 3.7, using the maximum values of Bsat and Pcumax for Bmax and Pcu respectively, shows that there is an optimum value of meff, to be designated mopt: mopt ¼ rBﬃﬃsﬃaﬃﬃtﬃlﬃﬃcﬃKﬃﬃﬃﬃiﬃﬃﬃﬃﬃﬃﬃﬃﬃ m0 Pcu maxNAw rwMLT ð3:11Þ The expression for mopt in Equation 3.11 shows that for a given core with a known window winding area and mean turn length, the value of the optimum value of the effective permeability is determined by the maximum ﬂux density Bsat in the core and the maximum dissipation PD. 3.1.5 Core Loss In inductor design, the core or iron loss is often negligible compared to the winding loss. This is a common situation where there is a small ripple in the presence of a DC current. The iron loss occurs as a result of the ripple current and, as a ﬁrst estimation, may be treated with the Steinmetz equation. The time-average core loss per unit volume is usually stated as: Pfe ¼ K c f a DBb 2 ð3:12Þ In this case, DB is the peak to peak ﬂux density ripple, Typical values for kc, a and b for sinusoidal excitation are given in Table 1.1. The Improved General Steinmetz Equation (iGSE) [1] treats the general case of ﬂux ripple, which will be described in Chapter 7. For present purposes, we will state the core loss as a fraction of the winding loss: Pfe ¼ gPcu ð3:13Þ In an inductor with negligible ﬂux ripple, g is taken to be 0. 3.1.6 The Thermal Equation The combined losses in the windings and core must be dissipated through the surface of the wound transformer. Heat transfer is dominated by conduction within the core and winding, and from the transformer surface by convection. Newton’s equation of Inductor Design 59 convection relates heat ﬂow to temperature rise (DT), surface area (At) and the coefﬁcient of heat transfer hc, by: Q ¼ hcAtDT ð3:14Þ where Q represents the total power loss, i.e. the combined winding loss and the core loss. In terms of an equivalent circuit analogy with Ohm’s law, the heat ﬂow is linearly propor- tional to the temperature difference and inversely proportional to the thermal resistance, so: DT ¼ RuQ ¼ 1 hcAt Q ð3:15Þ In the electrical analogy, Q represents current (and not power), and DT represents the potential difference. Manufacturers will sometimes provide the thermal resistance for a core, but normally the value is based on empirical data. For example, the thermal resistance may be related to the volume of the core Vc: Ru ¼ p0:0ﬃﬃﬃ6ﬃﬃ Vc ð3:16Þ In this empirical equation, Ru is in C/W for Vc in m3. There are many empirical formulas for estimating the hc value in natural convection for different conﬁgurations. For a vertical object of height H, one such formula is [2]: hc ¼ 1:42 DT !0:25 H ð3:17Þ For an ETD55 core, H ¼ 0.045 m and hc ¼ 8.2 W/m2 C for a 50 C temperature rise. Evi- dently, the position of the transformer relative to other components will have a profound effect on the value of hc. In fact, the value of hc is probably the most uncertain parameter in the entire design. However, the typical value of hc ¼ 10 W/m2 C is often used for cores encountered in switching power supplies [3]. In the case of forced convection with fanassisted airﬂow, the value of hc will be much higher – in the range of 10–30 W/m2 C. 3.1.7 Current Density in the Windings The window utilization factor, ku, is deﬁned as the ratio of the total conduction area Wc, for all conductors in all windings to the total window winding area Wa of the core: ku ¼ Wc Wa : ð3:18Þ The window utilization factor may vary from 0.2–0.8. In a core with a bobbin, it is possible to wind the coil very tightly, and ku may be as high as 0.8. On the other hand, in the case of a 60 Transformers and Inductors for Power Electronics toroidal core, automatic winding may necessitate a large free window winding area for the winding arm, and ku may be as low as 0.2. Insulation spacing to achieve creepage and clearance distances required by the relevant safety standards are included in the speciﬁcation of the window utilization factor. The total conduction area may be expressed in terms of the individual conductor area and the number of turns: Wc ¼ NAw ¼ kuWa ð3:19Þ Substituting Equation 3.19 into Equation 3.11 yields an expression for the optimum value of the effective relative permeability mopt, i.e. point 1 in Figure 3.1: mopt ¼ rBﬃﬃﬃsﬃaﬃﬃtﬃlﬃﬃcﬃKﬃﬃﬃﬃiﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m0 Pcu maxkuW a rwMLT ð3:20Þ In this form, the optimum value of the effective relative permeability is a function of the saturation ﬂux density and the maximum dissipation. The current density in the winding is: Jo ¼ Irms Aw ð3:21Þ Noting that the volume of the windings (fully wound ku ¼ 1) is Vw ¼ MLT Â Wa, then the copper loss given by Equation 3.6, using Equation 3.19 with Equations 3.21 and 3.3, may be expressed as: Pcu ¼ rw N2 MLTðJoAwÞ2 NAw ¼ rw V w ku J 2o ð3:22Þ Combining Equations 3.13 and 3.22 in Equation 3.14: Q ¼ Pcu þ Pfe ¼ ð1 þ Â gÞ rwV w kuJ20 Ã ¼ hcAtDT ð3:23Þ Extracting the current density: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Jo ¼ 1 hcAtDT 1 þ g rwVwku ð3:24Þ Inductor Design 61 3.1.8 Dimensional Analysis The physical quantities Vw, Vc and At may be related to the product of the core window winding area and the cross-sectional area (core window area product). It follows by dimensional analysis that: 3 Vw ¼ kwA4p ð3:25Þ 3 V c ¼ kcA4p ð3:26Þ 1 At ¼ kaA2p ð3:27Þ The coefﬁcients kw, kc and ka are dimensionless. The exponents of Ap are chosen so that the dimensions are consistent; for example Vw has the dimensions of m3 when Ap has the dimensions of m4, and so on. The values of ka, kc and kw vary for different types of cores [4]. However, the combinations required for the inductor design are approximately constant. Further accuracy in establishing these general constants is somewhat offset by the lack of accuracy around the heat transfer coefﬁcient in Equation 3.14. Based on extensive studies of several core types and sizes, it was found that typically ka ¼ 40, kc ¼ 5.6 and kw ¼ 10. An exception is the pot core, for which kw is typically 6. Of course, it is perfectly reasonable and straightforward to carry out a detailed study of a partic- ular core type, or for the cores of a particular manufacturer to establish more accurate values of these coefﬁcients. The dimensional analysis resulting in Equations 3.25 and 3.27, substituted into Equation 3.24, gives an expression for the current density in terms of the temperature rise in the windings and the core-window winding area product: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Jo ¼ Kt DT p1ﬃﬃﬃﬃﬃ kuð1 þ gÞ 8 Ap ð3:28Þ where sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Kt ¼ hcka : rwkw ð3:29Þ Inserting typical values: rw ¼ 1.72 Â 10À8 V-m, hc ¼ 10 W/m2 C, ka ¼ 40, kw ¼ 10 gives Kt ¼ 48.2 Â 103, Ap is in m4 and the current density is in A/m2. 3.2 The Design Methodology From Section 2.3, Equation 2.33, the energy stored in an inductor with a gap of length g is: Wm ¼ B2 V c 2mrm0 þ B2 V g 2m0 62 Transformers and Inductors for Power Electronics The volume of the core is Vc ¼ Aclc and the volume of the gap is Vg ¼ Agg. Adopting the notation from Section 3.1, and assuming that the cross-sectional area of the gap is equal to the cross-sectional area of the core, then: 1 L^I2 ¼ B2maxAc ! lc þ g 2 2m0 mr ð3:30Þ Ampere’s law for the gapped core is: ! Ni ¼ Hclc þ Hgg ¼ B m0 lc þ g mr ð3:31Þ For peak current ^I and for maximum ﬂux density Bmax, Equation 3.31 becomes, after rearranging: lc mr þ g ¼ N^I Bmax m0 ð3:32Þ Substituting the result in Equation 3.32 into Equation 3.30 yields: 1 2 L^I2 ¼ 1 2 BmaxAcN^I ð3:33Þ Invoking the deﬁnition for current density in Equation 3.21 with Equation 3.3, together with Equation 3.19: L^I2 ¼ BmaxAckuJ0Wa Ki ð3:34Þ Taking Jo from Equation 3.28 yields an expression for the core window winding area Â cross sectional area product Ap: Ap ¼ " pﬃ1ﬃﬃﬃþﬃﬃﬃﬃﬃgpﬃﬃKﬃﬃiﬃﬃLﬃﬃ^Iﬃﬃ2ﬃﬃﬃ#87 BmaxKt kuDT ð3:35Þ The Ap value determines the selection of the core. The parameters of the selected core determine mopt given by Equation 3.11, and the design may proceed from there. The overall design methodology is shown in ﬂow-chart form in Figure 3.2. The core manufacturer normally supplies the core data: cross-section, Ac, core length lc, window winding area, Wa, the mean length of a turn, MLT and the core volume Vc. The selected core from standard designs may not correspond exactly to the value of Ap given by Equation 3.35, and therefore the actual current density based on the selected core may be calculated using Equation 3.28. There are two basic designs: a core with a discrete gap or a core with a distributed gap. The effective permeability for the core with the discrete gap determines the gap length; the manufacturer normally supplies cores with pre-selected gaps. In the case of cores with Inductor Design 63 Specifications : L,Î,Ki,f,ku,ΔT Select the materials Calculate µopt Yes Calculate gm Select g Calculate Ap Select Ap Gapped core Ac Wa MLT lc No Calculate µmax Select µeff Calculate Turns N Calculate Copper Loss Calculate Core Loss Calculate High Frequency Losses Figure 3.2 Flow chart of design process. distributed gaps, the manufacturer normally supplies a set of cores with pre-selected values of effective permeability. The correct choice for gap length or effective permeability will be illustrated in the designs to follow. The next step is to calculate the number of turns N in the winding. The manufacturer will normally supply the AL value of the core for the corresponding gap length, i.e. the inductance per turn: rﬃﬃﬃﬃﬃ N¼ L AL ð3:36Þ Alternatively, it may be supplied in the form of inductance per 1000 turns. The following step is to select the conductor, based on the current density. The resistivity of the conductor at the maximum operating temperature is given by: rw ¼ r20½1 þ a20ðTmax À 20CÞ; ð3:37Þ 64 Transformers and Inductors for Power Electronics where Tmax is the maximum temperature, r20 is the resistivity at 20 C, and a20 is the temperature coefﬁcient of resistivity at 20 C. The wire sizes are selected from standard wire tables, which normally specify resistance in V/m at 20 C. The winding loss is then: Pcu ¼ MLT Â N Â ðV=mÞ Â ½1 þ a20ðTmax À 20CÞ Â I2: ð3:38Þ The design procedure may be summarized as follows: Speciﬁcations: Inductance DC current Frequency of operation Maximum core temperature or temperature rise Ambient temperature Window utilization factor Core selection: Core material Maximum ﬂux density Core window winding area product Winding design: Number of turns Current density Wire selection Losses: Copper loss Winding loss if applicable In Chapter 6, we will reﬁne the design to take high-frequency skin and proximity effects into account. 3.3 Design Examples 3.3.1 Example 3.1: Buck Converter with a Gapped Core Design an inductor for a buck converter with the speciﬁcations listed in Table 3.1. Circuit Parameters The circuit for a buck converter is shown in Figure 3.3 and the associated current and voltage waveforms are shown in Figure 3.4. Inductor Design 65 Table 3.1 Speciﬁcations Input voltage Output voltage Inductance DC Current Frequency, f Temperature rise, DT Ambient temperature, Ta Window utilization factor 12 V 6V 34 mH 20 A 80 kHz 15 C 70 C 0.8 is Q1 + Vs _ Control + iL eL L _ iL + Dm vC + iC io C Load vo _ _ Figure 3.3 Buck converter circuit diagram. vL Vs-Vo 0 t -Vo iL Imax ΔI Imin 0 t B Bmax ΔB Bmin 0 t DT T 2T Figure 3.4 Buck converter voltage and current waveforms. 66 Transformers and Inductors for Power Electronics The basic operation of the buck converter is treated in power electronics text, and the main features are summarized here for clarity. In the time period between 0 and DT, the switch Q is closed and the diode D is open and the voltage Vi À Vo is applied across the inductor. Between DT and T the applied voltage across the inductor is – Vo, resulting in the current waveform in Figure 3.4 with the current ripple DI. When Q is closed and D is open: vL ¼ Vs À Vo ¼ L DILþ DT When Q is open and D is closed: vL ¼ ÀV o ¼ L DILÀ ð1 À DÞT In steady state, the net change in ﬂux in the inductor is zero and there the increase in current while Q is closed must be equal to the decrease in current when Q is open. It follows that: jDILþj ¼ jDILÀj ¼ DI and: Vo ¼ DVs In this example, we are assuming that the current ripple is small, and therefore g in Equation 3.13 is negligible. Core Selection Ferrite would normally be used for this type of application at the speciﬁed frequency. The material speciﬁcations for EPCOS N87 Mn-Zn are listed in Table 3.2. The amplitude of the ripple current is: DIL ¼ ðV s À VoÞDT L ¼ ð34 ð12 À 6Þð0:5Þ Â 10À6Þð80 Â 103Þ ¼ 1:1 A Table 3.2 Material speciﬁcations Kc 16.9 a 1.25 b 2.35 Bsat 0.4T Inductor Design 67 Table 3.3 Core and winding speciﬁcations Ac lc Wa Ap Vc kf ku Ki MLT r20 a20 2.09 cm2 11.4 cm 2.69 cm2 5.62 cm4 23.8 cm3 1.0 0.8 1.0 8.6 cm 1.72 mV-cm 0.00393 The peak current is: ^I ¼ Idc þ DI 2 ¼ 20:0 þ 1:1 2 ¼ 20:55 A L^I2 ¼ ð34 Â 10À6Þð20:55Þ2 ¼ 0:0144 J For Mn-Zn ferrite, select Bmax ¼ 0.25T. This type of core will allow a very tight winding, so therefore the window utilization factor is set to ku ¼ 0.8. Ap from Equation 3.35 with g ¼ 0 is: Ap ¼ " pﬃ1ﬃﬃﬃþﬃﬃﬃﬃﬃgpﬃﬃKﬃﬃiﬃﬃLﬃﬃ^Iﬃﬃ2ﬃﬃﬃ#8=7 BmaxKt kuDT ¼ " ð0:25Þð48:2 #8=7 0:0144pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Â 103Þ ð0:8Þð15Þ Â 108 ¼ 4:12 cm4 The ETD49 core is suitable. The core speciﬁcations are given in Table 3.3. The thermal resistance of this core according to the manufacturer is 11 C/W. Therefore, the maximum dissipation given by Equation 3.8 is: PD ¼ DT Ru ¼ 15 11 ¼ 1:36 W The optimum value of the effective permeability for this core is given by Equation 3.20: mopt ¼ rBﬃﬃmﬃﬃﬃaﬃxﬃﬃlﬃﬃcﬃKﬃﬃﬃﬃiﬃﬃﬃﬃﬃﬃﬃﬃﬃ m0 Pcu maxkuW a rwMLT ¼ ð4p Â ð0:25sÞðﬃﬃ1ﬃﬃ1ﬃﬃ:ﬃﬃ4ﬃﬃﬃÂﬃﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃﬃÀﬃﬃ2ﬃﬃÞﬃﬃðﬃﬃ1ﬃﬃ:ﬃﬃ0ﬃﬃÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 10À7Þ ð1:36Þð0:8Þð2:69 Â 10À4Þ ð1:72 Â 10À8Þð8:6 Â 10À2Þ ¼ 51 68 Transformers and Inductors for Power Electronics Winding Design Gap g Referring to Figure 3.1, increasing the value of effective permeability above the optimum value means that we can operate the core at its maximum ﬂux density and below the maximum dissipation. This makes sense, since we have not included any core loss at this point. The maximum gap length is found from Section 2.1: gmax ¼ lc mmin ¼ 11:4 Â 10À2 51 Â 103 ¼ 2:24 mm The manufacturer provides a standard core set with g ¼ 2 mm, a corresponding AL value of 188 nH and an effective permeability of 81. Turns For the core, the manufacturer supplies the AL value (i.e. the inductance per turn): N ¼ rﬃﬃﬃﬃﬃ L AL ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð34 Â 10À6Þ ð188 Â 10À9Þ ¼ 13:5 turns Select 13 turns. Wire Size The current density is: pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Jo ¼ Kt pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDﬃﬃﬃﬃTﬃﬃﬃﬃpﬃﬃﬃﬃﬃ kuð1 þ gÞ 8 Ap ¼ 48:2 Â 103 pﬃðﬃ0ﬃﬃﬃ:ﬃ8ﬃﬃÞﬃﬃq8 ﬃðﬃ5ﬃðﬃ1ﬃ:ﬃ6ﬃ5ﬃ2ﬃÞﬃﬃﬃÂﬃﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃÀﬃﬃﬃ8ﬃﬃÞﬃ Â 10À4 ¼ 168 A=cm2 The cross-sectional area of the conductor is (for the purposes of this calculation, we are neglecting the current ripple): Aw ¼ Irms=J ¼ 20=168 ¼ 0:119 cm2 An 8 mm Â 2 mm wire meets this speciﬁcation, with a DC resistance of 10.75 Â 10À6 V/cm at 70 C. Copper Loss Use Equation 3.37 to correct the winding resistance for temperature: Tmax ¼ 70 þ 15 ¼ 85 C: Rdc ¼ ð13Þð8:6Þð10:75 Â 10À6Þ½1 þ ð0:00393Þð85 À 20Þ Â 103 ¼ 1:51 mV Inductor Design 69 The copper loss is: Pcu ¼ Rdc I 2 rms ¼ ð1:51 Â 10À3Þð20:0Þ2 ¼ 0:604 W Core Loss The ﬂux density ripple DB may be calculated by using Faraday’s law, noting that the voltage across the inductor in the time period 0 to DT is (Vi À Vo): DB ¼ ðV i À VoÞDT NAc ¼ ð12 À 6Þð0:5Þ ð13Þð2:09 Â 10À4Þð80 Â 103Þ ¼ 0:014 T Using the General Steinmetz Equation (GSE) (Equation 1.29), the peak value of the ﬂux density Bmax is DB/2 and the core loss is: Pfe ¼ VcKcf aBmaxb ¼ ð23:8 Â 10À6Þð16:9Þð80 000Þ1:25ð0:014=2Þ2:35 ¼ 0:005 W As expected, the core loss is much smaller than the copper loss and well within the margins allowed by our core selection. Total Losses : Copper loss Core loss Total losses 0:604 W 0:005 W 0:609 W 3.3.2 Example 3.2: Forward Converter with a Toroidal Core Speciﬁcations Design an inductor for a forward converter with the speciﬁcations listed in Table 3.4. Circuit Parameters The circuit for a forward converter is shown in Figure 3.5 and the associated current and voltage waveforms are shown in Figure 3.6. Table 3.4 Speciﬁcations Input voltage Output voltage Inductance DC Current Frequency, f Temperature rise, DT Ambient temperature, Ta Window utilization factor 12 V 9V 1.9 mH 2.1 A 60 kHz 20 C 60 C 0.2 70 Transformers and Inductors for Power Electronics + Vs _ Np : Ns is D1 iL Lo + + + Nt vp _ vs vd D2 _ _ D3 Q io + Co Vo _ Figure 3.5 Forward converter circuit diagram. vp Vs 0 t –⎧⎩NNpt ⎫⎭Vs vL ⎧Ns ⎩ Nt ⎫⎭Vs – Vo 0 t -Vo iL ÎL IL ΔIL 0 t B Bmax ΔB Bmin 0 DT T 2T t Figure 3.6 Forward converter current and ﬂux waveforms. Inductor Design 71 Table 3.5 Material speciﬁcations Kc a b Bsat 231.8 1.41 2.56 0.5 T The operation of the forward converter is very similar to the buck converter, with electrical isolation provided by the transformer. We will examine its operation in detail in Chapter 5 (transformer design). For present purposes, we are interested in the output inductor and we will assume the transformer turns ratio is 1: 1. From the analysis of the buck converter in Example 3.1: Vo ¼ DVi; The duty cycle in this case is D ¼ 9/12 ¼ 0.75. We assume the ﬂux ripple in the output inductor is negligible, therefore g ¼ 0. Core Selection In this case, we select an MPP powder iron core. The material speciﬁcations for the MPP core are listed in Table 3.5. The amplitude of the ripple current is: DIL ¼ ðV s À VoÞDT L ¼ ð12 À 9Þð0:75Þ ð1:6 Â 10À3Þð60 Â 103Þ ¼ 0:0234 A I peak ¼ Idc þ DI 2 ¼ 1:9 þ 0:0234 2 ¼ 1:912 A L^I2 ¼ ð1:6 Â 10À3Þð1:912Þ2 ¼ 0:0058 J In this example, Bmax ¼ 0.35 T. Ap, from Equation 3.35, with g ¼ 0, is: Ap ¼ " pﬃ1ﬃﬃﬃþﬃﬃﬃﬃﬃgpﬃﬃKﬃﬃiﬃﬃLﬃﬃ^Iﬃﬃ2ﬃﬃﬃ#8=7 BmaxKt kuDT ¼ " ð0:35Þð48:2 #8=7 0:0058pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Â 103Þ ð0:2Þð20Þ Â 108 ¼ 1:875 cm4 The Magnetics MPP toroidal core with dimensions given in Table 3.6 is suitable. The core speciﬁcations are given in Table 3.6, Ap ¼ 2.58 cm4. In this case, the manufacturer did not provide the thermal resistance. We can estimate it from Equation 3.16: Ru ¼ p0:0ﬃﬃﬃ6ﬃﬃ Vc ¼ pﬃﬃﬃﬃﬃﬃ0ﬃﬃ:ﬃﬃ0ﬃﬃ6ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6:09 Â 10À6 ¼ 24:3 C=W 72 Transformers and Inductors for Power Electronics Table 3.6 Core and winding speciﬁcations Ac Wa Ap lc Vc kf ku Ki MLT r20 a20 0.678 cm2 3.8 cm2 2.58 cm4 8.98 cm 6.09 cm3 1.0 0.2 1.0 5.27 cm 1.72 mV-cm 0.00393 and the maximum dissipation is: PD ¼ DT Ru ¼ 20 24:3 ¼ 0:823 W The optimum value of the effective permeability for this core is given by Equation 3.20: mopt ¼ rBﬃﬃmﬃﬃﬃaﬃﬃxﬃlﬃﬃcﬃﬃkﬃﬃiﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m0 Pcu maxkuW a rwMLT ¼ ð4p Â ð0:3s5Þﬃðﬃﬃ8ﬃﬃ:ﬃ9ﬃﬃﬃ8ﬃﬃﬃÂﬃﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃﬃÀﬃﬃﬃ2ﬃÞﬃﬃðﬃﬃ1ﬃﬃ:ﬃﬃ0ﬃﬃÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 10À7Þ ð0:823Þð0:2Þð3:8 Â 10À4Þ ð1:72 Â 10À8Þð5:27 Â 10À2Þ ¼ 95 In this case, there is no provision for core loss. Taking Equation 3.9, with the core speciﬁcations in Table 3.6, yields the maximum value of the effective relative permeability. This is based on point 3 of Figure 3.1: meff ¼ B2maxAclc m0L^I2 ¼ ð0:35Þ2ð0:678 ð4p Â Â 10À4Þð8:98 Â 10À7Þð0:0058Þ 10À2Þ ¼ 101 The manufacturer provides a core with effective permeability of 125. Winding Design The MPP core with effective permeability of 125 has an inductance of 117 mH per 1000 turns. Turns rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃ N ¼ 1000 L ¼ 1 000 1:6 ¼ 117 Turns L1000 117 Inductor Design 73 The magnetic ﬁeld intensity at the maximum value of the DC bias is: H ¼ NI lc ¼ ð117Þð1:9Þ ð8:98 Â 10À2Þ ¼ 2 490 A=m ¼ 31:3 oersted: At this value of H, the inductance has fallen to 80%. Wire Size The current density is: pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Jo ¼ Kt pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDﬃﬃﬃﬃTﬃﬃﬃﬃpﬃﬃﬃﬃﬃ kuð1 þ gÞ 8 Ap ¼ ð48:2 Â 103Þ ðpﬃ0ﬃﬃ:ﬃ2ﬃﬃÞðq8 ﬃ2ðﬃﬃ:2ﬃ5ﬃ0ﬃﬃ8ﬃÞﬃﬃÂﬃﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃﬃÀﬃﬃ8ﬃﬃÞﬃﬃ Â 10À4 ¼ 428:1 A=cm2 The cross-sectional area of the conductor is: Aw ¼ Idc=Jo ¼ 1:9=428:1 ¼ 0:0044 cm2 This corresponds to a 0.79 mm diameter. A 1 mm diameter copper wire with a DC resistance of 218 Â 10À6 V/cm at 20 C (Table A.1) will sufﬁce. Copper Loss Use Equation 3.37 to correct the winding resistance for temperature: Tmax ¼ 60 þ 20 ¼ 80 C: Rdc ¼ ð117Þð5:27Þð218 Â 10À6Þ½1 þ ð0:00393Þð80 À 20Þ Â 103 ¼ 166 mV The copper loss is: Pcu ¼ RdcI 2 rms ¼ ð0:166Þð1:9Þ2 ¼ 0:600 W Core loss The ﬂux density ripple DB may be calculated by using Faraday’s law, noting that the voltage across the inductor in the time period 0 to DT is (Vi À Vo): DB ¼ ðV i À VoÞDT NAc ¼ ð12 À 9Þð0:75Þ ð117Þð0:678 Â 10À4Þð60 Â 103Þ ¼ 0:005 T Using the General Steinmetz Equation (GSE) (Equation 1.29), the peak value of the ﬂux density Bmax is DB/2 and the core loss is: Pfe ¼ VcKcf aBbmax ¼ ð6:09 Â 10À6Þð231:8Þð60 000Þ1:41ð0:005=2Þ2:56 ¼ 0:002 W 74 Transformers and Inductors for Power Electronics As expected, the core loss is much smaller than the copper loss and well within the margins allowed by our core selection. Total Losses : Copper loss 0:600 W Core loss 0:002 W Total losses 0:602 W 3.4 Multiple Windings In many applications, such as a ﬂyback inductor, there are two windings. The total current is divided in the ratio m:(1 À m) and the areas are distributed in the ratio n:(1 À n). In other words, mI ﬂows in an area nWc and current (1 À m)I ﬂows in an area (1 À n)Wc. Taking the individual I2R loss in each winding, using Equation 3.6, the total copper loss becomes: " # Pcu ¼ rw lw Wc m2 n þ ð1 À mÞ2 1Àn I 2 rms ð3:39Þ Taking Po as the loss for the current I in area Wc, Equation 3.39 may be normalized as: " # Pcu ¼ Po m2 n þ ð1 À mÞ2 1Àn ð3:40Þ The optimum value of Pcu in relation to m is obtained by taking the partial derivative with respect to m and setting it to zero: ! @Pcu @m ¼ Po 2m n À 2ð1 1 À À mÞ n ¼0 ð3:41Þ The minimum loss occurs when m ¼ n. The current density in each of the windings is: mI nW c ¼ ð1 À mÞI ð1 À nÞWc ¼ I Wc ¼ J ð3:42Þ The result in Equation 3.42 shows that the optimum distribution of current in the available area is to have the same current density in each winding. Of course, this result may be generalized by further subdivision of any of the sections above so that, for any number of windings, the optimum distribution of current is set to the same current density in each winding. Consider winding i in an inductor with multiple windings and with current density Jo Jo ¼ I rmsi Awi ¼ N i I rmsi N i Awi ¼ N i I rmsi Wci ð3:43Þ Inductor Design 75 For two windings with the same current density, we can deduce from Equation 3.43: W ci W c1 þ W c2 ¼ Wc1 Wc ¼ N 1 I rms1 N1Irms1 þ N2Irms2 ¼ 1 1 þ Irms2 : aIrms1 ð3:44Þ Where a is the ratio N1/N2, and Wc is the total winding area. This means that the area assigned to each winding is directly related to the rms current in that winding. Note that from Equation 3.19, it follows that the window utilization factor for each individual winding may be related to the overall window utilization factor: W c1 Wc ¼ W c1 Wa Wa Wc ¼ ku1 ku ¼ 1 þ 1 I rms2 aIrms1 ð3:45Þ and: ku ¼ ku1 þ ku2 ð3:46Þ 3.4.1 Example 3.3: Flyback Converter The ﬂyback converter is the isolated version of the buck-boost converter. The circuit diagram is shown in Figure 3.7 and the main voltage and current waveforms are shown in Figure 3.8. The inductor has Np turns in the primary winding and Ns turns in the secondary winding. We assume continuous conduction. + Vs _ ip is D1 + Np Ns vs _ + Q1 _vQ1 + C RL vo R1 Control R2 _ Figure 3.7 Flyback converter circuit diagram. 76 vp Vs 0 –⎧⎩NNps⎫⎭Vo ip Îp Ip Transformers and Inductors for Power Electronics t ΔIp t is Îs Is ΔIs 0 t B Bmax ΔB Bmin 0 DT T T+DT 2T t Figure 3.8 Flyback converter current and ﬂux waveforms. The detailed description of the operation is available in most textbooks on power electronics and are summarized here. When switch Q is closed, from 0 to DT the input voltage is applied to the primary winding and: Vi ¼ Lp DILþ DT ¼ N p Ac DBþ DT ; The current in the primary winding increases as shown in Figure 3.9. Diode D is off and there is no current in the second winding. Inductor Design 77 12 10 Vp Ip Po 8 6 4 2 Indirect Direct 0.2 0.4 0.6 0.8 1.0 Vo Vi Figure 3.9 Switch stress factors for direct and indirect converters [5]. Kassakian, John G.; Schlecht, Martin F.; Verghese, George C., Principles of Power Electronics, 1st Edition, # 1991. Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ. At time t ¼ DT, Q is opened and D is closed so that: v2 ¼ ÀV o ¼ Ls DILÀ ð1 À DÞT ¼ N s Ac DBÀ ð1 À DÞT ; In steady state, the net change in ﬂux in the core is zero and, therefore, the increase in ﬂux while Q is closed must be equal to the decrease in ﬂux when Q is open. It follows that: jDBþj ¼ jDBÀj; and Vo ¼ Ns Np 1 D À D Vi; The characteristic trapezoidal current waveforms for the primary winding and the secondary winding are shown in Figure 3.8. The rms values of these waveforms are related to the ripple as: Ip rms ¼ Ip v u u tﬃDﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃþﬃﬃﬃﬃﬃxﬃﬃﬃ2pﬃﬃ!ﬃﬃﬃ; 12 xp ¼ DIp Ip Is rms ¼ IpsﬃDﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃþﬃﬃﬃﬃﬃ1xﬃﬃ22sﬃﬃﬃﬃﬃﬃ; xs ¼ DIs Is 78 Transformers and Inductors for Power Electronics The current waveform factors, as deﬁned in Equation 3.3, are: Kip ¼ v u u tﬃDﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃÀﬃﬃﬃﬃﬃyﬃﬃpﬃﬃﬃþﬃﬃﬃﬃﬃyﬃ3ﬃ2pﬃﬃ!ﬃﬃﬃ; yp ¼ DIp I^p ¼ 2xp 2 þ xp K is ¼ sﬃDﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃÀﬃﬃﬃﬃﬃyﬃﬃsﬃﬃþﬃﬃﬃﬃﬃyﬃ3ﬃ2sﬃﬃﬃﬃﬃ; ys ¼ DIp I^s ¼ 2xs 2 þ xs Note that the transferred energy per cycle is: P ¼ DViIp ¼ ð1 À DÞVoIs so that: Is ¼ ð1 P À DÞVo ¼ Np Ns Ip Ip ¼ P DV i DIs ¼ Np DIp Ns P is the transferred energy per cycle in the ﬂyback converter. The value of inductance that ensures continuous conduction is: Lg ¼ V 2 i D2 P T 2 ¼ Vo Io Np 2 Ns T 2 ð1 À DÞ2 This proof of this formula is available in texts on power electronics. Speciﬁcations The design speciﬁcations for inductor are listed in Table 3.7. Table 3.7 Speciﬁcations Input voltage Output voltage Output current Inductance Frequency, f Temperature rise, DT Ambient temperature, Ta Window utilization factor 325.3 V 24 V 10 A 700 mH 70 kHz 30 C 60 C 0.235 Inductor Design 79 Circuit Parameters For continuous conduction, the minimum value of inductance is: Lg ¼ V2i D2 P T 2 ¼¼ ð325:3Þ2ð0:314Þ2 ð240Þð2Þð70 Â 103Þ Â 106 ¼ 310 mH pﬃﬃ The input DC voltage is based on a rectiﬁed 240 V mains, i.e. 2 Â 240 V. The turns ratio is chosen to ensure that the switch stress is minimized, and the ratio Vo/Vi should be approximately 0.5, as shown in Figure 3.9 [5], so that Np/Ns % 325.3/48 ¼ 6.7. The value chosen was 6.2, so therefore D is: D ¼ 1 1 þ Vi aV o ¼ 1 þ p1ﬃﬃ ð 2Þð230Þ ð6:2Þð24Þ ¼ 0:314 The output power is P ¼ 24 V Â 10 A ¼ 240 W. The primary current is: Ip ¼ P DV i ¼ 2p40ﬃﬃ ð0:314Þð 2Þð230Þ ¼ 2:351 A The ripple of the primary winding current is found from: Vi ¼ Lp DIp DT pﬃﬃ DIp ¼ V i DT Lp ¼ ð ð700 2Þð230Þð0:314Þ Â 10À6Þð70 Â 103Þ ¼ 2:084 A The peak value of the primary current is: ^Ip ¼ Ip þ DIp 2 ¼ 2:351 þ 2:084 2 ¼ 3:393 A The current waveform factor is: yp ¼ DIp ^Ip ¼ 2:084 3:393 ¼ 0:614 Kip ¼ v u u tﬃDﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃÀﬃﬃﬃﬃﬃyﬃﬃpﬃﬃﬃþﬃﬃﬃﬃﬃyﬃ3ﬃ2pﬃﬃ!ﬃﬃﬃ ¼ v u u tﬃðﬃ0ﬃﬃﬃ:ﬃ3ﬃﬃ1ﬃﬃﬃ4ﬃﬃÞﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃÀﬃﬃﬃﬃﬃ0ﬃﬃ:ﬃﬃ6ﬃﬃ1ﬃﬃﬃ4ﬃﬃﬃþﬃﬃﬃﬃﬃðﬃﬃ0ﬃﬃ:ﬃ6ﬃﬃﬃ1ﬃﬃ4ﬃﬃﬃÞﬃ2ﬃﬃ!ﬃﬃﬃﬃ 3 ¼ 0:4 80 Transformers and Inductors for Power Electronics The rms value of the primary current is: Ip rms ¼ Kip ^Ip ¼ ð0:4Þð3:393Þ ¼ 1:357 A The secondary current is: Is ¼ ð1 P À DÞVo ¼ ð1 À 240 0:314Þð24Þ ¼ 14:577 A The ripple of the secondary winding current is: DIs ¼ Np Ns DIp ¼ ð6:2Þð2:084Þ ¼ 12:92 A The peak value of the secondary current is: ^Is ¼ Is þ DIs 2 ¼ 14:577 þ 12:92 2 ¼ 21:037 A The current waveform factor is: ys ¼ DIs ^Is ¼ 12:92 21:037 ¼ 0:614 Kis ¼ sﬃðﬃ1ﬃﬃﬃﬃÀﬃﬃﬃﬃﬃDﬃﬃﬃÞﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃÀﬃﬃﬃﬃﬃyﬃﬃsﬃﬃþﬃﬃﬃﬃﬃyﬃ3ﬃ2sﬃﬃﬃﬃﬃ ¼ v u u tﬃðﬃ1ﬃﬃﬃﬃÀﬃﬃﬃﬃﬃ0ﬃﬃ:ﬃ3ﬃﬃﬃ1ﬃﬃ4ﬃﬃÞﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃÀﬃﬃﬃﬃ0ﬃﬃﬃ:ﬃ6ﬃﬃ1ﬃﬃﬃ4ﬃﬃﬃþﬃﬃﬃﬃﬃðﬃﬃ0ﬃﬃ:ﬃﬃ6ﬃﬃ1ﬃﬃ4ﬃﬃﬃÞﬃﬃ2ﬃﬃ!ﬃﬃﬃ 3 ¼ 0:592 The rms value of the secondary current is: Isrms ¼ Kis^Is ¼ ð0:592Þð21:037Þ ¼ 12:454 A We are now in a position to calculate kup using Equation 3.46. Select the total window utilization factor as 0.235, due to the voltage insulation requirement: kup ¼ ku 1 þ 1 Is rms aIp rms ¼ ð0:235Þ 1 þ 1 12:454 ð6:2Þð1:359Þ ¼ ð0:235Þð0:4035Þ ¼ 0:0948 Core Selection Ferrite would normally be used for this type of application at the speciﬁed frequency. The material speciﬁcations for EPCOS N87 Mn-Zn are listed in Table 3.8. Inductor Design 81 Table 3.8 Material speciﬁcations Kc 16.9 a 1.25 b 2.35 Bsat 0.4 T The maximum stored energy in the primary is: L^I2p ¼ ð700 Â 10À6Þð3:393Þ2 ¼ 0:0081 J For Mn-Zn N87 ferrite, select Bmax ¼ 0.2 T. At this point, we do not know the core loss but, since we are dealing with a large current ripple, we can expect large core loss, so a good design rule is to set the iron loss at two times the copper loss, that is g ¼ 2. Ap, from Equation 3.35, is corrected for the two windings (see Problem 1): Ap ¼ " pﬃ1ﬃÀﬃﬃþﬃﬃﬃﬃﬃgﬃﬃKpipﬃLﬃﬃﬃpﬃÁ^Ip2p ﬃﬃﬃﬃﬃﬃ#8=7 "Bmax K t kup= ku pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DT #8=7 ¼ ð1 þ 2Þð0:4Þð0:p00ﬃ8ﬃﬃﬃ1ﬃﬃÞﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ Â 108 ¼ 6:89 cm4 ð0:2Þð48:2 Â 103Þð0:0948= 0:235Þð 30Þ The E55/28/21 core with Ap ¼ 9.72 cm4 is suitable. The core speciﬁcations are given in Table 3.9. The thermal resistance of this core, according to the manufacturer, is 10 C/W; therefore, the maximum dissipation, given by Equation 3.8, is: PD ¼ DT Ru ¼ 30 10 ¼ 3:0 W Since g ¼ 2, the total copper loss in 1.0 W and half of this is assigned to the primary winding giving Pcupmax ¼ 0.5 W. Table 3.9 Core and winding speciﬁcations Ac lc Wa Ap Vc kf ku MLT r20 a20 3.51 cm2 12.4 cm 2.77 cm2 9.72 cm4 43.5 cm3 1.0 0.235 11.3 cm 1.72 mV-cm 0.00393 82 Transformers and Inductors for Power Electronics The optimum value of the effective permeability for this core is given by Equation 3.20, corrected for two windings (see Problem 2): mopt ¼ sBﬃﬃmﬃﬃﬃaﬃxﬃﬃlﬃﬃcﬃﬃKﬃﬃﬃiﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m0 PcupmaxkupW a rwMLT ¼ ð4p Â ð0:2sÞðﬃﬃ1ﬃﬃ2ﬃﬃ:ﬃﬃ4ﬃﬃﬃÂﬃﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃﬃÀﬃﬃ2ﬃﬃÞﬃﬃðﬃﬃ0ﬃﬃ:ﬃﬃ4ﬃﬃÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 10À7Þ ð0:5Þð0:155Þð2:77 Â 10À4Þ ð1:72 Â 10À8Þð11:3 Â 10À2Þ ¼ 75 Winding Design Gap g Referring to Figure 3.1, increasing the value of effective permeability above the optimum value means that we can operate the core at its maximum ﬂux density and below the maximum dissipation. This makes sense, since we have not included any core loss at this point. The maximum gap length is found from Section 2.1: gmax ¼ lc mmin ¼ ð12:4 Â 10À2Þ 75 103 ¼ 1:65 mm The manufacturer provides a standard core set, with g ¼ 1 mm, a corresponding AL value of 496 nH and an effective permeability of 138. Turns For the core, the manufacturer supplies the AL value (i.e. the inductance per turn): N ¼ rﬃﬃﬃﬃﬃ L AL ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð700 Â 10À6Þ ð496 Â 10À9Þ ¼ 37:6 T urns Select 38 primary turns. The number of turns in the secondary winding is 38/6.2 ¼ 6 turns Wire Size The current density is: pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ Jo ¼ Kt pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDﬃﬃﬃﬃTﬃﬃﬃﬃpﬃﬃﬃﬃﬃ kuð1 þ gÞ 8 Ap ¼ 48:2 Â 103 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3ﬃq0 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð0:235Þð1 þ 2Þ 8 ð9:72 Â 10À8Þ Â 10À4 ¼ 236:6 A=cm2 The cross-sectional area of the conductor in the primary winding is: Awp ¼ Ip rms=Jo ¼ 1:357=236:6 ¼ 0:00574 cm2 Inductor Design 83 This corresponds to four 0.428 mm diameter copper wires in parallel. A 0.5 mm diameter wire has a DC resistance of 871 Â 10À6 V/cm at 20 C (Table A.1). The operating frequency is 70 kHz, corresponding to 0.25 mm skin depth. The radius of the wire is less than 1.5 skin depths, as required (see Figure 6.3). The cross-sectional area of the conductor in the secondary winding is: Aws ¼ Is rms=Jo ¼ 12:454=236:6 ¼ 0:0526 cm2 This corresponds to a copper foil 25.4 Â 0.2 mm, with a DC resistance of 33.86 Â 10À6 V/cm at 20 C (Table A.1). Copper Loss Primary winding: Use Equation 3.37 to correct the winding resistance for temperature: Tmax ¼ 60 þ 30 ¼ 90 C: Rdc ¼ ð38Þð11:3Þðð871=4Þ Â 10À6Þ½1 þ ð0:00393Þð90 À 20Þ Â 103 ¼ 119:2 mV The copper loss in the primary winding is: Pcu ¼ Rdc I 2 rms ¼ ð119:2 Â 10À3Þð1:357Þ2 ¼ 0:220 W Secondary winding: Use Equation 3.38 to correct the winding resistance for temperature: Tmax ¼ 60 þ 30 ¼ 90 C: Rdc ¼ ð6Þð11:3Þð33:86 Â 10À6Þ½1 þ ð0:00393Þð90 À 20Þ Â 103 ¼ 2:927 mV The copper loss in the secondary winding is: Pcu ¼ RdcI2rms ¼ ð2:927 Â 10À3Þð12:454Þ2 ¼ 0:454 W The total copper loss is 0.674 W. Core Loss The ﬂux density ripple DB may be calculated by using Faraday’s law, noting that the voltage across the inductor in the time period 0 to DT is Vi: DB ¼ V i DT NpAc ¼ 38 Â 325:3 Â 0:314 3:51 Â 10À4 Â 70 Â 103 ¼ 0:109 T 84 Transformers and Inductors for Power Electronics Using the General Steinmetz Equation (GSE), (Equation 1.29), the peak value of the ﬂux density Bmax is DB/2 and the core loss is: Pfe ¼ VcKcf aBbmax ¼ ð43:5 Â 10À6Þð16:9Þð70000Þ1:25ð0:109=2Þ2:35 ¼ 0:898 W Total Losses : Copper loss Core loss Total losses 0:674 W 0:898 W 1:572 W These losses are less than PD max of 3 W. 3.5 Problems 3.1 Derive the formula for Ap for a the two winding inductor. 3.2 Derive the formula for mopt for a two winding inductor. 3.3 Calculate the H ﬁeld in the dielectric of a coaxial cable with the following dimensions: the radius of the inner conductor is ri, and the inner and outer radii of the outer conductor are roi and roo respectively. 3.4 Describe the three types of power loss in a magnetic component. 3.5 A buck converter with 25A 100 ms pulse of output current operates at 80 kHz with a DC input voltage of 180 V and an output voltage of 24 V DC. Calculate the inductance required for a current ripple of 5%. Select a toroidal powder iron core to meet these requirements. The maximum allowable temperature rise in the inductor is 35 C. The transient thermal impedance of the inductor is 0.5 C/W for the 100 ms pulse. 3.6 A zero current switching quasi-resonant converter (ZCS-QRC) operates at 85 kHz with a nominal load current of 0.8A, and the DC input voltage is 48 V. The resonant inductor is 86 mH and the resonant capacitor is 33 nF. Select a ferrite core to meet these requirements, with the inductor isolation material not exceeding a temperature rise of 110 C with an ambient temperature of 65 C. 3.7 Design a common-mode choke for an induction hob which delivers 3000 W at 230 V. The purpose of the common-mode choke is to attenuate the common-mode current which generates EMI noise. The attenuation should be effective at 20 kHz, with common-mode impedance of 1000 V. The peak value of the common-mode current is 100 mA. The ambient temperature is 40 C and the maximum allowable temperature rise is 70 C. Use Mn-Zn ferrite. 3.8 A 30 W ﬂyback converter operates at 70 kHz with a DC input of 325 V, rectiﬁed mains and a DC output voltage of 5 V. Calculate the minimum value of the primary coil inductance to ensure continuous conduction. For a primary coil inductance of 620 mH, select a ferrite ETD core (N87 material in Table 1.1) and calculate the copper loss and core loss. The maximum temperature rise allowed in the winding is 30 C and the ambient temperature is 50 C. Inductor Design 85 MATLAB Program for Example 3.1 %example 3.1: Buck Converter with a gapped core alpha = 1.25 alpha20 = 0.00393 beta = 2.35 deltaT = 15 gamma = 0 muo = 4*pi*10^-7 row = 1.72e-8; Ac = 2.09e-4 AL = 188e-9 Ap2 = 5.62*10^-8 Bmax = 0.25 Bsat = 0.4 D = 0.5 DT = D/(80e3) f = 80e3 Idc = 20 Kc = 16.9 Ki = 1.0 Kt = 48.2e3 ku = 0.8 L = 34e-6 lc = 11.4e-2 MLT = 8.6e-2 N1 = 13 Pcumax = 1.36 Rteta = 11 Tmax = 70+15 Vc = 23.8*10^-6 Vi = 12 Vo = 6 Vs = 12 Wa = 2.69e-4 wire_Rdc=10.75*10^-6; deltaIL = (Vs-Vo)*DT/L Ipeak = Idc+deltaIL/2 LIpeak2 = L*Ipeak^2 Ap = [(sqrt(1+gamma)*Ki*L*Ipeak^2)/(Bmax*Kt*sqrt(ku*deltaT))]^(8/7) *10^8 86 Transformers and Inductors for Power Electronics PD = deltaT/Rteta muopt =(Bmax*lc*Ki)/(muo*sqrt((Pcumax*ku*Wa)/(row*MLT))) gmax = lc/muopt N = sqrt(L/AL) Jo = Kt*(sqrt(deltaT))/(sqrt(ku*(1+gamma))*Ap2^(1/8))*10^-4 Aw = Idc/Jo Rdc = N1*MLT*wire_Rdc*[1+alpha20*(Tmax-20)]*10^5 Pcu =(Rdc*Idc^2)*10^-3 deltaB =((Vi-Vo)*DT)/(N1*Ac) Bm = deltaB/2 Pfe = Vc*Kc*f^alpha*Bm^beta Ptot = Pfe+Pcu]]> MATLAB Program for Example 3.2 %example 3.2 Forward Converter Toroidal Core alpha = 1.41 alpha20 = 0.00393 beta = 2.56 deltaT = 20 gamma = 0 muo = 4*pi*10^-7 row = 1.72e-8; Ac = 0.678e-4 Ap2 = 2.58e-8 Bmax = 0.35 Vo = 9 Vi = 12 D = Vo/Vi f = 60e3 DT = D/f Idc = 1.9 Kc = 231.8 Ki = 1 Kt = 48.2*10^3 ku = 0.2 L = 1.6e-3 L1000 = 117e-3 lc = 8.98e-2 MLT = 5.27e-2 N1 = 117 Pcumax = 0.823 Inductor Design 87 Tmax = 60+20 Vc = 6.09e-6 Vs = 12 Wa = 3.8e-4 wire_Rdc = 218e-6; deltaIL = (Vs-Vo)*DT/L Ipeak = Idc+deltaIL/2 LIpeak2 = L*Ipeak^2 Ap = [(sqrt(1+gamma)*Ki*L*Ipeak^2)/(Bmax*Kt*sqrt(ku*deltaT))]^(8/7) *10^8 Rteta = 0.06/sqrt(Vc) PD = deltaT/Rteta muopt =(Bmax*lc*Ki)/(muo*sqrt((Pcumax*ku*Wa)/(row*MLT))) mueff =(Bmax^2*Ac*lc)/(muo*L*Ipeak^2) N = 1000*sqrt(L/L1000) H = N*Ipeak/lc Jo = Kt*(sqrt(deltaT))/(sqrt(ku*(1+gamma))*Ap2^(1/8))*10^-4 Aw = Idc/Jo Rdc = N1*MLT*wire_Rdc*[1+alpha20*(Tmax-20)]*10^5 Pcu = (Rdc*Idc^2)*10^-3 deltaB =((Vi-Vo)*DT)/(N*Ac) Bm = deltaB/2 Pfe = Vc*Kc*f^alpha*Bm^beta Ptot = Pfe+Pcu]]> MATLAB Program for Example 3.3 %example 3.3 Flyback Converter alpha = 1.25 alpha20 = 0.00393 beta = 2.35 deltaT = 30 gamma = 2 mumin = 67 muo = 4*pi*10^-7 row = 1.72e-8; a = 6.2 Ac = 3.51e-4 AL = 496e-9 Ap2 = 9.72e-8 Bm = 0.109/2 88 Transformers and Inductors for Power Electronics Bmax = 0.2 Bsat = 0.4 D = 0.314 f = 70e3 Kc = 16.9 kc = 16.9 Kt = 48.2e3 ku = 0.235 kup2 = 0.155 lc = 12.4e-2 Lp = 700e-6 MLT = 11.3e-2 N1 = 38 N2 = 6 Np = 38 NpoverNs = 6.2 P = 24*10 Pcumax = 0.5 Rteta = 10 T = 1/70e3 Vc = 43.5e-6 Vi = sqrt(2)*230 Vo = 24 Wa = 2.77e-4 Tmax = 60+30 wire_Rdc = (871/4)*10^-6 wire_Rdc2 = 33.86e-6; Lg = ((Vi^2*D^2)/P)*(T/2) D = 1/(1+Vi/(a*Vo)) Ip = P/(D*Vi) deltaIp = (Vi*D*T)/Lp Ippeak = Ip+deltaIp/2 yp = deltaIp/Ippeak Kip = sqrt(D*(1-yp+yp^2/3)) Iprms = Kip*Ippeak Is = P/((1-D)*Vo) deltaIs = NpoverNs*deltaIp Ispeak = Is+deltaIs/2 ys = deltaIs/Ispeak Kis = sqrt((1-D)*(1-ys+ys^2/3)) Isrms = Kis*Ispeak kup = ku*1/(1+Isrms/(a*Iprms)) LIppeak2 = Lp*Ippeak^2 Ap = [(sqrt(1+gamma)*Kip*Lp*Ippeak^2)/(Bmax*Kt*(kup/sqrt(ku))*sqrt (deltaT))]^(8/7) Inductor Design 89 PD = deltaT/Rteta muopt = (Bmax*lc*Kip)/(muo*sqrt((Pcumax*kup2*Wa)/(row*MLT))) gmax = lc/mumin N = sqrt(Lp/AL) Jo = Kt*sqrt(deltaT)/(sqrt(ku*(1+gamma))*Ap2^(1/8)) Awp = Iprms/Jo Aws = Isrms/Jo Rdc1 = N1*MLT*wire_Rdc*[1+alpha20*(Tmax-20)]*10^5 Pcu1 = (Rdc1*Iprms^2)*10^-3 Rdc2 = N2*MLT*wire_Rdc2*[1+alpha20*(Tmax-20)]*10^5 Pcu2 = (Rdc2*Isrms^2)*10^-3 deltaB = (Vi*D*T)/(Np*Ac) Pfe = Vc*kc*f^alpha*Bm^beta Ptot = Pcu1+Pcu2+Pfe]]> References 1. Venkatachalam, K., Sullivan, C.R., Abdallah, T., and Tacca, H. (2002) Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters. Proceedings of IEEE Workshop on Computers in Power Electronics, COMPEL, pp. 36–41. 2. McAdams, W.H. (1954) Heat Transmission, 3rd edn, McGraw-Hill, New York. 3. Judd, F. and Kressler, D. (1977) Design optimization of small low-frequency power transformers. IEEE Transac- tions on Magnetics, 13 (4), 1058–1069. 4. McLyman, C.W.T. (2004) Transformer and Inductor Design Handbook, 3rd edn, Marcel Dekker Inc., New York. 5. Kassakian, J.G., Schlecht, M.F., and Verghese, G.C. (1991) Principles of Power Electronics (Addison-Wesley Series in Electrical Engineering), Prentice Hall, Reading, MA. Further Reading 1. Bartoli, M., Reatti, A., and Kazimierczuk, M.K. (1994) High-frequency models of ferrite core inductors. Proceedings of the IEEE Industrial Electronics, Control and Instrumentation, IECON, pp. 1670–1675. 2. Bartoli, M., Reatti, A., and Kazimierczuk, M.K. (1994) Modelling iron-powder inductors at high frequencies. Proceedings of the IEEE Industry Applications Conference, IAS, pp. 1225–1232. 3. Bennett, E. and Larson, S.C. (1940) Effective resistance to alternating currents of multilayer windings. Transactions of the American Institute of Electrical Engineers, 59 (12), 1010–1017. 4. Blume, L.F. (1982) Transformer Engineering, John Wiley & Sons, New York. 5. Bueno, M.D.A. (2001) Inductance and Force Calculations in Electrical Circuits, Nova Science Publishers, Huntington. 6. Carsten, B. (1986) High frequency conductor losses in switchmode magnetics. Proceedings of the High Frequency Power Converter Conference, pp. 155–176. 7. Cheng, K.W.E. and Evans, P.D. (1994) Calculation of winding losses in high-frequency toroidal inductors using single strand conductors. IEE Proceedings on Electric Power Applications, B, 141 (2), 52–62. 8. Cheng, K.W.E. and Evans, P.D. (1995) Calculation of winding losses in high frequency toroidal inductors using multistrand conductors. IEE Proceedings on Electric Power Applications, B, 142 (5), 313–322. 9. Del Vecchio, R.M., Poulin, B., Feghali, P.T. et al. (2001) Transformer Design Principles: With Applications to Core-Form Power Transformers, 1st edn, CRC Press, Boca Raton, FL. 10. Dowell, P.L. (1966) Effects of eddy currents in transformer windings. Proceedings of the Institution of Electrical Engineers, 113 (8), 1387–1394. 11. Erickson, R.W. (2001) Fundamentals of Power Electronics, 2nd edn, Springer, Norwell, MA. 90 Transformers and Inductors for Power Electronics 12. Evans, P.D. and Chew, W.M. (1991) Reduction of proximity losses in coupled inductors. IEE Proceedings on Electric Power Applications, B, 138 (2), 51–58. 13. Ferreira, J.A. (2010) Electromagnetic Modelling of Power Electronic Converters (Power Electronics and Power Systems), 1st edn, Springer, Norwell, MA. 14. Fitzgerald, A.E., Kingsley, C., and Umans, S.D. (2002) Electric Machinery, 6th edn, McGraw-Hill, New York. 15. Flanagan, W.M. (1992) Handbook of Transformer Design and Application, 2nd edn, McGraw-Hill, New York. 16. Georgilakis, P.S. (2009) Spotlight on Modern Transformer Design (Power Systems), 1st edn, Springer, New York. 17. Hanselman, D.C. and Peake, W.H. (1995) Eddy-current effects in slot-bound conductors. IEE Proceedings on Electric Power Applications, B, 142 (2), 131–136. 18. Hoke, A.F. and Sullivan, C.R. (2002) An improved two-dimensional numerical modeling method for E-core trans- formers. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 151–157. 19. Hurley, W.G., Wolﬂe, W.H., and Breslin, J.G. (1998) Optimized transformer design: inclusive of high-frequency effects. IEEE Transactions on Power Electronics, 13 (4), 651–659. 20. Jieli, L., Sullivan, C.R., and Schultz, A. (2002) Coupled-inductor design optimization for fast-response low- voltage. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 817–823. 21. Kassakian, J.G. and Schlecht, M.F. (1988) High-frequency high-density converters for distributed power supply systems. Proceedings of the IEEE, 76 (4), 362–376. 22. Kazimierczuk, M.K. (2009) High-Frequency Magnetic Components, John Wiley & Sons, Chichester. 23. Krein, P.T. (1997) Elements of Power Electronics (Oxford Series in Electrical and Computer Engineering), Oxford University Press, Oxford. 24. Kulkarni, S.V. (2004) Transformer Engineering: Design and Practice, 1st edn, CRC Press, New York. 25. B.H.E. Limited (2004) Transformers: Design, Manufacturing, and Materials (Professional Engineering), 1st edn, McGraw & -Hill, New York. 26. McLyman, C.W.T. (1997) Magnetic Core Selection for Transformers and Inductors, 2nd edn, Marcel Dekker Inc., New York. 27. McLyman, C.W.T. (2002) High Reliability Magnetic Devices, 1st edn, Marcel Dekker Inc., New York. 28. E.S. MIT (1943) Magnetic Circuits and Transformers (MIT Electrical Engineering and Computer Science), The MIT Press, Cambridge, MA. 29. Muldoon, W.J. (1978) Analytical design optimization of electronic power transformers. Proceedings of Power Electronics Specialists Conference, PESC, pp. 216–225. 30. Pentz, D.C. and Hofsajer, I.W. (2008) Improved AC-resistance of multiple foil windings by varying foil thick- ness of successive layers. The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 27 (1), 181–195. 31. Perry, M.P. (1979) Multiple layer series connected winding design for minimum losses. IEEE Transactions on Power Apparatus and Systems, PAS-98 (1) 116–123. 32. Petkov, R. (1996) Optimum design of a high-power, high-frequency transformer. IEEE Transactions on Power Electronics, 11 (1), 33–42. 33. Pollock, J.D., Lundquist, W., and Sullivan, C.R. (2011) Predicting inductance roll-off with dc excitations. Proceedings of the IEEE Energy Conversion Congress and Exposition, ECCE, pp. 2139–2145. 34. Pressman, A.I., Bellings, K., and Morey, T. (2009) Switching Power Supply Design, 3rd edn, McGraw-Hill, New York. 35. Ramo, S., Whinnery, J.R., and Van Duzer, T. (1984) Fields and Waves in Communication Electronics, 2nd edn, John Wiley & Sons, New York. 36. Smith, B. (2009) Capacitors, Inductors and Transformers in Electronic Circuits (Analog Electronics Series), Wexford College Press, Wexford. 37. Snelling, E.C. (1988) Soft Ferrites: Properties and Applications, 2nd edn, Butterworths, London. 38. Sullivan, C.R. (1999) Optimal choice for number of strands in a litz-wire transformer winding. IEEE Transactions on Power Electronics, 14 (2), 283–291. 39. Sullivan, C.R. and Sanders, S.R. (1996) Design of microfabricated transformers and inductors for highfrequency power conversion. IEEE Transactions on Power Electronics, 11 (2), 228–238. 40. Urling, A.M., Niemela, V.A., Skutt, G.R., and Wilson, T.G. (1989) Characterizing high-frequency effects in transformer windings-a guide to several signiﬁcant articles. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 373–385. Inductor Design 91 41. Van den Bossche, A. (2005) Inductors and Transformers for Power Electronics, 1st edn, CRC Press, New York. 42. Vandelac, J.P. and Ziogas, P.D. (1988) A novel approach for minimizing high-frequency transformer copper losses. IEEE Transactions on Power Electronics, 3 (3), 266–277. 43. Venkatraman, P.S. (1984) Winding eddy current losses in switch mode power transformers due to rectangular wave currents. Proceedings of the 11th National Solid-State Power Conversion Conference, Powercon 11, pp. A1.1–A1.11. 44. Williams, R., Grant, D.A., and Gowar, J. (1993) Multielement transformers for switched-mode power supplies: toroidal designs. IEE Proceedings on Electric Power Applications, B, 140 (2), 152–160. 45. Ziwei, O., C. Thomsen, O., and Andersen, M. (2009) The analysis and comparison of leakage inductance in different winding arrangements for planar transformer. Proceedings of the IEEE Power Electronics and Drive Systems, PEDS, pp. 1143–1148. Section Two Transformers 4 Transformers1 The transformer is used in three broad areas of application: Mains power transmission, which involves raising or lowering voltage in an AC circuit with a corresponding decrease or increase in current. Signals transmission with impedance matching for maximum power transfer. In power electronics, for energy conversion and control. In power generation and transmission, the generator may operate in the 10–20 kV range, whereas high voltage transmission is normally above 200 kV; distribution will be at, say, 10 kV, being further stepped down to 110 V or 230 V for residential supplies. Transformers are also ideally suited to impedance matching for transfer maximum power in, say, an audio system, where the speaker load resistance might be 8 V. This would be matched to the output impedance of an ampliﬁer measured in kV. An important application for power electronics is electrically isolating one circuit from another to satisfy safety regulatory requirements. A fundamental principle of transformer operation is that the size is inversely proportional to the operating frequency (up to a point, as we shall see later in Chapter 5), and this has opened up the role of the transformer from its more traditional role at power frequencies. When the transformer is used in power electronics applications that incorporate electrical isolation, the voltage that appears across a switch can be adjusted by the transformer to reduce the stresses on the switch. A transformer consists of two or more mutually coupled windings. An alternating voltage source is connected to one of the windings – usually referred to as the primary winding – and 1 Parts of this chapter are reproduced with permission from [1] Hurley, W.G., Wilcox, D.J., and McNamara, P.S. (1991) Calculation of short circuit impedance and leakage impedance in transformer windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 651–658; [2] Hurley, W.G. and Wilcox, D.J. (1994) Calculation of leakage inductance in transformer windings. IEEE Transactions on Power Electronics 9 (1), 121–126; [3] Hurley, W.G., Wolﬂe, W.H., and Breslin, J.G. (1998) Optimized transformer design: inclusive of highfrequency effects. IEEE Transactions on Power Electronics 13 (4), 651–659. Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 96 Transformers and Inductors for Power Electronics this produces a changing magnetic ﬂux ﬁeld in accordance with the laws outlined in Chapter 2. The resultant ﬂux will depend on the number of turns in the primary winding. Normally, the windings are wound on a core of magnetic material without an air gap to obtain high ﬂux levels, so therefore the ﬂux will depend on the reluctance of the core, including the physical dimensions of the core length and cross-sectional area in addition to the number of turns. The manufacturing process may produce a very small air gap, which has the advantage of being able to control the magnitude of the inrush current. The magnetic ﬂux is coupled to the other winding – called the secondary winding – and a voltage is induced in accordance with laws of electromagnetic induction. An inductor stores energy, whereas, in a transformer, the energy is transferred from the primary to the secondary load. Normally, iron laminations are used in the construction of large mains transformers to reduce eddy current loss in the core. Compressed ferromagnetic alloys (ferrites) are used in power electronic circuit applications for high frequency operation. 4.1 Ideal Transformer A basic two-winding transformer is shown in Figure 4.1, where the windings are wound on a magnetic core. Sinusoidal excitation is applied to the input winding and the second winding is on opencircuit. These windings are usually referred to as the primary and secondary windings respectively. Drawing on the development of the concept of inductance in Chapter 2, the primary winding has an inductance Lm called the magnetizing inductance. This is given by: Lm ¼ N 2 1 Rc ð4:1Þ and the reluctance of the core is Rc ¼ lc mrm0Ac ð4:2Þ where lc is the mean length of the magnetic path around the closed core and Ac is the crosssectional area of the core. Evidently, as the relative permeability of the core increases the reluctance becomes smaller, this in turn means that the mmf (N1I1) required to establish the ﬂux in the core also iΦ + + v1 e1 _ _ φ N1 N2 Figure 4.1 Two-winding transformer: no load conditions. Transformers 97 becomes smaller. For this reason, it is normally assumed that the magnetizing current to establish the ﬂux in the core is inﬁnitesimally small. 4.1.1 No Load Conditions The secondary winding in Figure 4.1 is in open circuit under no load conditions. A magnetizing current if ﬂows in the primary winding, which establishes the alternating ﬂux f in the magnetic core.The basic relationship between the applied voltage and the ﬂux in the core follows from Ampere’s law and Faraday’s law: e1 ¼ d l1 dt ¼ N1 d f1 dt ð4:3Þ By Lenz’s law, e1 is a counter-emf to v1 and, in accordance with Kirchhoff’s voltage law, v1 ¼ e1. At this point, we shall assume that we are dealing with sinusoidal excitation at frequency f (v ¼ 2pf) and the amplitude of the ﬂux is fmax: fðtÞ ¼ fmaxsin vt e1 ¼ N1 df dt ¼ vN 1 fmax cos vt ð4:4Þ ð4:5Þ or, in terms of the sine function: e1ðtÞ ¼ vN1fmaxsin vt þ p 2 which shows that the ﬂux lags the applied voltage by 90. The amplitude of the primary EMF is ð4:6Þ E1max ¼ 2pf N1fmax ð4:7Þ The magnetic ﬂux may be expressed in terms of the ﬂux density: fmax ¼ BmaxAc ð4:8Þ and it follows that: E1rms ¼ Ep1mﬃﬃax 2 ¼ 4:44 f N1BmaxAc ð4:9Þ Since V1rms ¼ E1rms, the rms value of the input voltage is related to the number of turns in the input or primary winding, the maximum ﬂux density and core cross-sectional area by: V 1rms ¼ 4:44 f N1BmaxAc ð4:10Þ This is the celebrated transformer equation for sinusoidal excitation. 98 Transformers and Inductors for Power Electronics i1 + + v1 e1 _ _ Hc N1 N2 i2 + + e2 v2 _ _ Figure 4.2 Two-winding transformer: load conditions. 4.1.2 Load Conditions At this point, we shall apply a load to the secondary winding with N2 turns, causing a current i2 to ﬂow, as shown in Figure 4.2. The voltage v1 applied to the primary winding establishes the ﬂux f as before: v1 ¼ e1 ¼ N1 df dt ð4:11Þ The common core ﬂux links the secondary winding and induces an emf e2 in the secondary winding and a voltage v2 across the load: v2 ¼ e2 ¼ N2 df dt ð4:12Þ Taking the ratio v1/v2 from Equations 4.11 and 4.12 yields: v1 ¼ N1 v2 N2 ð4:13Þ In terms of the rms values of the voltages: V1rms ¼ N1 ¼ a V 2rms N2 ð4:14Þ In summary, the voltage transformation ratio is directly proportional to the transformer turns ratio a. The next step is to look at the effect of the load current. The mmf corresponding to the load current is N2i2. Ampere’s law dictates that the integral of the magnetic ﬁeld intensity around a closed loop that links the primary and secondary windings is equal to the net mmf. With the direction of the mmf given by the right hand rule, then referring to Figure 4.2, we have: Hclc ¼ N1i1 À N2i2 ð4:15Þ The negative sign of N2i2 arises because the current in the secondary winding opposes the ﬂux f by the right hand screw rule convention. Transformers 99 The ﬂux density Bc inside the core is related to Hc by the magnetic permeability: Bc ¼ mrm0Hc ð4:16Þ The ﬂux inside the core is: fc ¼ BcAc ð4:17Þ Combining Equations 4.2, 4.15, 4.16 and 4.17 yields: N1i1 À N2i2 ¼ fc Â Rc ð4:18Þ In the ideal transformer, we assume the core has inﬁnite permeability (mr ! 1), that the resistance of the windings is negligible and that there is no core loss (sc ! 0). Inﬁnite permeability in the core means that the magnetic reluctance is negligible, which, in turn, means that the mmf required to establish the ﬂux is negligibly small. Thus, if the secondary mmf N2i2 is established by the load, it must be countered by an mmf N1i1 in the primary to satisfy Equation 4.18. In the hypothetical ideal transformer, Equation 4.18 becomes: N1i1 À N2i2 ¼ 0 ð4:19Þ so: N1i1 ¼ N2i2 ð4:20Þ or: i1 ¼ N2 i2 N1 ð4:21Þ And in terms of rms values: I1rms ¼ N2 ¼ 1 I2rms N1 a ð4:22Þ Thus, an ideal transformer changes currents in the inverse of the turns ratio of its windings. There are no losses in an ideal transformer and the input power is equal to the output power: v1i1 ¼ v2i2 ð4:23Þ Therefore: i1 ¼ v2 ¼ N2 ¼ 1 i2 v1 N1 a ð4:24Þ 4.1.3 Dot Convention The windings in Figure 4.2 show the input current into the positive voltage terminal of the primary winding and the load current out of the positive voltage terminal of the secondary winding. This conveniently meets the conditions imposed on the mmf by Ampere’s law. However, we could just as easily draw the windings as shown in Figure 4.3, and again we 100 Transformers and Inductors for Power Electronics i1 + + v1 _ e1 _ N1 N2 _ _ e2 v2 + + i2 Figure 4.3 The ideal transformer: load conditions, alternative winding. I1 + I2 + V1 V2 Z2 N1:N2 _ _ Figure 4.4 Electrical circuit symbol for a transformer. can judiciously select the positive voltage terminal and the positive direction of current, so the relationship in Equation 4.15 holds. Obviously, great care must be taken in drawing the windings and in selecting the voltage and current polarities; adding more windings makes the situation more complex. In reality, most transformers are enclosed and it is not possible by inspection to tell the direction in which each coil is wound. In order to avoid any confusion, the dot convention is adopted. The dot markings in Figures 4.2 and 4.3 indicate terminals of corresponding polarity. If one follows through either winding, beginning at the dotted terminal, both windings encircle the core in the same direction with respect to ﬂux (in accordance with the right hand screw rule). Using this convention, the voltages at the dotted terminals are of the same instantaneous polarity for the primary and secondary windings. Similarly, the currents as shown are in phase. According to the convention, the instantaneous currents are in opposite directions through the windings, so therefore their mmfs cancel. In the electrical circuit symbol for a transformer, we can deduce the physical direction of the winding from the dot convention. Thus, the transformers of Figures 4.2 and 4.3 are represented by the electrical circuit symbol in Figure 4.4. The parallel bars between the two windings represent a common ferromagnetic core. 4.1.4 Reﬂected Impedance When signals are transmitted in a circuit, the maximum power transfer theorem dictates that maximum power is transferred from a source to a load when the source impedance is equal to the load impedance. The transformer may be used to match the impedances between the source and load. Transformers 101 I1 + I2 + I1 + V1 V2 Z2 V1 a2Z2 N1:N2 _ _ _ (a) (b) Figure 4.5 Reﬂected impedance in a transformer winding. The ratio V1/I1 is the impedance seen by the input terminals of the transformer, recalling: V1 ¼ aV2 ð4:25Þ I1 ¼ 1 a I2 ð4:26Þ Z 1 2 ¼ V1 I1 ¼ a2 V2 I2 ¼ a2 Z 2 ð4:27Þ where Z2 is the impedance of the load. Thus, the impedance Z2 in the secondary may be replaced by the equivalent impedance Z12, as seen from the primary terminals. The transformer equivalent circuit of Figure 4.5(a) is shown in Figure 4.5(b). Reﬂected impedance is commonly used in electronic circuits to achieve maximum power transfer. 4.1.5 Summary In an ideal transformer: 1. Voltages are transformed in the direct ratio of turns: V1 ¼ N1 N2 V2 ¼ aV 2 2. Currents are transformed in the inverse ratio of turns: I1 ¼ N2 N1 I2 ¼ 1 a I2 3. Impedances are transformed in the direct ratio squared: Z 1 2 ¼ N N 1 2 2 Z 2 ¼ a2 Z 2 The notation Z12 means the secondary impedance Z2 reﬂected in the primary. ð4:28Þ ð4:29Þ ð4:30Þ 102 Transformers and Inductors for Power Electronics The voltage impressed on a winding is related to the frequency, the number of turns, the maximum ﬂux density and the core cross-sectional area. Vrms ¼ 4:44 f NBmaxAc ð4:31Þ In building up the equivalent electrical circuit for the transformer, we can refer quan- tities in one winding to another winding so that the secondary voltage reﬂected into the primary winding is V 1 2 and the secondary voltage reﬂected into the primary winding is V21. The relationships are: V 1 2 ¼ N1 N2 V2 ¼ aV 2 V 2 1 ¼ N2 N1 V1 ¼ 1 a V1 ð4:32Þ ð4:33Þ 4.2 Practical Transformer So far, we have idealized the transformer to simplify its analysis. However, in a practical transformer, the following factors must be taken into account: magnetizing current and core loss; winding resistance; magnetic leakage ﬂux. In power electronics applications, winding capacitance may be an issue because a resonance condition can occur at high frequency. We will deal with winding capacitance in Chapter 8. 4.2.1 Magnetizing Current and Core Loss The current in the primary winding of a transformer plays two roles: 1. It sets up the mutual ﬂux in accordance with Ampere’s law. 2. It balances the demagnetizing effect of the load current in the secondary winding. The net mmf is N1I1 À N2I2 and, in terms of the magnetic circuit law, this may be related to the reluctance of the transformer core: N1I1 À N2I2 ¼ fmR ð4:34Þ Transformers 103 I1 I21 + Iφ I2 + V1 Xm =ωLm _ V2 Z2 N1:N2 _ Figure 4.6 Electrical circuit for a transformer with the magnetizing inductance. We had previously assumed that, in the ideal transformer, the core had inﬁnite permeabil- ity. In reality, however, there is a ﬁnite permeability and there is an inductance associated with the reluctance of the core. We call this the magnetizing inductance Lm, as described in Section 4.1 and shown in Figure 4.6. Thus, the primary current has two components: the magnetizing component If and a load component reﬂected into the primary I12: I 1 2 ¼ 1 a I2 ð4:35Þ The instantaneous magnetizing current, if, which establishes the ﬂux in the ferromagnetic core, is determined by the magnetic properties of the core. Let us examine the establishment of the core ﬂux in more detail. Returning to no-load conditions and assuming as before that the winding resistances are negligible, according to Equations 4.4 and 4.6, the applied voltage leads the ﬂux in the core by 90. At this point, we need to turn our attention to the magnetizing current. To simplify the construction of the magnetizing current curve, we will use a single value normal magnetization curve for ﬂux versus current, as shown in Figure 4.7. This assumption, in effect, neglects hysteresis. The ﬂux corresponding to the current oa is ab and the value of the current oa on the horizontal current axis of the magnetizing curve is drawn vertically at b in the time domain to give the vertical value of the current at that instant. In this manner, the magnetizing current on the time graph is generated and the resultant is shown, reﬂecting the effect of saturation. The ﬁrst observation is that the magnetizing current and ﬂux are in phase as expected, because mmf is the product of ﬂux and reluctance. The second observation is that the distorted magnetizing current contains harmonics, and Fourier analysis shows that these are odd harmonics. This further shows that the percentage of third and ﬁfth harmonics will increase with increased distortion as the core goes further into saturation. Evidently, the peak value of the magnetizing current will increase rapidly as the transformer goes further into saturation. We assumed a single valued magnetization curve and neglected the hysteresis loss; the construction in Figure 4.7 may be repeated for hysteresis by noting the rising and falling values of ﬂux, 104 Flux b Transformers and Inductors for Power Electronics i b φ o oa Current i a Time Figure 4.7 Magnetizing current wave shape. which introduces further distortion of the magnetization curve, but the overall effect on harmonics is not radically altered. Hysteresis is a power loss in the core and, therefore, it will introduce a component of the magnetizing current that is in phase with the applied voltage. In power electronics applications, eddy current loss in the core will also add to the hysteresis loss, and the current representing these losses will also be in phase with the applied voltage. As a ﬁrst approximation, the magnetizing current can therefore be split into two components: one in phase with the applied voltage for the core loss Ic; and the other in phase with the ﬂux Im. This approach allows us to construct phasor diagrams for the transformer. The harmonic components of the magnetizing current could, in some circumstances, lead to resonant conditions with capacitive components of connected circuits. The magnetizing inductance Lm or the magnetizing reactance Xm may represent the ﬂux in the core, and Rc may represent the core loss; the components of current through these circuit elements combine to form the magnetizing current. The magnetizing current is now represented by a shunt branch connected across V1, consisting of Rc and Xm in parallel, as shown in Figure 4.8. I1 + Ic V1 Rc I21 Iφ Im Xm I2 + V2 Z2 N1:N2 _ _ Figure 4.8 Electrical circuit for a transformer with the magnetizing branch. Transformers 105 4.2.2 Winding Resistance Winding resistance can be represented by the resistances of the wires used in the windings, R1 and R2 for primary and secondary, respectively. The AC resistance due to the internal ﬂux in the conductor may be approximated by 2 ro4 3 Rac ¼ Rdc641 þ 48 þ 0d:8ro475 d ð4:36Þ Where d is the skin depth in the conductor, as deﬁned in Equation 1.21, and ro is the radius of the conductor. For high-frequency operation, we have to take AC loss in the form of skin effect and proximity effects into account. We will deal with these effects in Chapter 6. 4.2.3 Magnetic Leakage In the ideal transformer, the same ﬂux links both the primary and secondary circuits. However, in practice, there is always some leakage ﬂux which links only one winding. Leakage inductance is a property of one winding relative to another. If there is a third winding on the transformer core, the primary-secondary leakage will be distinctly different from the primary-tertiary leakage and so on. Consider the two elementary coils in air presented in Figure 4.9, which constitute an elementary transformer. In Figure 4.9(a), coil 1 has an alternating current i1 applied and coil 2 is open-circuited. This produces a magnetic ﬁeld described by the ﬂux lines in the diagram. Some of this ﬂux links the second coil and is thus termed the mutual ﬂux. The common or mutual ﬂux is represented by dotted lines. The remaining ﬂux does not link the secondary, and is termed leakage ﬂux (depicted by solid lines). In Figure 4.9(b), coil 1 is open-circuited and a current i2 is applied to coil 2. Again, leakage ﬂux is represented by solid ﬂux lines and mutual ﬂux is denoted by dotted lines. Clearly, the nature of the two leakage ﬁelds is quite different. Transformer action occurs leakage flux common flux Coil 2 Coil 1 Coil 2 Coil 1 i1 + _ (a) i2 + _ (b) Figure 4.9 Leakage inductance in a transformer. 106 Transformers and Inductors for Power Electronics when current ﬂows in both coils. In this case, the ﬂux linking both coils has three components. If w11 is the total ﬂux linking coil 1 and w22 is the total ﬂux linking coil 2, then [1,2]: f11 ¼ fl1 þ f21 þ f12 ð4:37Þ f22 ¼ fl2 þ f12 þ f21 ð4:38Þ Where wl1 is the leakage ﬂux associated with coil 1 due to i1 in coil 1, w21 is the ﬂux linking both coils due to the current i1, w12 is the mutual ﬂux due to the current i2 in coil 2 and wl2 is the leakage ﬂux of coil 2. Each of these ﬂux elements represents an inductance.From the ﬂux equa- tions, we can derive equations for the voltages on each coil: V1 ¼ ½Ll1 þ L1 di1 dt þ M12 di2 dt ð4:39Þ V2 ¼ ½Ll2 þ L2 di2 dt þ M21 di1 dt ð4:40Þ From the form of these equations, we can extract the self inductances of coil 1 and coil 2 as L11 and L22 respectively, where: L11 ¼ Ll1 þ L1 L22 ¼ Ll2 þ L2 ð4:41Þ ð4:42Þ Of course, as is always the case with mutual inductance, M12 ¼ M21 ¼ M. Current in coil 1 sets up ﬂux, some of which links coils 2. By deﬁnition, the mutual induc- tance is the ratio of the ﬂux linking one coil due to the current in the other coil, so that: L1 ¼ N1 N2 M L2 ¼ N2 N1 M ð4:43Þ ð4:44Þ Thus: pﬃﬃﬃﬃﬃﬃﬃﬃﬃ M ¼ L1L2 ð4:45Þ The leakage inductance terms in Equations 4.41 and 4.42 may be obtained using Equations 4.43 and 4.44: Ll1 ¼ L11 À N1 N2 M ð4:46Þ Ll2 ¼ L22 À N2 N1 M ð4:47Þ Deﬁne: k1 ¼ L1 L11 ¼ 1 À Ll1 L11 ð4:48Þ Transformers 107 φ common flux + + _ _ primary leakage flux secondary leakage flux Figure 4.10 Leakage inductance in a transformer. and: k2 ¼ L2 L22 ¼ 1 À Ll2 L22 ð4:49Þ It follows that: pﬃﬃﬃﬃﬃﬃﬃﬃﬃ k ¼ k1k2 ð4:50Þ and: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ M ¼ k L11L22 ð4:51Þ k is called the coupling coefﬁcient. Taking this deﬁnition of k and the circuit relationships in Equations 4.39 and 4.40, with the appropriate dot convention, yields the classical equivalent electrical circuit representation of the transformer – the model that is normally used in circuit simulations of the transformer. Figure 4.10 shows the physical layout of the winding with the dot convention and the equivalent electrical circuit with coupled inductors. Leakage inductance is affected by high frequency operation; this will be discussed in Chapter 6. The leakage affects can be represented by primary and secondary leakage inductors or primary and secondary leakage reactances Xl1 and Xl2, respectively. 4.2.4 Equivalent Circuit The equivalent circuit model is now complete and shown in Figure 4.11(a). For an ideal transformer: E1 ¼ a E2 ð4:52Þ As a further step, we can refer all quantities in the secondary to primary in order to obtain the equivalent circuit of Figure 4.11(b). The leakage reactance of the secondary winding referred to the primary winding is: X1l2 ¼ a2Xl2 ð4:53Þ 108 Transformers and Inductors for Power Electronics R1 Xl 1 + I1 Ic (a) V1 Rc _ R1 Xl1 + I1 Ic (b) V1 Rc I21 Iφ + Im XcE1 N1:N2 _ Xl 2 + I2 E2 _ Iφ Im Xc Xl 1 2 I2′ R2 + V2 ZL _ R21 + V21 ZL1 _ + I1 (c) V1 Iφ Ic Im Rc Xc Xeq1 I2′ _ Req1 + V21 ZL1 _ _ Figure 4.11 Transformer equivalent circuits. The resistance of the secondary winding referred to the primary winding is: R12 ¼ a2R2 ð4:54Þ The voltage across the secondary winding referred to the primary is given by Equation 4.32: V 1 2 ¼ aV 2 ð4:55Þ And the current in the secondary referred to the primary is: I12 ¼ 1 a I2 ð4:56Þ Finally, we can combine corresponding quantities such as winding resistances and leakage reactances: Req ¼ R1 þ a2R2 ð4:57Þ Xeq ¼ Xl1 þ a2Xl2 ð4:58Þ Transformers 109 The shunt branch representing the core loss and the core magnetization may be moved to the input terminal with little loss of accuracy to obtain Figure 4.11(c). This is an approximation to Figure 4.11(b), since If will change slightly, but it is very small, so the error is negligible. However, it greatly simpliﬁes circuit calculations. This is the usual equivalent electrical circuit that is used to represent transformers and, by traditional circuit analysis load voltage, regulation and transformer efﬁciency may be determined. 4.3 General Transformer Equations So far, we have treated the transformer in its traditional role with sinusoidal excitation. For power electronics applications, we have to expand the analysis to include non-sinusoidal excitation and deal with frequencies well above the typical mains frequencies. We will begin by generalizing the equations for voltage, power and losses. The dissipation of the losses will determine the temperature rise in the windings, which will lead to an optimization of the transformer core size. In Chapter 6, we will show that further optimization will result from the analysis of high frequency loss in the windings. 4.3.1 The Voltage Equation Faraday’s law relates the impressed voltage on a winding v to the rate of change of ﬂux density B, recalling Equation 4.11: v ¼ N df dt ¼ NAm dB dt ð4:59Þ where N is the number of turns and Am is the effective cross-sectional area of the magnetic core. In the case of laminated and tape-wound cores, this is less than the physical area, Ac, due to interlamination space and insulation. The layout of a typical transformer is shown in Figure 4.12 and the physical parameters are illustrated. The two areas are related by the core stacking factor, kf (Am ¼ kf Ac). Typically, kf is 0.95 for laminated cores. Mean Length of a Turn, MLT Window area,Wa Volume of windings,VW Cross-sectional area, AC Volume of 2ro core,VC Figure 4.12 Typical layout of a transformer. 110 Transformers and Inductors for Power Electronics The average value of the applied voltage during the interval t from the point where the ﬂux density is zero to the point where it is at its maximum value (Bmax) is hvi. This may be found by integrating Equation 4.59 [3]: hvi ¼ 1 ðt vðtÞdt ¼ t0 1 t NAm ð Bmax 0 dB ð4:60Þ ¼ 1 t N Am Bmax We want to relate this to the rms value of the applied voltage waveform. The form factor k is deﬁned as the ratio of the rms value of the applied voltage waveform to the average value hvi: k ¼ V rms hvi ð4:61Þ Combining Equations 4.60 and 4.61 yields: V rms ¼ k t f N BmaxAm ¼ Kvf N BmaxAm ð4:62Þ T with: Kv ¼ k t ¼ k tf ð4:63Þ T where f ¼ 1 is the frequency of the periodic applied voltage v(t), and T is the period of v(t). T Equation 4.62 has the same form as the classical transformer voltage equation, as given in Equation 4.9, with K, the waveform factor, deﬁned by k, t and T (or f ). For a sinusoidal waveform, K ¼ 4.44 and, for a square waveform, K ¼ 4.0. The calculation of K for typical power electronic applications will be given in later examples. Example 4.1 Establish the value of Kv for a square waveform. Figure 4.13 shows the voltage and ﬂux distribution in a transformer winding with a square wave of voltage applied to the winding. The ﬂux rises from 0 to Bmax in time t ¼ T/4 and, therefore, t/T ¼ 0.25. The form factor for a square wave is 1 since the average value over the time t is Vdc and the rms value of the waveform is Vdc. Kv is 1/0.25 ¼ 4.0. Example 4.2 Establish the value of Kv for the input voltage waveform in a forward converter. The input voltage and ﬂux waveforms for a forward converter are shown in Figure 4.14. The ratio (Np/Nt) is ﬁxed such that: Np Nt ¼ 1 D À D so that the volt-seconds balance in the winding is maintained. Transformers 111 e1 0 t B Bmax 0 t -Bmax τ T Figure 4.13 Square-wave voltage and ﬂux waveforms. The ﬂux density increases from 0 to its maximum value in the time t ¼ DT. The rms value of the voltage waveform in Figure 4.14 is: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Vrms ¼ D 1 À V D dc The average value of the voltage waveform during the time t is Vdc: hvi ¼ Vdc Vp Vdc 0 t DT T – Np Nt Vdc B Bmax 0 t τ T Figure 4.14 Voltage and ﬂux waveforms with duty cycle D. 112 and thus, from Equation 4.61: and from Equation 4.63: Transformers and Inductors for Power Electronics rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k ¼ V rms hvi ¼ D 1ÀD Kv ¼ pﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Dð1 À DÞ 4.3.2 The Power Equation Equation 4.62 applies to each winding of the transformer. It is straightforward to calculate the voltage Â current product or VA rating of each winding in the transformer. Taking the sum of the VA products in an n winding transformer and taking the voltage given by Equation 4.62: X Xn VA ¼ Kvf BmaxAm NiIi i¼1 ð4:64Þ Ni is the number of turns in winding i that carries a current with rms value Ii. The window utilization factor ku is the ratio of the total conduction area Wc for all conduc- tors in all windings to the total window winding area Wa of the core: ku ¼ Wc Wa ð4:65Þ The total conduction area is related to the number of conductors (turns) and the area of each conductor summed over all the windings: Xn Wc ¼ NiAwi i¼1 ð4:66Þ where Awi is the conducting area of the wire in winding i. Substituting Equation 4.66 in Equation 4.65: ku ¼ Pn i¼1 N i Awi Wa ð4:67Þ Thus: Xn NiAwi ¼ kuWa i¼1 ð4:68Þ Transformers 113 The current density in each winding is Ji ¼ Ii/Awi. Normally, the wire area and the conduction area are taken as the area of bare conductor. However, we can account for skin effect in a conductor and proximity effect between conductors by noting that the increase in resistance due to these effects is manifested by reducing the effective conduction area. The skin effect factor, ks, is the increase in resistance (or decrease in conduction area) due to skin effect, and likewise for the proximity effect factor, kx: ks ¼ Rac Rdc kx ¼ R0ac Rdc ð4:69Þ ð4:70Þ Incorporating these deﬁnitions into the window utilization factor: ku ¼ kb kskx ð4:71Þ where kb is the ratio of bare conductor total area to the window winding area. The deﬁnition in Equation 4.71 makes allowance in the window utilization factor for skin and proximity effects. At this point, we do not have analytical expression for skin and proximity effects; we will deal with these effects in detail in Chapter 6. Typically, kb ¼ 0.7, ks ¼ 1.3 and kx ¼ 1.3, giving ku ¼ 0.4. Combining Equations 4.64 and 4.68, with the same current density Jo in each winding, yields the total VA for all the windings. In Section 3.3, we showed that the optimum distribu- tion of current between multiple windings is achieved when the same current density is applied to each winding: X VA ¼ Kvf Bmaxkf AcJokuWa ð4:72Þ The product of the core cross-sectional area and the window winding area AcWa appears in Equation 4.72 and is an indication of the core size, and is designated window-cross-section product Ap. Rearranging Equation 4.72 relates the summation of the VA ratings of all the windings to the physical, electrical and magnetic properties of the transformer: X VA ¼ Kvf BmaxJokf kuAp ð4:73Þ 4.3.3 Winding Loss The ohmic or I2R loss in any of the windings is: I2R ¼ rw lwi Awi I 2 i ¼ rw N i MLTðJ o Awi Þ2 Awi ð4:74Þ The electrical resistivity of the conductor is rw and the length of the conductor in the winding is lwi, i.e. the product of the number of turns Ni and the mean length of a turn (MLT). 114 Transformers and Inductors for Power Electronics The current in the winding is expressed in terms of the current density.The total resistive loss for all the windings is: Pcu ¼ X RI2 ¼ rw Xn NiMLTðJoAwiÞ2 i¼1 Awi ð4:75Þ Incorporating the deﬁnition of window utilization factor, ku (Equation 4.67), and noting that the volume of the windings (fully wound ku ¼ 1) is Vw ¼ MLT Â Wa, then: Pcu ¼ rwV wkuJ 2 o ð4:76Þ 4.3.4 Core Loss In general, the core loss per unit volume are given in W/m3 in accordance with the general Steinmetz equation (Equation 1.29, reproduced below): Pfe ¼ Kcf aBbmax where Kc, a and b are constants. Typical values are given in Table 1.1. The core loss includes hysteresis and eddy current losses. The manufacturer’s data is normally measured for sinusoidal excitation. In the absence of test data on the design core, the manufacturer’s data must be used in establishing the constants in Equation 1.29. The constants may also be deduced from measurements of the core loss; this will be dealt with in Chapter 8. 4.3.5 Optimization Eliminating the current density in Equation 4.76 using Equation 4.73 yields an expression for the copper or winding loss: Pcu ¼ P VA 2 rwV wku Kvf Bmaxkf kuAp ¼ a f 2B2max ð4:77Þ It is written in this form to show that the copper loss is inversely proportional to both the frequency squared and the ﬂux density squared. Rewriting Equation 1.29 shows that the core loss is also dependent on the frequency and the ﬂux density: Pfe ¼ V cKcf aBbmax ¼ bf aBbmax ð4:78Þ The total loss is made up of the combined core and winding losses: P ¼ f a 2 B2max þ bf aBbmax ð4:79Þ Transformers 115 The domain of P is in the ﬁrst quadrant of the f-Bmax plane. Pis positive everywhere and it is singular along the axes. If a ¼ b, P has a global minimum dP dðf BmÞ ¼ 0 at the frequency ﬂux density product: 1 f oBo ¼ 2a b þ 2 bb ð4:80Þ For a ¼ b ¼ 2, with Equations 4.77 and 4.78, the frequency ﬂux density product is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃsﬃﬃﬃﬃPﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f oBo ¼ 4 rwV wku rc V c K c VA Kvkf kuAp ð4:81Þ Given that Bo must be less than the saturation ﬂux density Bsat, there is a critical frequency, given by Equation 4.81 above, by which the total loss may be minimized by selecting an optimum value of ﬂux density which is less than the saturation value (Bo < Bsat). Equation 4.81 shows that foBo is related to power density in the transformer, since Ap is related to core size. In the more general case (a 6¼ b), there is no global minimum. The minimum of P at any given frequency is obtained by taking the partial derivative with respect to Bmax and setting it to zero: @P @Bmax ¼ À f 2a 2B3max þ bbf aBbmÀax1 ¼ 0 ð4:82Þ The minimum loss occurs when: Pcu ¼ b 2 Pfe ð4:83Þ for a ﬁxed frequency f. The minimum value of P at any given ﬂux density is obtained by taking the partial deriva- tive with respect to f and setting it to zero. The minimum loss occurs when: Pcu ¼ a 2 Pfe ð4:84Þ for a ﬁxed ﬂux density Bmax. Evaluation of Equation 4.84 at Bo ¼ Bsat gives the critical frequency above which the total loss is minimized by operating at an optimum value of ﬂux density that is less than the satu- ration value (Bo < Bsat): f aþ2 o Bboþ2 ¼ 2 rwVwku a rcVcKc P VA 2 Kvkf kuAp ð4:85Þ The nature of Equation 4.85 is illustrated in Figure 4.15. The two sets of curves shown are for low frequency (say, 50 Hz) and high frequency (say, 50 kHz). At 50 Hz, the optimum ﬂux density (at point B) is greater than the saturation ﬂux density, so therefore the minimum loss achievable is at point A. However, the winding and core losses are not equal. At 50 kHz, the optimum ﬂux density is less than the saturation ﬂux density and the 116 Transformers and Inductors for Power Electronics Losses 50Hz 50kHz P Pcu D Pfe P C A Pfe B Pcu BoD Bsat BoB Flux density Figure 4.15 Winding, core, and total losses at different frequencies. core and winding losses are equal at the optimum point D. The ﬁrst step in a design is to establish whether the optimum ﬂux density given by the optimization criterion in Equation 4.83 is greater or less than the saturation ﬂux density. We will return to this in Chapter 5, when we will set out the design methodology based on the analysis presented above. 4.4 Power Factor The VA rating of each winding is required in order to proceed with the transformer design, as indicated by Equation 4.73. Traditionally, the concept of power factor was applied to sinusoidal waveforms of current and voltage, and the power factor is simply the cosine of the phase angle u between the waveforms. In power electronics, waveforms are very often a mixture of sinusoids and square waves. The deﬁnition of power factor is: kp ¼ hpi V rmsIrms ð4:86Þ where hpi is the average power delivered at the terminals where Vrms and Irms are the rms values of voltage and current respectively. Consider a case where the voltage is a sinusoid and the current is a square wave. The average power is: hpi ¼ 1 T ðT 0 vðtÞiðtÞ ¼ V rms I 1rms cosðuÞ ð4:87Þ Since v(t) is a pure sinusoid, only the fundamental of the current waveform will yield a DC or average value of the product v(t)i(t). Vrms is the rms value of voltage and I1rms is the rms value of the fundamental of the current waveform. u is the phase angle between the voltage waveform and the fundamental of the current waveform. Equation 4.87 can be rewritten: hpi ¼ V rmsIrms I 1rms Irms cosðuÞ ¼ V rmsIrmskd ku ð4:88Þ Transformers 117 is + Vosin ωt _ vs Vo + is _ Io π 2π ωt π ωt Figure 4.16 Half-wave rectiﬁer with a resistive load. Where kd is called the distortion factor and ku is called the displacement factor. Note that Irms is the rms value of the total waveform including all the harmonics. Clearly, for pure sinusoids, kd is unity and ku is equal to cos u, the classical deﬁnition of power factor. Example 4.3 Calculate the power factor of the source in a half-wave rectiﬁer with a resistive load. The circuit and associated voltage vs and current is waveforms are shown in Figure 4.16. We can write down the rms values of the source voltage and current using the relationships: Vsrms ¼ pVﬃﬃ 2 I srms ¼ I 2 where V is the peak value of the voltage waveform and I is the peak value of the current waveform. The average power delivered to the load is: ðp hpi ¼ VI sinðvtÞ2dðvtÞ ¼ VI 2p 4 0 The power factor is: kp ¼ hpi V rmsIrms ¼ p1ﬃﬃ 2 Example 4.4 P Calculate the VA rating for a transformer with a sinusoidal input. The transformer has a 1 : 1 turns ratio, and associated waveforms for voltage and current are shown in Figure 4.17. The load power factor is denoted by kp and the output power by Po, where Po ¼ VsIskp. The power factor is simply kp ¼ cosu, because the voltage and current waveforms are sinusoidal. For the overall efﬁciency of a power electronics converter, transformer losses play a minor role in most cases and, for practical designs, we shall assume that the efﬁciency is 100%. The VA rating of the primary winding is: VAp ¼ VpIp ¼ Po kp kp is the power factor of the primary winding and is the same as the power factor of the load, since the reﬂected voltage and current from the secondary to the primary maintain the phase angle u. 118 Transformers and Inductors for Power Electronics + Vosin ωt _ ip + vp _ vp,vs is Vo 0 π + vs ip,is _ Io 1:1 0 θ π 2π ωt ωt 2π 2π + θ Figure 4.17 Transformer and associated waveforms. In the secondary winding, the VA rating is: VAs ¼ VsIs ¼ Po kp The total VA rating for both the primary and secondary windings is now: X VA ¼ ð1 þ 1Þ Po kp ¼ 2 kp Po Example 4.5 Centre-tapped rectiﬁer P Calculate the VA rating for a centre tapped rectiﬁer shown in Figure 4.18. The waveforms are shown for voltage in each of the output windings and for the input winding with the notation shown in the transformer diagram. The waveforms are shown for both a resistive load and an inductive load, where the inductance is assumed to be very large. The simplest approach is to calculate the power factor of each winding, recognizing that each of the secondary windings handles half the throughput of power. The average power delivered through each secondary winding is: hpi ¼ Po : 2 (a) Resistive load The average output power is deﬁned as: Secondary windings Po ¼ pV oﬃﬃ 2 pIoﬃﬃ 2 ¼ VoIo 2 The rms value of the secondary voltage is: V s1 ¼ V s2 ¼ pV oﬃﬃ 2 Is1 ¼ Is2 ¼ Io 2 The average power through each of the secondary windings is: hpsi ¼ Po 2 Transformers 119 + Vosin ωt _ is1 ip + vp _ + vs1 _ is2 + vs2 1 :1 :1 _ + io vo _ (a) Resistive Load (b) Inductive Load vp,vs1,vs2 Vo 0 vp,vs1,vs2 Vo π 2π ωt 0 π 2π ωt ip ip Io Io 0 π 2π ωt vo Vo 0 π 2π ωt vo Vo 0 π 2π ωt io Io 0 is1 π 2π ωt Io 0 π 2π ωt io Io 0 is1 π 2π ωt Io 0 π 2π ωt is2 0 π 2π ωt -Io 0 π 2π ωt is2 0 π 2π ωt -Io Figure 4.18 Centre-tapped rectiﬁer and associated waveforms for (a) resistive and (b) inductive loads. and the power factor of each secondary winding is: kp ¼ hpsi VsIs ¼ p1ﬃﬃ 2 The VA rating of each secondary winding is now: VAs ¼ hpsi kp ¼ pPoﬃﬃ 2 Primary winding 120 Transformers and Inductors for Power Electronics The rms value of the primary voltage is: Vp ¼ pVoﬃﬃ 2 The rms value of the primary current is: Ip ¼ pIoﬃﬃ 2 and the power factor of the primary winding is: kp ¼ hpi VpIp ¼ 1 The total VA rating of the centre tapped transformer is now: X VA ¼ 1 þ p1ﬃﬃ þ p1ﬃﬃ Po ¼ 1 þ pﬃﬃ 2 Po 22 (b) Inductive load The solution for the inductive load follows the steps above taking the shape of the current waveform into account. The average output power is deﬁned as: ðp Po ¼ VoIo p sinðvtÞdðvtÞ ¼ 2VoIo p 0 Secondary windings Vs1 ¼ Vs2 ¼ pVoﬃﬃ 2 Is1 ¼ Is2 ¼ pIoﬃﬃ 2 The average power through each of the secondary windings is Poand the power factor of each secondary winding is: 2 kp ¼ hpi VsIs ¼ 2 p The VA rating of each secondary winding is now: VA ¼ hpi kp ¼ p 4 Po Primary winding Vp ¼ pVoﬃﬃ 2 Ip ¼ Io pﬃﬃ kp ¼ hpi VpIp ¼ 22 p ¼ 0:9 Transformers 121 The total VA rating of the centre tapped transformer with an inductive load is now: X VA ¼ 1 0:9 þ p 4 þ p 4 Po ¼ 1 0:9 þ p 2 Po 4.5 Problems 4.1 List ﬁve different types of transformers and describe their applications. 4.2 Explain why the rms value of the primary current of a transformer is not zero when the secondary current is zero. 4.3 Derive k and Kv for a sine wave input voltage waveform in a transformer. 4.4 Derive k and Kv for a pulsed waveform with a duty cycle D input voltage waveform in a transformer. 4.5 Derive k and Kv for a triangular waveform with a duty cycle D input voltage waveform in a transformer. 4.6 Calculate the power factor for the full-wave bridge rectiﬁer with an inductive load shown in Figure 4.19. 4.7 Calculate the input power factor for the full-wave bridge rectiﬁer with capacitive ﬁlter shown in Figure 4.20. Vosin ωt ip is + + vp vs _ _ 1:1 io + vo _ Figure 4.19 Full-wave bridge rectiﬁer. Vosin ωt ip is + + vp vs _ _ 1:1 io + vo _ Figure 4.20 Full-wave bridge rectiﬁer with a capacitive ﬁlter. 122 Transformers and Inductors for Power Electronics References 1. Hurley, W.G., Wilcox, D.J., and McNamara, P.S. (1991) Calculation of short circuit impedance and leakage impedance in transformer windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 651–658. 2. Hurley, W.G. and Wilcox, D.J. (1994) Calculation of leakage inductance in transformer windings. IEEE Transactions on Power Electronics 9 (1), 121–126. 3. Hurley, W.G., Wolﬂe, W.H., and Breslin, J.G. (1998) Optimized transformer design: inclusive of high-frequency effects. IEEE Transactions on Power Electronics 13 (4), 651–659. Further Reading 1. Blume, L.F. (1982) Transformer Engineering, John Wiley & Sons, New York. 2. Bueno, M.D.A. (2001) Inductance and Force Calculations in Electrical Circuits, Nova Science Publishers, Huntington. 3. Del Vecchio, R.M., Poulin, B., Feghali, P.T. et al. (2001) Transformer Design Principles: With Applications to Core-Form Power Transformers, 1st edn, CRC Press, Boca Raton, FL. 4. Dowell, P.L. (1966) Effects of eddy currents in transformer windings. Proceedings of the Institute of Electrical and Electronic Engineers 113 (8), 1387–1394. 5. Erickson, R.W. (2001) Fundamentals of Power Electronics, 2nd edn, Springer, Norwell, MA. 6. Fitzgerald, A.E., Kingsley, C. Jr, and Umans, S.D. (2002) Electric Machinery, 6th edn, McGraw-Hill, New York. 7. Flanagan, W.M. (1992) Handbook of Transformer Design and Application, 2nd edn, McGraw-Hill, New York. 8. Georgilakis, P.S. (2009) Spotlight on Modern Transformer Design (Power Systems), 1st edn, Springer, New York. 9. Hoke, A.F. and Sullivan, C.R. (2002) An improved two-dimensional numerical modeling method for E-core transformers. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 151–157. 10. Jieli, L., Sullivan, C.R., and Schultz, A. (2002) Coupled-inductor design optimization for fast-response low-voltage. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 817–823. 11. Judd, F. and Kressler, D. (1977) Design optimization of small low-frequency power transformers. IEEE Transactions on Magnetics 13 (4), 1058–1069. 12. Kazimierczuk, M.K. (2009) High-Frequency Magnetic Components, John Wiley & Sons, Chichester. 13. Krein, P.T. (1997) Elements of Power Electronics (Oxford Series in Electrical and Computer Engineering), Oxford University Press, Oxford. 14. Kulkarni, S.V. (2004) Transformer Engineering: Design and Practice, 1st edn, CRC Press, New York. 15. B.H.E. Limited (2004) Transformers: Design, Manufacturing, and Materials (Professional Engineering), 1st edn, McGraw-Hill, New York. 16. McAdams, W.H. (1954) Heat Transmission, 3rd edn, McGraw-Hill, New York. 17. McLyman, C.W.T. (1997) Magnetic Core Selection for Transformers and Inductors, 2nd edn, Marcel Dekker Inc., New York. 18. McLyman, C.W.T. (2002) High Reliability Magnetic Devices, 1st edn, Marcel Dekker Inc., New York. 5 Transformer Design Transformers used in power electronics applications normally serve to provide isolation from the input mains and to reduce voltage stress on switching components by more closely matching the operating voltage to the switch voltage ratings. We saw in Chapter 4 that the size of the transformer is reduced as the frequency of operation is increased. These three objectives may be simultaneously achieved in a DC-DC converter. While it is true to say that the design of conventional power transformers is well documented, the additional issues that arise in high-frequency operation need special attention. One approach is to develop empirical design rules, but these tend to lead to conservative designs that often mean core sizes that are too large for the intended application. The switching waveforms encountered in modern converters mean that non-sinusoidal excitation occurs at high frequencies. We have seen in Chapters 1 and 4 that the losses are frequency dependant: hysteresis loss and eddy current loss in ferromagnetic cores and skin and proximity effects in windings. Traditionally, the starting point for transformer design is that winding loss is approximately equal to the core loss. However, in a typical power frequency transformer, the ratio may be as high as 6 : 1. This was explained in Section 4.3.5 by virtue of the fact that ﬂux density is limited by its saturation value. When the optimization is not limited by the saturation ﬂux density, then as the frequency increases, the actual ﬂux density decreases and, in many cases, the operating ﬂux density may be a fraction of the saturation ﬂux density. It is not sufﬁcient to conﬁne the design to electrical issues, because the heat generated in the transformer must be dissipated through the surface of the transformer (consisting of the exposed core surface and the winding surface). The dissipation of the losses will ultimately determine the maximum temperature inside the transformer. The maximum temperature may be limited by the Curie temperature in the core material, or by the temperature rating of the insulation used in the conductors. The design of a modern transformer must incorporate elements of circuit analysis, magnetic circuit laws and heat transfer. In Chapter 4, we restated the traditional equations for transformers to take high frequency, power factor and non-sinusoidal excitation into account – the fundamental relationships as presented form the bedrock of the approach to the design methodology. Dimensional analysis of the various relationships is invoked to establish a set of Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 124 Transformers and Inductors for Power Electronics robust design rules. Finally, the overall approach is informed by the avoidance of unnecessary design factors that often tend to cloud the main design criteria. The purpose of a transformer is to transfer energy from the input winding to the output winding through electromagnetic induction. The overall aim of the design methodology aim is to optimize this energy transfer in a given application, in terms of minimizing the core and winding losses. Optimization can be based on cost, weight or volume; all of these criteria lead to the conclusion that the core loss is approximately equal to the winding loss in an optimized design. The main factors effecting transformer operation are the operating temperature, the electrical frequency and the maximum ﬂux density. We demonstrated in Chapter 4 that there is a critical frequency above which the total loss can be minimized by selecting a value of maximum ﬂux density that is less than the saturation ﬂux density. Below the critical frequency, the transfer of energy through the transformer is restricted by the limitation that maximum operating ﬂux density cannot be greater than the saturation value for the ferromagnetic material used in the core construction. In this chapter, we will establish the design rules to select the core and winding in a transformer. Several examples from power electronics are presented to illustrate the robust nature of the methodology. The approach is based on design rules that are derived from ﬁrst principles, thus ensuring the general applicability of the design algorithms. The main speciﬁcations are power rating, frequency and temperature rise. When the core and winding are selected, the overall transformer is evaluated for temperature rise and efﬁciency. Several examples are included to illustrate the design methodology, including a centre-tapped rectiﬁer transformer, a forward converter and a push-pull converter. 5.1 The Design Equations In Chapter 4, we established the basic electrical relationships in the transformer. In the case of the inductor design, we established an expression for Ap in terms of the stored energy, and it is possible to establish an expression for Ap in the case of the transformers in terms of the power transferred. 5.1.1 Current Density in the Windings The optimum value of current density in the windings may be found from the optimum criterion (Equation 4.83) using the equations for copper loss (Equation 3.22), core loss (Equation 1.29) and thermal heat transfer (Equation 3.14). From Equation 4.83, for a ﬁxed frequency: Pcu þ Pfe ¼ b þ b 2 Pcu ð5:1Þ Transformer Design 125 and: Pcu þ Pfe ¼ b þ b 2 Â rwV w ku J2o Ã ¼ hcAtDT ð5:2Þ Extracting the current density: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Jo ¼ b b þ 2 hcAtDT : rwV wku ð5:3Þ Employing the dimensional analysis equations (Equations 3.25 to 3.27) and taking b ¼ 2 gives an expression for the current density in terms of the temperature rise in the windings and the core-window winding area product: sﬃﬃﬃﬃﬃﬃﬃ Jo ¼ Kt DT p1ﬃﬃﬃﬃﬃ 2ku 8 Ap ð5:4Þ where Kt is deﬁned in Equation 3.29. 5.1.2 Optimum Flux Density unlimited by Saturation The optimum design is at point A in Figure 5.1, and the optimum ﬂux density in the core is not limited by saturation. The optimum conditions established by Equation 5.1 may be exploited to establish a formula for Ap in terms of the design speciﬁcations: output power, frequency and temperature rise. Taking Bo as the ﬂux density at the optimum operating point and J0 as the corresponding value of the current density given by Equation 5.4, and substituting these values in the power equation (Equation 4.73), gives us an expression for the core window winding area product as: Ap ¼ pﬃﬃ 2 P VpAﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!8=7 Kvf Bokf Kt kuDT ð5:5Þ Losses P Pfe A Pcu Bo Bsat Flux density Figure 5.1 Winding, core, and total losses unlimited by saturation. 126 Transformers and Inductors for Power Electronics At this point, we do not know the value of Bo, but this is required to size the core. We can ﬁnd Bo by taking a closer look at the optimum conditions. Taking b ¼ 2 as before means that the core and winding losses are equal and the copper loss is equal to half the total loss. Therefore, with P as the total of the copper and iron losses (the same as Q in Equation 3.14): !2 P3 2 1 7 ¼1 P1cu2P1fe2 ð5:6Þ The rather unusual format of Equation 5.6 is designed to extract Bo. Substituting for P with Equation 5.2, Pcu with Equation 4.83 and Pfe with Equation 1.29, and incorporating the dimensional analysis given by Equations 3.25 to 3.27, yields the fol- lowing equation 2 ½hc ka DT 3 2 23 ½rw 1 kwku12 Â kcK cf aB2oÃ172ÀJo Á1 Ap 6 ¼ 1 ð5:7Þ Extracting (JAp)1/6 from the power equation (Equation 4.73) yields an expression for Bo: 2 !1 Bo ¼ ½hc ka DT 3 2 1 23½rwkwku12½kcK cf a172 Kvf kf ku 6 SVA ð5:8Þ The optimum ﬂux density may be found from the speciﬁcations of the application and the material constants. 5.1.3 Optimum Flux Density limited by Saturation The optimum design is at point B in Figure 5.2, and the optimum ﬂux density in the core is limited by saturation. The value of Bmax in the voltage equation is ﬁxed by Bsat. The initial estimate of Ap is found by assuming that the total loss is equal to twice the copper loss (at point C in Figure 5.2, the core loss is smaller than the winding loss). Assuming that the total loss is double the winding loss means that the core is oversized. We will reﬁne this later. The initial value of Ap is given by Equation 5.5, with the maximum ﬂux density given by Bsat: pﬃﬃ P !8 Ap1 ¼ 2 VpAﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 Kvf Bsatkf Kt kuDT ð5:9Þ The current density is found by combining the winding loss (Equation 4.76) and the core loss (Equation 1.29), and then using the thermal equation (Equation 5.2): J 2 o ¼ hcAtDT rwV wku À VcKcf aBbmax rwV wku ð5:10Þ Transformer Design 127 Losses P Pfe CB Pcu Bsat Bo Flux density Figure 5.2 Winding, core and total losses unlimited by saturation. Extracting J2o from the power equation (Equation 4.73) and invoking the dimensional analysis for At, Vc and Vw in Equations 3.24 to 3.26 gives an expression for Ap: kcKcf aBbmax rwkwku A2p À hckaDT rwkwku A7p=4 þ SVA !2 ¼0 Kvf Bmaxkf ku ð5:11Þ We now have an expression for Ap in the form: 7 f ðApÞ ¼ a0Ap2 À a1A4p þ a2 ¼ 0 ð5:12Þ The roots of f(Ap) are found numerically using the Newton Raphson Method: Apiþ1 ¼ Api À f ðApi Þ f 0ðApi Þ ¼ Api À a0A2pi À 7 a1Api 4 þ a2 3 2a0Api À 7 4 a1 A4pi ð5:13Þ Api is given by Equation 5.9, and one iteration should be sufﬁcient. Finally, we need to calculate the corresponding value of the current density. Proceeding as before, with dimensional analysis for At in Equation 5.10, gives us an expression for Jo in terms of the transformer speciﬁcations and material properties: Jo ¼ shﬃﬃﬃcﬃﬃkﬃﬃaﬃﬃpﬃﬃﬃﬃﬃﬃAﬃﬃﬃﬃﬃpﬃﬃﬃﬃDﬃﬃﬃTﬃﬃﬃﬃÀﬃﬃﬃﬃﬃVﬃﬃﬃcﬃﬃKﬃﬃﬃcﬃﬃfﬃﬃaﬃﬃBﬃﬃﬃbmﬃﬃﬃaﬃxﬃﬃ rwV wku ð5:14Þ Note that the volume of the winding (fully wound ku ¼ 1) is given by Vw ¼ MLT Â Wa, and the volume of the core is Vc ¼ lc Â Ac. 128 Transformers and Inductors for Power Electronics 5.2 The Design Methodology The overall design methodology is shown in ﬂowchart form in Figure 5.3. The core manufacturer normally supplies the core data: cross-section, Am (or Ac with no laminations), window winding area Wa, the mean length of a turn MLT and the core volume. The number of turns in each winding is found from the voltage equation (Equation 4.62): N ¼ Vrms : Kvf BmaxAm ð5:15Þ In this equation, Bmax is interpreted as Bo or Bsat, depending on which is lower. The selected core from standard designs may not correspond exactly to the value of Ap given by Equation 5.5 or Equation 5.12 and, therefore, the current density should be calculated using Equation 5.14. The calculation of the core and winding losses follows the procedures established in Chapter 3. The design procedure may be summarized as follows: Speciﬁcations: Input voltage and current Output voltage and current/power Frequency of operation Maximum core temperature or temperature rise Ambient temperature Circuit parameters: Waveform factor Power factor Core Selection: Core material Maximum ﬂux density Core window winding area product Winding design: Number of turns Current density Wire selection Losses: Copper loss Winding loss Efﬁciency Transformer Design 129 Specifications : ∑VA,K,f,ku,ΔT Select Material : Bsat,ρc,Kc,α,β Calculate Bo Yes Calculate Ap Select Ap Ac Wa MLT m Bo < Bsat No Bmax ≤ Bsat Calculate Api Ac Wa MLT m Select Api+1 Select Ap Calculate Turns Calculate Jo Select Wires Calculate Copper Loss Calculate Core Loss Calculate High Frequency Losses Calculate Efficiency, η Figure 5.3 Flow chart of design process. In Chapter 6, we will reﬁne the design to take high-frequency skin and proximity effects into account. 5.3 Design Examples 5.3.1 Example 5.1: Centre-Tapped Rectiﬁer Transformer Speciﬁcations The design speciﬁcations for the transformer are listed in Table 5.1. 130 Transformers and Inductors for Power Electronics Table 5.1 Speciﬁcations Input Output Frequency, f Temperature rise, DT Ambient temperature, Ta 230 Vrms, sine wave 100 Vrms, 10 A 50 Hz 55 C 40 C Circuit Parameters The centre-tapped transformer is shown in Figure 5.4. The waveform factor Kv ¼ 4.44 for sine wave excitation. The power factors and VA ratings of the input and output windings were established in Example 4.5. Core Selection Laminated grain orientated steel would normally be used for this type of application. Typical material speciﬁcations are listed in Table 5.2. The output power of the transformer is Po ¼ (100 þ 1) Â 10 ¼ 1010 W, assuming a forward voltage drop of 1 V for each diode. The VA ratings of the windings were established in Example 4.5, giving: X pﬃﬃ pﬃﬃ VA ¼ 1 þ 2 Po ¼ 1 þ 2 1010 ¼ 2438 VA The optimum ﬂux density (Equation 5.8) is: Bo ¼ ½ð10Þð40Þð55Þ2=3 22=3Âð1:72 Â 10À8Þð10Þð0:4ÞÃ1=12hð5:6Þð3:388Þð50Þ1:7i7=12 Á ð4:44Þð50Þð0:95Þð0:4Þ!1=6 ¼ 4:1 T 2438 + V0sin ωt _ is1 ip + vp _ + vs1 _ is2 + vs2 1::11 _ + io vo _ Figure 5.4 Centre-tapped rectiﬁer with a resistive load. Transformer Design 131 Table 5.2 Material speciﬁcations Kc a b Bsat 3.388 1.7 1.9 1.5 T Since Bo > Bsat, the design is saturation-limited. 1st iteration: Bmax ¼ 1.5 T and, from Equation 5.9, Ap1 is: " pﬃﬃ #8=7 Ap1 ¼ 2ð2438Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð4:44Þð50Þð1:5Þð0:95Þð48:2 Â 103Þ ð0:4Þð55Þ Â 108 ¼ 1166 cm4 2nd iteration: From Equation 5.11: a0 ¼ ð5:6Þð3:388Þð50Þ1:7ð1:5Þ1:9 ð1:72 Â 10À8Þð10Þð0:4Þ ¼ 4:606 Â 1011 a1 ¼ ð10Þð40Þð55Þ ð1:72 Â 10À8Þð10Þð0:4Þ ¼ 3:198 Â 1011 a2 ¼ 2438 !2 ð4:44Þð50Þð1:5Þð0:95Þð0:4Þ ¼ 371:3 and from Equation 5.13: Ap2 ¼ 1166 Â 10À8 À ð4:606 Â 1011Þð1166 Â 10À8Þ2 À ð3:198 Â 1011Þð1166 Â 10À8Þ7=4 þ 371:3 2ð4:606 Â 1011Þð1166 Â 10À8Þ À 7 4 ð3:198 Â 1011Þð1166 Â 10À8Þ3=4 ¼ 859 cm4 A third iteration is not necessary. A tape-wound toroidal core with 0.23 mm laminations is suitable. The manufacturer’s data for this core is summarized in Table 5.3 Table 5.3 Core and winding speciﬁcations Ac Wa Ap Vc kf ku MLT r20 a20 19.5 cm2 50.2 cm2 979 cm4 693 cm3 0.95 0.4 28 cm 1.72 mV-cm 0.00393 132 Transformers and Inductors for Power Electronics Winding Design Primary turns Np ¼ Vp K v Bmax Ac f ¼ 230 ð4:44Þð1:5Þð19:5 Â 10À4Þð50Þ ¼ 354 turns Secondary turns The rms value of each secondary voltage is (100 þ 1) ¼ 101 V, which includes a 1 V for- ward voltage drop in the diode. Ns ¼ Np Vs Vp ¼ 354 101 230 ¼ 155 turns Wire size The resistivity of the copper must be adjusted for temperature using Equation 3.37: Tmax ¼ 40 þ 55 ¼ 95 C and the resistivity is: rw ¼ ð1:72 Â 10À8Þ½1 þ ð0:00393Þð95 À 20Þ ¼ 2:23 Â 10À8 Vm The current density (Equation 5.14) for the chosen core is: Jo ¼ shﬃﬃﬃkﬃﬃﬃaﬃﬃpﬃﬃﬃﬃﬃAﬃﬃﬃﬃﬃpﬃﬃﬃﬃDﬃﬃﬃTﬃﬃﬃﬃÀﬃﬃﬃﬃVﬃﬃﬃﬃcﬃKﬃﬃﬃﬃcﬃfﬃﬃaﬃﬃﬃBﬃﬃﬃbmﬃﬃ rwV wku ¼ sðﬃﬃ1ﬃﬃﬃ0ﬃﬃÞﬃﬃðﬃﬃ4ﬃﬃ0ﬃﬃﬃÞﬃﬃpﬃﬃﬃﬃ9ﬃﬃﬃﬃﬃ7ﬃﬃﬃﬃ9ﬃﬃﬃﬃﬃﬃÂﬃﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃ0ﬃﬃﬃﬃÀﬃﬃﬃﬃﬃﬃ8ﬃﬃﬃðﬃﬃ5ﬃﬃ5ﬃﬃÞﬃﬃﬃÀﬃﬃﬃﬃðﬃﬃ6ﬃﬃ9ﬃﬃﬃ3ﬃﬃﬃÂﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃﬃÀﬃﬃ6ﬃﬃÞﬃﬃðﬃﬃ3ﬃﬃ:ﬃﬃ3ﬃﬃ8ﬃﬃ8ﬃﬃﬃÞﬃﬃðﬃ5ﬃﬃﬃ0ﬃﬃÞﬃﬃ1ﬃﬃ:ﬃ7ﬃﬃðﬃ1ﬃﬃﬃ:ﬃ5ﬃﬃﬃÞﬃ1ﬃﬃ:ﬃ9ﬃ ð2:23 Â 10À8Þð28 Â 10À2Þð50:2 Â 10À4Þð0:4Þð0:4Þ ¼ 2:277 Â 106 A=m2 Remember that Vw ¼ MLT Â Wa. Primary copper loss Ip ¼ Po kppV p ¼ 1010 ð1Þð230Þ ¼ 4:39 A The cross-sectional area of the conductor is: Aw ¼ Ip=Jo ¼ 4:39=2:277 ¼ 1:929 mm2 This is equivalent to a 1.57 mm bare diameter wire. Select a 1.6 mm diameter wire with a DC resistance of 8.50 mV/m at 20 C (Table A.1). Use Equation 3.37 to correct the winding resistance for temperature: Rdc ¼ ð28 Â 10À2 À Þð354Þ 8:50 Â 10À3 Á ½1 þ ð0:00393Þð95 À 20Þ ¼ 1:091 V Transformer Design 133 The primary copper loss is: Pcu ¼ I2rmsRdc ¼ ð4:39Þ2ð1:091Þ ¼ 21:04 W Secondary copper loss The rms value of the load current, which is a full wave-rectiﬁed sine wave, is 10 A. The current appears in each secondary winding as a half-wave-rectiﬁed sine wave, as shown in Figure 5.5. From Table 6.1: Is1rms ¼ Is2rms ¼ 10/2 ¼ 5 A. Is ¼ 5 A. Aw ¼ Is/Jo ¼ 5/2.277 ¼ 2.196 mm2 This is equivalent to a 1.67 mm bare diameter wire. Select a 1.8 mm diameter wire with a DC resistance of 6.72 mV/m at 20 C (Table A.1). Use Equation 3.37 to correct the winding resistance for temperature: Ru ¼ ð28 Â 10À2 À Þð155Þ 6:72 Â 10À3 Á ½1 þ ð0:00393Þð95 À 20Þ ¼ 0:378 V The secondary copper loss (two windings) is: Pcu ¼ I2rmsRdc ¼ ð5Þ2ð0:378Þð2Þ ¼ 18:88 W vp,vs1,vs2 V0 0 ip I0 0 is1 I0 vo V0 π 2π ωt 0 io I0 π 2π ωt 0 π 2π ωt π 2π ωt 0 π 2π ωt is2 0 π 2π ωt -I0 Figure 5.5 Centre-tapped rectiﬁer voltage and current waveforms. 134 Transformers and Inductors for Power Electronics pﬃﬃﬃﬃﬃ The skin depth at 50 Hz in copper is d ¼ 66= 50 ¼ 9:3 mm, and therefore does not create additional loss, since the radii of both primary and secondary conductors are less than the skin depth (see Figure 6.3). Core loss Pfe ¼ VcKcf aBbmax ¼ ð693 Â 10À6Þð3:388Þð50Þ1:7ð1:5Þ1:9 ¼ 3:92 W Total losses: Efficiency: Primary copper Secondary copper 21.04 W 18.88 W 3.92 W 43.84 W Efficiency ¼ 1010 1010 þ 43:84 100 ¼ 95:8 % 5.3.2 Example 5.2: Forward Converter Speciﬁcations The design speciﬁcations for the transformer are listed in Table 5.4. Circuit Parameters In a forward converter, the transformer provides electrical isolation and adjusts the input/output voltage ratio for correct component stresses. The circuit is shown in Figure 5.6. Np, Ns and Nt are the number of turns in the primary, secondary and reset windings respectively. When switch Q in Figure 5.6 turns on, ﬂux builds up in the core, as shown in Figure 5.7. When the switch is turned off at t ¼ DT, this core ﬂux must be reset, otherwise core creep takes place and eventually the core will saturate. Assume at t ¼ 0, the core ﬂux is 0, so that, at the end of the switching period, DT, the ﬂux is fmax. Table 5.4 Speciﬁcations Input Output Frequency, f Temperature rise, DT Ambient temperature, Ta 12 ! 36 V 9 V, 7.5 A 25 kHz 35 C 40 C Transformer Design 135 + Vs _ Np:Ns is D1 iL Lo + + + Nt vp _ vs vd D2 _ _ D3 Q io + Co Vo _ Figure 5.6 Forward converter. vp Vs 0 t –⎧⎩NNpt ⎫⎭Vs vL ⎧⎩NNst ⎫⎭Vs – Vo 0 t -Vo iL ÎL IL ΔIL 0 t B Bmax ΔB Bmin 0 DT T 2T t Figure 5.7 Forward converter voltage and ﬂux waveforms. 136 Transformers and Inductors for Power Electronics By Faraday’s law, the applied DC voltage is related to the linear rise of ﬂux in the core: Vs ¼ Np df dt Integrating for the ﬂux, with the initial value at 0: ð f¼ 1 Np Vsdt ¼ Vs Np t 0 t DT The maximum ﬂux is at the end of the on period at t ¼ DT: fmax ¼ Vs Np DT At t ¼ DT, resetting of the core ﬂux begins through the action of the reset winding, this must be achieved in the time (1 À D)T. During this time period, the ﬂux in the core is: f ¼ fmax À Vs Nt ðt À DT Þ DT t ð1 À DÞT If the ﬂux is to reset to zero at the end of this period, then: Np Nt ¼ 1 D À D For a duty cycle of 75%, the ratio of primary turns to reset turns ratio is 3. The voltage waveform factor was derived in Example 4.2: Kv ¼ pﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Dð1 À DÞ The power factor, kp of the input and output windings can be found from the voltage and current waveforms in Figure 5.7: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Vrms ¼ D 1 À V D s Irms ¼ pﬃﬃﬃ D Ns Np Io The average power through the winding is: Pav ¼ hpi ¼ D Ns Np VsIo The power factor kp is: kp ¼ hpi V rmsIrms ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1ÀD in both the primary and secondary windings. Transformer Design 137 Table 5.5 Material speciﬁcations Kc 37.2 a 1.13 b 2.07 Bsat 0.4T Core Selection Ferrite would normally be used for this type of application at the speciﬁed frequency. The material speciﬁcations for Mn-Zn are listed in Table 5.5. The output power of the transformer is Po ¼ (9 þ 1) Â 7.5 ¼ 75 W, assuming a forward voltage drop of 1 V for the diode. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The duty cycle is D ¼ 9/12 ¼ 0.75 and the power factor is then 1 À D ¼ 0:5: Kv ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:75ð1 À 0:75Þ ¼ 2:31 The power factor and VA ratings of the windings are established above giving X VA ¼ 1 kpp þ 1 kps Po ¼ ð2 þ 2Þð75Þ ¼ 300 VA P Adding 5% for the reset winding gives VA ¼ 315 VA. Set ku ¼ 0.4. The optimum ﬂux density is found using Equation 5.8. However, Bo is the amplitude of the ﬂux waveform. On the other hand, Bmax in Equation 4.63 is the maximum ﬂux density, and in the case of the forward converter, this is at least equal to the ﬂux ripple DB (in continuous conduction) and the amplitude of the ﬂux used in the core loss calculation Equation 1.29 is DB/2. In calculating Bo using Equation 5.8, 2 Kv may be used to properly account for these effects. Bo ¼ 22=3 Â ð1:72 Â ½ð10Þð40Þð35Þ2=3 10À8Þð10Þð0:4ÞÃ1=12 h ð5:6Þð37:2Þð25 000Þ1:13i7=12 Á ð2 Â 2:31Þð25 000Þð1:0Þð0:4Þ!1=6 ¼ 0:186 T 315 Bmax ¼ 2Bo ¼ 0.372 T. This is less than Bsat, and Ap from Equation 5.5 is: " pﬃﬃ #8=7 Ap ¼ 2ð315Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2:31Þð25 000Þð0:372Þð1:0Þð48:2 Â 103Þ ð0:4Þð35Þ Â 108 ¼ 1:173 cm4 Select the ETD39 core. The core speciﬁcations are given in Table 5.6. 138 Transformers and Inductors for Power Electronics Table 5.6 Core and winding speciﬁcations Ac Wa Ap Vc kf ku MLT r20 a20 1.25 cm2 1.78 cm2 2.225 cm4 11.5 cm3 1.0 0.4 6.9 cm 1.72 mV-cm 0.00393 Winding Design The ratio Vrms/K is given by DVs from the analysis above. Thus, in calculating the number of turns, take D at its voltage waveform ims Vaxrmims u¼mrva1ﬃﬃlﬃﬃuÀﬃDﬃeﬃﬃ,ﬃDﬃﬃiﬃ.Ve.dcD¼¼p0.ﬃ37ﬃV5 and dc. Kv ¼ 2.31. The RMS value of the input Primary turns pﬃﬃ Np ¼ Vp K v Bmax Ac f ¼ ð2:31Þð2 3ð12Þ Â 0:186Þð1:25 Â 10À4Þð25 000Þ ¼ 7:7 turns rounded up to 9 turns. Secondary turns In this design, the number of secondary turns is equal to the number of primary turns, i.e. 9 turns. The number of turns in the reset winding is: Nt ¼ 1 ÀD D Np ¼ 1 À 0:75 0:75 ð9Þ ¼ 3 turns Wire size The current density (Equation 5.4) is: Jo ¼ ð48:2 Â 103Þ pﬃﬃ3ﬃﬃ5ﬃﬃﬃﬃﬃﬃﬃﬃ 2ð0:4Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 ð2:225 Â 10À8Þ ¼ 2:885 Â 106 A=m2 Primary current Ip ¼ Po kppV p ¼ p75ﬃﬃ ð0:5Þ 3ð12Þ ¼ 7:22 A Aw ¼ Ip=Jo ¼ 7:22=2:885 ¼ 2:502 mm2 This corresponds to a 1.79 mm diameter. A 1.8 mm diameter wire with a dc resistance of 6.72 mV/m at 20 C will sufﬁce (Table A.1). Transformer Design 139 Primary copper loss Use Equation 3.37 to correct the winding resistance for temperature: Tmax ¼ 40 þ 35 ¼ 75 C: Rdc ¼ 9 Â 6:9 Â 10À2 Â 6:72 Â 10À3 Â ½1 þ 0:00393 Â ð75 À 20Þ Â 103 ¼ 5:08 mV The copper loss in the primary winding is: Pcu ¼ RdcI2rms ¼ 5:08 Â 10À3 Â ð7:22Þ2 ¼ 0:264 W Secondary current pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Is ¼ DIo ¼ 0:75ð7:5Þ ¼ 6:50 A Aw ¼ Is=Jo ¼ 6:50=2:885 ¼ 2:252 mm2 This corresponds to a 1.69 mm diameter. A 1.8 mm diameter wire with a DC resistance of 6.72 mV/m at 20 C will sufﬁce (Table A.1). Secondary copper loss Use Equation 3.37 to correct the winding resistance for temperature: Rdc ¼ 9 Â 6:9 Â 10À2 Â 6:72 Â 10À3 Â ½1 þ 0:00393 Â ð75 À 20Þ Â 103 ¼ 5:08 mV The copper loss in the secondary winding is: Pcu ¼ RdcI2rms ¼ 5:08 Â 10À3 Â ð6:50Þ2 ¼ 0:214 W High frequency effects pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ These are dc values. At 25 kHz, the skin depth d ¼ 66= 25 000 ¼ 0:42 mm. This is less than the radius of either primary or secondary conductor, so it increases the resistance. The correction factor to account for skin effect is given by Equation 1.22. Primary AC resistance: " # Rpac ¼ 5:08 1 þ 48 ð0:9=0:42Þ4 þ 0:8ð0:9=0:42Þ4 ¼ 6:73 mV I2pRpac ¼ ð7:22Þ2ð6:73 Â 10À3Þ ¼ 0:350 W Secondary AC resistance: " # Rsac ¼ 5:08 1 þ 48 ð0:9=0:42Þ4 þ 0:8ð0:9=0:42Þ4 ¼ 6:73 mV I2s Rsac ¼ ð6:50Þ2ð6:73 Â 10À3Þ ¼ 0:284 W 140 Transformers and Inductors for Power Electronics Core loss The ﬂux density ripple DB can be calculated by using Faraday’s law: DB ¼ V s DT N P Ac ¼ ð12Þð0:75Þ ð9Þð1:25 Â 10À4Þð25 000Þ ¼ 0:320 T Pfe ¼ VcKcf aðDB=2Þb ¼ ð11:5 Â 10À6Þð37:2Þð25 000Þ1:13ð0:16Þ2:07 ¼ 0:898 W Total losses: Primary copper Secondary copper Core 0.350 W 0.284 W 0.898 W 1.532 W Efficiency Efficiency ¼ 75 75 þ 1:532 ¼ 98% 5.3.3 Example 5.3: Push-Pull Converter Speciﬁcations The speciﬁcations for the push-pull converter are given in Table 5.7. Circuit Parameters The circuit diagram for the push-pull converter is shown in Figure 5.8 and its associated voltage and current waveforms are shown in Figure 5.9. We assume for simplicity that the turns ratio is 1 : 1. In Figure 5.8, switch 1 turns on at t ¼ 0 and turns off at time DT 0. By deﬁning the duty cycle in this manner, the combined on-time of the two switches is DT and the output voltage is DVs. The switching period is T and each switch controls the voltage waveform for T 0 ¼ T/2. The ﬂux density increases from 0 to its maximum value in the time t ¼ DT 0/2 ¼ DT/4. The RMS value of the input voltage waveform in Figure 5.9 is found by: pﬃﬃﬃ V rms ¼ DV s Table 5.7 Speciﬁcations Input Output Frequency, f Temperature rise, DT Ambient temperature, Ta 36 ! 72 V 24 V, 12.5 A 50 kHz 35 C 45 C Transformer Design 141 + Vs _ S2 Np : Ns is1 D1 + + vp2 vs2 _ _ + + vp1 vs1 _ _ S1 is2 D2 Lo + Co Vo _ Figure 5.8 Push-pull converter circuit. The average value of the input voltage waveform during the time t is Vs: hvi ¼ Vs and thus, from Equation 4.61: k ¼ V rms hvi ¼ pﬃﬃﬃ DV s Vs ¼ pﬃﬃﬃ D t ¼ DT0=2 ¼ DT=4 ¼ D TT T4 pﬃﬃﬃ Kv ¼ k t=T ¼ D D=4 ¼ p4ﬃﬃﬃ D : For D ¼ 1.0, K ¼ 4.0, as expected for a square waveform. When both switches are off in the circuit of Figure 5.8, half the output current circulates in each of the secondary windings. This circulating current contributes to heating, but there is no transfer of power through the transformer. We can include this effect by correctly deﬁning the power factor of the windings. The RMS value of the secondary current (neglecting the ripple), as shown in Figure 5.9, is given by: I srms ¼ Io 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ DÞ: The RMS value of the secondary voltage is: V srms ¼ pﬃﬃﬃ DV s ¼ pVﬃoﬃﬃ : D 142 Transformers and Inductors for Power Electronics Vp,Vs Vs 0 τ Φ t ip1 ip2 Io 0 t is1 Io Io 2 0 t is2 Io Io 2 0 DT’ T’ T t Figure 5.9 Push-pull converter voltage and current waveforms. The VA rating of each secondary winding is now: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V srmsIsrms ¼ 1 2 1pþﬃﬃﬃ D VoIo ¼ D 1pþﬃﬃﬃ D Po : D2 Recalling the deﬁnition of power factor and noting that, for each winding, the average power kpi ¼ Po/2, where Po is the total output power, the power factor of each second- ary winding is: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kps ¼ hpi V srmsIsrms ¼ 1 D þ : D pﬃﬃ For D ¼ 1, kps ¼ 1= 2 as expected. Transformer Design 143 The RMS values of the input voltage and current are: pﬃﬃﬃ V prms ¼ DVs; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Iprms ¼ ðD=2ÞIs: The average power delivered by each of the primary windings is: hpi ¼ ð DT 0 0 vðtÞiðtÞdt ¼ 1 T V s I s DT 0 ¼ D 2 VsIs For a 1 : 1 turns ratio, Idc ¼ I0 and the power factor in each primary winding is then given by: kpp ¼ hpi V prmsIprms ¼ p1ﬃﬃ 2 : We can now sum the VA ratings over the two input windings and the two output windings. The average power through each secondary winding is Po/2 and the average power through each primary winding is Po/2. Thus we have: X VA ¼ 1 Po þ Po þ 1 Po þ Po ¼ kpp pﬃﬃ 2 þ 2rﬃﬃﬃﬃﬃﬃ2ﬃﬃﬃﬃﬃﬃ! kps 1þD D Po 2 2 : For the input voltage range 36 V to 72 V, the duty cycle can vary between 33% and 67%. For an inputpﬃvﬃﬃoltage of 36 V, the duty cycle is 24/36 ¼ 67%. The waveform factor is Kv ¼ 4= D ¼ 4:88. Core Selection Ferrite would normally be used for this type of application at the speciﬁed frequency. The material speciﬁcations for EPCOS N67 Mn-Zn ferrite are listed in Table 5.8. The output power of the transformer is Po ¼ (24 þ 1.0) Â 12.5 ¼ 312.5 W, assuming a for- ward voltage drop of 1.5 V for the diode. In terms of core selection, the maximum dissipation occurs at maximum duty cycle, that is D ¼ 0.67: X VA ¼ pﬃﬃ 2 þ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 þ 0:67 0:67 ð312:5Þ ¼ 935 VA Table 5.8 Material speciﬁcations Kc a b Bsat 9.12 1.24 2.0 0.4 T 144 Transformers and Inductors for Power Electronics Table 5.9 Core and winding speciﬁcations Ac Wa Ap Vc kf ku MLT r20 a20 1.73 cm2 2.78 cm2 4.81 cm4 17.70 cm3 1.0 0.4 7.77 cm 1.72 mV-cm 0.00393 The optimum ﬂux density Equation 5.8 is: Bo ¼ 22=3 Â ð1:72 Â ½ð10Þð40Þð35Þ2=3 10À8Þð10Þð0:4ÞÃ1=12 h ð5:6Þð9:12Þð50000Þ1:24 i7=12 Á ð4:88Þð50000Þð1:0Þð0:4Þ!1=6 ¼ 0:126 T 935 The optimum ﬂux density is less than Bsat and Ap from Equation 5.5 and is: " pﬃﬃ #8=7 Ap ¼ 2ð935Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð4:88Þð50 000Þð0:126Þð1:0Þð0:4Þð48:2 Â 103Þ ð0:4Þð35Þ Â 108 ¼ 2:693 cm4 The EPCOS ETD44 core is suitable. The core speciﬁcations are given in Table 5.9. Winding Design Primary turns pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Np ¼ Vp K v Bmax Ac f ¼ 0:67ð36Þ ð4:88Þð0:126Þð1:73 Â 10À4Þð50000Þ ¼ 5:5 turns Choose six turns. Secondary turns We assumed a 1 : 1 turns ratio. so the number of secondary turns is 6. Wire size The current density (Equation 5.4) is: sﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Jo ¼ Kt DT p1ﬃﬃﬃﬃﬃ ¼ ð48:2 Â 103Þ 2ku 8 Ap 35 2ð0:4Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 ð4:81 Â 10À8Þ ¼ 2:620 Â 106 A=m2 Transformer Design 145 Primary current Ip ¼ Po=2 kpp V p ¼ 312:5=2 ð0:707Þð29:5Þ ¼ 7:5 A Aw ¼ Ip=Jo ¼ 2:863 mm2 Standard 0.1 Â 30 mm copper foil with a DC resistance of 5.8 mV/m at 20 C meets this requirement. Primary copper loss Use Equation 3.37 to correct the winding resistance for temperature: Tmax ¼ 45 þ 30 ¼ 75 C: Rdc ¼ ð6Þð7:77 Â 10À2Þð5:80 Â 10À3Þ½1 þ ð0:00393Þð75 À 20Þ Â 103 ¼ 3:29 mV The copper loss in each the primary winding is: Pcu ¼ RdcI2rms ¼ ð3:29 Â 10À3Þð7:5Þ2 ¼ 0:185 W Secondary current Is ¼ Io 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þD ¼ 12:5 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 0:67 2 ¼ 8:08 A Aw ¼ Is=Jo ¼ 3:083 mm2 Again, standard 0.1 Â 30 mm copper foil meets this requirement. Secondary copper loss Use Equation 3.37 to correct the winding resistance for temperature: Rdc ¼ ð6Þð7:77 Â 10À2Þð5:80 Â 10À3Þ½1 þ ð0:00393Þð75 À 20Þ Â 103 ¼ 3:29 mV The copper loss in each of the secondary winding is: Pcu ¼ Rdc I 2 rms ¼ ð3:29 Â 10À3Þð8:08Þ2 ¼ 0:215 W High frequency effects The skin depth in copper at 50 kHz is 0.295 mm, which is greater than the thickness of the foil and therefore does not present a problem. See Example 7.6. 146 Transformers and Inductors for Power Electronics Core loss The peak value of the ﬂux density in the selected core is: pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Bmax ¼ DV d c Kvf NpAc ¼ ð4:88Þð50 0:67ð36Þ 000Þð6Þð1:73 Â 10À4Þ ¼ 0:116 T Pfe ¼ VcKcf aBbmax ¼ ð17:7 Â 10À6Þð9:12Þð50 000Þ1:24ð0:116Þ2:0 ¼ 1:466 W Total losses: Efficiency Primary copper Secondary copper Core 0.185 W 0.185 W 0.215 W 0.215 W 1.466 W 2.266 W Efficiency ¼ 312:5 312:5 þ 2:266 ¼ 99:3% 5.4 Transformer Insulation The window utilization factor in a transformer is typically 40%, which arises due to the requirements to isolate windings from each other. In general, there is a distinction between the insulation required for the proper functioning of a transformer (operational insulation) and the safe isolation of a circuit from a hazardous voltage (double insulation or reinforced insulation). These are three basic means of providing insulation: By insulator; this consists of a dielectric material (e.g. mylar tape) separating two conductors. By creepage; this is the distance along an insulating surface (e.g. from one solder pin to another on a bobbin) between two conductors on the surface. By clearance, which is normally understood to be the air gap between two conducting bodies. The insulation capability achieved by dielectric materials is rated in kV/mm or (kV/sec)/ mm. In the case of creepage, the roughness of the surface and the degree of pollution deﬁnes the insulation capability. Clearance normally applies to air, and the breakdown voltage is a function of ionization and conduction of the air in question, depending on pressure, temperature, humidity and pollution. To quantify the quality of the insulation, the voltages that are applied to the insulation barriers need to be deﬁned clearly. There are three main voltage deﬁnitions required for insulation barriers: The rms working voltage is the rms value of the highest voltage to which the insulation can be subjected when the transformer is operating under conditions of normal use. The peak working voltage is the peak value of the working voltage and it may be an order of magnitude greater than the rms working voltage, depending on the shape of the voltage waveform. The rms and peak working voltages are steady state voltages that occur in normal operation. Transformer Design 147 Transient voltages must also be considered. These are irregular and represent abnormal operation. There are different categories of overvoltage, and the applicable values depend on the types of equipment and the location of the installation. 5.4.1 Insulation Principles The wire used in the winding of a transformer has single or double temperature-resistant lacquer insulation on its surface, which meets the usual turn-to-turn voltage insulation requirements. In a multilayer transformer, the additive effect of the turns means that the wire insulation might not be sufﬁcient to insulate all the turns from each other. In this case, additional insulation is required between the layers (intra-winding insulation). Basic insulation meets the operating insulation requirements in a transformer. The basic insulation should be compatible with the rms working voltage, the peak working voltage and the relevant overvoltage category of the applied circuits. Transformers are often used for isolating hazardous high-voltage circuits from low-voltage circuits that may be exposed to human contact. The hazard to humans means that additional insulation is required to meet the safe extra low voltage (SELV) requirements. The additional insulation requirement results from the tolerance of a single failure and therefore requires double basic insulation. The additional insulation, combined with the basic insulation, is referred to as double insulation or reinforced insulation in the relevant standards. In addition to the insulation requirements, the insulation materials in a transformer must withstand the highest possible operating temperatures. During abnormal operations, such as a fault inside the transformer or a short-circuit condition, the internal temperature within the coil could rise to a sufﬁciently high level to cause ignition of the insulation materials. Insulation materials are therefore classiﬁed under different ﬂammability categories, depending on the maximum temperature to which the material will be exposed. 5.4.2 Practical Implementation The main insulation elements of a transformer are illustrated in Figure 5.10. The bobbin isolates the windings from the core; there is lacquer insulation on the wire; there are Insulation spacer Double wire insulation Basic insulation (tube) Secondary II Wire Secondary I Wire insulation Intra-layer insulation Primary Bobbin Double isolation Figure 5.10 Transformer insulation. 148 Transformers and Inductors for Power Electronics insulation tapes between the winding layers; there are sleeves over the wires; and the insulation spacer achieves the required creepage distances. The cross-section of the bobbin assembly for an E-type core, with the various types of insulation, is shown in Figure 5.10. The example shows one primary (hazardous voltage) winding with two layers and two secondary windings. The intra-layer insulation (basic insulation), consisting of tapes, isolates the working voltages between the wires, while the inter-layer insulation (double insulation) tape provides the required isolation between the windings. Tape that provides basic insulation is always required between windings. The low-voltage secondary winding is separated from the primary winding by a double layer of basic insulation tape which achieves double insulation, or reinforced insulation. This achieves the safe separation of the hazardous voltage on the primary to the low voltage on the secondary side (SELV). Space holders (or insulation spacers) on each side of a winding increase the required creepage distance by increasing the distance from the end wires in the primary windings to the end wires of the secondary windings. These space holders are often made up of several layers of insulation tape. The wire sleeves provide the double insulation for coil wires entering and exciting the bobbin. The ﬁnal assembly is impregnated with an insulating resin to prevent the ingress of pollution and moisture. The insulation requirements ultimately depend on the application and the relevant safety standards. The widely used standards that are relevant to the insulation requirements in power supplies and its transformers for ofﬁce and household applications are IEC60950 and IEC60335. Special transformers need to comply with speciﬁc standards, for example IEC6155-8, IEC6155-2 and IEC6155-16. These international standards form the basis for nationally adopted standards in various jurisdictions. 5.5 Problems 5.1 Rework Example 5.1 for a centre-tapped rectiﬁer with a highly inductive load. 5.2 Calculate the core size in a full-bridge converter with the speciﬁcations of Example 5.3. The circuit is shown in Figure 5.11 and the voltage and current waveforms are shown in Figure 5.12. S1 + Vs _ S3 ip + vp _ S4 S2 is1 D1 + vs2 _ + vs1 _ is2 D2 Lo + Co Vo _ Figure 5.11 Full-bridge converter circuit. Transformer Design Vp,Vs1,Vs2 Vs 0 -Vs ip Io 149 t 0 t -Io is1 Io Io 2 0 t is2 Io Io 2 0 DT’ T’ T t Figure 5.12 Full-bridge converter voltage and current waveforms. 5.3 Calculate the core size in a half-bridge converter with the speciﬁcations of Example 5.3. The circuit is shown in Figure 5.13 and the voltage and current waveforms are shown in Figure 5.14. 5.4 A 600 W forward converter operates at 80 kHz with a nominal DC input voltage of 325 V and a nominal DC output voltage of 24 V. Using an EE ferrite core (N87 material in Table 1.1), select the transformer core and calculate the copper loss in the windings and the core loss. The maximum allowed temperature on the transformer isolation materials is 110 C and the maximum temperature rise allowed in the winding is 35 C. 5.5 A 600 W push-pull transformer operates at 80 kHz with a rectiﬁed AC input voltage of 230 V and a DC output voltage of 24 V. Using an EE ferrite core (N87 material in Vdc + 2_ C1 S1 ip is1 D1 Lo + Vs _ + + vs2 _ vp + Vdc + 2_ C2 _ S2 vs1 _ is2 D2 Figure 5.13 Half-bridge converter circuit. Vp,Vs1,Vs2 Vs 2 0 - Vs 2 ip Io + Co Vo _ t 0 t -Io is1 Io Io 2 0 t is2 Io Io 2 0 DT’ T’ T t Figure 5.14 Half-bridge converter voltage and current waveforms. Transformer Design 151 Table 1.1), select the transformer core and calculate the copper loss in the windings and the core loss. The maximum ambient temperature of the transformer is 70 C and the maximum temperature rise allowed in the winding is 25 C. 5.6 A power supply with an AC input voltage of 110 V and active power factor correction boosts the input voltage to 400 V DC. This voltage is supplied to a half bridge converter with 200 W output power and DC output voltage of 24 V. Using a ferrite core (material similar to N87 in Table 1.1), select the core for a transformer with a centre-tapped output winding. Calculate the copper loss in the windings and the core loss for a nominal operating frequency of 90 kHz. The maximum allowed operating temperature of the transformer materials is 130 C, with a maximum temperature inside the power supply of 100 C. Hint: Use the waveforms in Example 4.5 to ﬁnd the power factor of the windings. MATLAB Program for Example 5.1 %example 5.1 Centre-Tapped Rectiﬁer Transformer alpha=1.7; alpha20=0.00393; beta=1.9; deltaT=55; ro20=1.72e-8; row=1.72e-8; Ac=19.5e-4; Ap=979e-8; Bm=1.5; Bsat=1.5; f=50; h=10; Is=5; ka=40; Kc=3.388; kc=5.6; kf=0.95; kpp=1; Kt=48.2e3; ku=0.4; Kv=4.44; kw=10; MLT=28e-2; N=354; N2=155; Po=1010; Tmax=95; 152 Transformers and Inductors for Power Electronics Vc=693e-6; Vp=230; Vs=101; Vw =0.28e-2*50.2e-4; wire_Rdc1=8.50e-3; wire_Rdc2=6.72e-3; sumVA=(1+sqrt(2))*Po Bo=([(h*ka*deltaT)^(2/3)]/[2^(2/3)*(row*kw*ku)^(1/12)* (kc*Kc*f^alpha)^(7/12 )])*[(Kv*f*kf*ku)/(sumVA)]^(1/6) Ap1=[(sqrt(2)*sumVA)/(Kv*f*Bsat*kf*Kt*sqrt(ku*deltaT))]^(8/7) a0 =(kc*Kc*f^alpha*Bsat^beta)/(row*kw*ku) a1 =(h*ka*deltaT)/(row*kw*ku) a2 =((sumVA)/(Kv*f*Bsat*kf*ku))^2 Ap2 = Ap1-(a0*Ap1^2-a1*Ap1^(7/4)+a2)/(2*a0*Ap1-(7/4)*a1*Ap1^(3/4)) Np=Vp/(Kv*Bm*Ac*f) Ns=Np*Vs/Vp Tmax=40+55; row=ro20*[1+alpha20*(Tmax-20)] Jo = sqrt((h*ka*sqrt(Ap)*deltaT-Vc*Kc*f^alpha*Bsat^beta)/ (row*Vw*ku)) Ip=Po/(kpp*Vp) Aw1=Ip/Jo Rdc1=MLT*N*wire_Rdc1*(1+alpha20*(Tmax-20)) Pcu1=((Ip^2)*Rdc1) Aw2=Is/Jo Rdc2=MLT*N2*wire_Rdc2*(1+alpha20*(Tmax-20)) Pcu2=(Is^2*Rdc2*2) Pfe=Vc*Kc*f^alpha*Bm^beta Ptot=Pfe+Pcu1+Pcu2 efﬁciency=Po/(Po+Ptot) MATLAB Program for Example 5.2 %example 5.2 Forward Converter Vs=12; alpha=1.13; alpha20=0.00393; beta=2.07; delta=0.42; deltaT = 35; Tamb = 40; row = 1.72e-8; Transformer Design 153 Bsat=0.4; D=0.75; f = 25e3; h = 10; Io=7.5; ka = 40; kc = 5.6; kw = 10; Kc = 37.2; kf = 1.0; Kt = 48.2e3; ku = 0.4; Po=75; rop=0.9; ros=0.9; Vp=sqrt(3)*Vs; kpp=sqrt(1-D); kps=sqrt(1-D); Kv=1/(sqrt(D*(1-D))) sumVA=(1/kpp+1/kps)*Po sumVA2=1.05*sumVA Bo=(((h*ka*deltaT)^(2/3))/((2^(2/3))*((row*kw*ku)^(1/12))* ((kc*Kc*f^alpha)^ (7/12))))*([(2*Kv*f*kf*ku)/sumVA2]^(1/6)) Ap1=([(sqrt(2)*sumVA2)/(Kv*f*2*Bo*kf*Kt*sqrt(ku*deltaT))]^(8/7)) *10^8 Ap2=2.2251e-8; Ac=1.25e-4; Vc=11.5e-6; MLT=6.9e-2; Np=Vp/(Kv*2*Bo*Ac*f) Np=9; Nt=(1-D)/D*Np Tmax=Tamb+deltaT; Jo=Kt*((sqrt(deltaT))/(sqrt(2*ku)))*(1/(Ap2)^(1/8)) Ip=Po/(kpp*Vp) Awp=Ip/Jo wire_Rdcp=6.72e-3; Rdcp=MLT*Np*wire_Rdcp*(1+alpha20*(Tmax-20)) Pcup=Rdcp*Ip^2 Is=sqrt(D)*Io Aws=Is/Jo wire_Rdcs=6.72e-3; Rdcs=MLT*Np*wire_Rdcs*(1+alpha20*(Tmax-20)) Pcus=Rdcs*Is^2 Rpac=Rdcp*[1+(rop/delta)^4/(48+0.8*(rop/delta)^4)] 154 Transformers and Inductors for Power Electronics Ip2Rpac=Ip^2*Rpac Rsac=Rdcs*[1+(ros/delta)^4/(48+0.8*(ros/delta)^4)] Is2Rsac=Is^2*Rsac deltaB=(Vs*D)/(Np*Ac*f) Pfe=Vc*Kc*(f^alpha)*((deltaB/2)^beta) Ptot=Pfe+Ip2Rpac+Is2Rsac Efﬁciency=Po/(Po+Ptot) MATLAB Program for Example 5.3 %example 5.3 Push-Pull Converter alpha=1.24; alpha20=0.00393; beta=2.0; deltaT=35; row=1.72e-8; Ac=1.73e-4; D=0.67; f=50e3; h=10; Io=12.5; ka=40; kc=5.6; Kc=9.12; Kc2=9.12; kf=1; kpp=0.707; Kt=48.2e3; ku=0.4; Kv=4.88; kw=10; MLT=7.77e-2; Po=312.5; Tmax=45+35; Vc=17.7e-6; Vdc=36; wire_Rdc=5.80e-3; wire_Rdc2=5.80e-3; sumVA=round((sqrt(2)+sqrt((1+D)/D))*Po) Bo=(((h*ka*deltaT)^(2/3))/((2^(2/3))*((row*kw*ku)^(1/12))* ((kc*Kc*f^alpha)^ (7/12))))*([(Kv*f*kf*ku)/sumVA]^(1/6)) Transformer Design 155 Ap1=([(sqrt(2)*sumVA)/(Kv*f*Bo*kf*Kt*sqrt(ku*deltaT))]^(8/7))*10^8 Vp=sqrt(0.67)*36 Np=round(Vp/(Kv*Bo*Ac*f)) Kt=48.2e3; Ap2=4.81e-8; Jo=Kt*((sqrt(deltaT))/(sqrt(0.8)))*(1/(Ap2)^(1/8)) Ip=(Po/2)/(kpp*Vp) Aw1=Ip/Jo Rdc=MLT*Np*wire_Rdc*(1+alpha20*(Tmax-20)) Pcu1=Rdc*Ip^2 Is=(Io/2)*sqrt(1+D) Aw2=Is/Jo Rdc2=MLT*Np*wire_Rdc2*(1+alpha20*(Tmax-20)) Pcu2=Rdc*Is^2 Bmax=(sqrt(D)*Vdc)/(Kv*f*Np*Ac) Pfe=Vc*Kc2*(f^alpha)*(Bmax^beta) Ptot=Pfe+2*Pcu1+2*Pcu2 efﬁciency=Po/(Po+Ptot) Further Reading 1. Bartoli, M., Reatti, A., and Kazimierczuk, M.K. (1994) High-frequency models of ferrite core inductors. Proceedings of the IEEE Industrial Electronics, Control and Instrumentation, IECON, pp. 1670–1675. 2. Bartoli, M., Reatti, A., and Kazimierczuk, M.K. (1994) Modelling iron-powder inductors at high frequencies. Proceedings of the IEEE Industry Applications Conference, IAS, pp. 1225–1232. 3. Bennett, E. and Larson, S.C. (1940) Effective resistance to alternating currents of multilayer windings. Transactions of the American Institute of Electrical Engineers 59 (12), 1010–1017. 4. Blume, L.F. (1982) Transformer Engineering, John Wiley & Sons, New York. 5. Bueno, M.D.A. (2001) Inductance and Force Calculations in Electrical Circuits, Nova Science Publishers, Huntington. 6. Carsten, B. (1986) High frequency conductor losses in switchmode magnetics. Proceedings of the High Fre- quency Power Converter Conference, pp. 155–176. 7. Cheng, K.W.E. and Evans, P.D. (1994) Calculation of winding losses in high-frequency toroidal inductors using single strand conductors. IEE Proceedings-Electric Power Applications B 141 (2), 52–62. 8. Cheng, K.W.E. and Evans, P.D. (1995) Calculation of winding losses in high frequency toroidal inductors using multistrand conductors. IEE Proceedings-Electric Power Applications B 142 (5), 313–322. 9. Del Vecchio, R.M., Poulin, B., Feghali, P.T. et al. (2001) Transformer Design Principles: With Applications to Core-Form Power Transformers, 1st edn, CRC Press, Boca Raton, FL. 10. Dowell, P.L. (1966) Effects of eddy currents in transformer windings. Proceedings of the Institution of Electri- cal Engineers 113 (8), 1387–1394. 11. Erickson, R.W. (2001) Fundamentals of Power Electronics, 2nd edn, Springer, Norwell, MA. 12. Evans, P.D. and Chew, W.M. (1991) Reduction of proximity losses in coupled inductors. IEE Proceedings- Electric Power Applications B 138 (2), 51–58. 13. Ferreira, J.A. (2010) Electromagnetic Modelling of Power Electronic Converters (Power Electronics and Power Systems), 1st edn, Springer, Norwell, MA. 156 Transformers and Inductors for Power Electronics 14. Fitzgerald, A.E., Kingsley, C. Jr, and Umans, S.D. (2002) Electric Machinery, 6th edn, McGraw-Hill, New York. 15. Flanagan, W.M. (1992) Handbook of Transformer Design and Application, 2nd edn, McGraw-Hill, New York. 16. Georgilakis, P.S. (2009) Spotlight on Modern Transformer Design (Power Systems), 1st edn, Springer, New York. 17. Hanselman, D.C. and Peake, W.H. (1995) Eddy-current effects in slot-bound conductors. IEE Proceedings- Electric Power Applications B 142 (2), 131–136. 18. Hoke, A.F. and Sullivan, C.R. (2002) An improved two-dimensional numerical modeling method for E-core transformers. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 151–157. 19. Hurley, W.G. and Wilcox, D.J. (1994) Calculation of leakage inductance in transformer windings. IEEE Transactions on Power Electronics 9 (1), 121–126. 20. Hurley, W.G., Wilcox, D.J., and McNamara, P.S. (1991) Calculation of short circuit impedance and leakage impedance in transformer windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 651–658. 21. Hurley, W.G., Wolﬂe, W.H., and Breslin, J.G. (1998) Optimized transformer design: inclusive of high-frequency effects. IEEE Transactions on Power Electronics 13 (4), 651–659. 22. Jieli, L., Sullivan, C.R., and Schultz, A. (2002) Coupled-inductor design optimization for fast-response lowvoltage. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 817–823. 23. Judd, F. and Kressler, D. (1977) Design optimization of small low-frequency power transformers. IEEE Transactions on Magnetics 13 (4), 1058–1069. 24. Kassakian, J.G. and Schlecht, M.F. (1988) High-frequency high-density converters for distributed power supply systems. Proceedings of the IEEE 76 (4), 362–376. 25. Kassakian, J.G., Schlecht, M.F., and Verghese, G.C. (1991) Principles of Power Electronics (Addison-Wesley Series in Electrical Engineering), Prentice Hall, Reading, MA. 26. Kazimierczuk, M.K. (2009) High-Frequency Magnetic Components, John Wiley & Sons, Chichester. 27. Krein, P.T. (1997) Elements of Power Electronics (Oxford Series in Electrical and Computer Engineering), Oxford University Press, Oxford. 28. Kulkarni, S.V. (2004) Transformer Engineering: Design and Practice, 1st edn, CRC Press, New York. 29. B. H. E. Limited (2004) Transformers: Design, Manufacturing, and Materials (Professional Engineering), 1st edn, McGraw-Hill, New York. 30. McAdams, W.H. (1954) Heat Transmission, 3rd edn, McGraw-Hill, New York. 31. McLyman, C.W.T. (1997) Magnetic Core Selection for Transformers and Inductors, 2nd edn, Marcel Dekker Inc., New York. 32. McLyman, C.W.T. (2002) High Reliability Magnetic Devices, 1st edn, Marcel Dekker Inc., New York. 33. McLyman, C.W.T. (2004) Transformer and Inductor Design Handbook, 3rd edn, Marcel Dekker Inc., New York. 34. E. S. MIT (1943) Magnetic Circuits and Transformers (MIT Electrical Engineering and Computer Science), The MIT Press, Cambridge, MA. 35. Muldoon, W.J. (1978) Analytical design optimization of electronic power transformers. Proceedings of Power Electronics Specialists Conference, PESC, pp. 216–225. 36. Pentz, D.C. and Hofsajer, I.W. (2008) Improved AC-resistance of multiple foil windings by varying foil thickness of successive layers. The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 27 (1), 181–195. 37. Perry, M.P. (1979) Multiple layer series connected winding design for minimum losses. IEEE Transactions on Power Apparatus and Systems PAS-98 (1), 116–123. 38. Petkov, R. (1996) Optimum design of a high-power, high-frequency transformer. IEEE Transactions on Power Electronics 11 (1), 33–42. 39. Pollock, J.D., Lundquist, W., and Sullivan, C.R. (2011) Predicting inductance roll-off with dc excitations. Proceedings of the IEEE Energy Conversion Congress and Exposition, ECCE, pp. 2139–2145. 40. Pressman, A.I., Bellings, K., and Morey, T. (2009) Switching Power Supply Design, 3rd edn, McGrawHill, New York. 41. Ramo, S., Whinnery, J.R., and Van Duzer, T. (1984) Fields and Waves in Communication Electronics, 2nd edn, John Wiley & Sons, New York. Transformer Design 157 42. Sagneri, A.D., Anderson, D.I., and Perreault, D.J. (2010) Transformer synthesis for VHF converters. Proceedings of International Power Electronics Conference, IPEC, pp. 2347–2353. 43. Smith, B. (2009) Capacitors, Inductors and Transformers in Electronic Circuits (Analog Electronics Series), Wexford College Press, Wexford. 44. Snelling, E.C. (1988) Soft Ferrites: Properties and Applications, 2nd edn, Butterworths, London. 45. Sullivan, C.R. (1999) Optimal choice for number of strands in a litz-wire transformer winding. IEEE Transac- tions on Power Electronics 14 (2), 283–291. 46. Sullivan, C.R. and Sanders, S.R. (1996) Design of microfabricated transformers and inductors for high- frequency power conversion. IEEE Transactions on Power Electronics 11 (2), 228–238. 47. Urling, A.M., Niemela, V.A., Skutt, G.R., and Wilson, T.G. (1989) Characterizing high-frequency effects in transformer windings-a guide to several signiﬁcant articles. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 373–385. 48. Van den Bossche, A. (2005) Inductors and Transformers for Power Electronics, 1st edn, CRC Press, New York. 49. Vandelac, J.P. and Ziogas, P.D. (1988) A novel approach for minimizing high-frequency transformer copper losses. IEEE Transactions on Power Electronics 3 (3), 266–277. 50. Venkatachalam, K., Sullivan, C.R., Abdallah, T., and Tacca, H. (2002) Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters. Proceedings of IEEE Workshop on Computers in Power Electronics, COMPEL, pp. 36–41. 51. Venkatraman, P.S. (1984) Winding eddy current losses in switch mode power transformers due to rectangular wave currents. Proceedings of the 11th National Solid-State Power Conversion Conference, Powercon 11, pp. A1.1–A1.11. 52. Williams, R., Grant, D.A., and Gowar, J. (1993) Multielement transformers for switched-mode power supplies: toroidal designs. IEE Proceedings-Electric Power Applications B 140 (2), 152–160. 53. Ziwei, O., Thomsen, O.C., and Andersen, M. (2009) The analysis and comparison of leakage inductance in different winding arrangements for planar transformer. Proceedings of the IEEE Power Electronics and Drive Systems, PEDS, pp. 1143–1148. 6 High Frequency Effects in the Windings1 It is clear from Chapter 2 that operating the transformer at high frequency reduces its size. However, there are additional loss mechanisms that come into play at high frequencies. These contribute to temperature rise, and a larger core may be required to increase the surface area to improve heat dissipation. A proper understanding of high-frequency effects is required to ensure a trade-off between reduced size and increased loss. It was shown in Section 4.3.5 that the optimum design has the core loss approximately equal to the copper (winding) loss. The methodology for transformer design developed in Chapter 5 assumed that the winding loss was due to the DC resistance of the windings. In reality, and in particular for power electronics applications, high-frequency operation leads to increased AC loss due to skin and proximity effects. The skin effect and the proximity effect give rise to increased loss in conductors, due to the non-uniform distribution of current in the conductors. These effects are a direct result of Faraday’s law, whereby eddy currents are induced to oppose the ﬂux created in the windings by the AC currents. Both of these effects are examined in detail in the sections to follow, and straightforward formulae are given to calculate them. Traditionally, Dowell’s celebrated formula is used to calculate the high-frequency proximity effect loss with sinusoidal excitation. We will expand Dowell’s formula [1] to take account of arbitrary current waveforms that are encountered in switch mode power supplies by calculating the loss at the individual frequencies in the Fourier series. The resultant loss can be calculated for a range of winding layer thicknesses, and an optimum layer thickness can be established graphically. We will also use a more convenient approach, based on the derivative of the current waveform, to simplify the calculations. This will lead to a simple expression for the optimum layer thickness in a multilayer winding. 1 Parts of this chapter are reproduced with permission from [2] Hurley, W.G., Gath, E., and Breslin, J.G. (2000) Optimizing the AC resistance of multilayer transformer windings with arbitrary current waveforms. IEEE Transactions on Power Electronics 15 (2), 369–376. Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 160 Transformers and Inductors for Power Electronics 6.1 Skin Effect Factor An isolated round conductor carrying AC current generates a concentric alternating magnetic ﬁeld which, in turn, induces eddy currents (Faraday’s law). Figure 6.1 shows the distribution of the magnetic ﬁeld at low frequencies. The nature of the ﬂux density inside and outside the conductor was analyzed in Section 1.2. Eddy currents oppose the ﬂux, and the resulting distribution of current means that eddy currents cancel some of the current at the centre of the conductor, while increasing the current near the surface, as shown in Figure 6.1. The overall effect is that the total current ﬂows in a smaller annular area. At high frequencies, the current ﬂows in an equivalent annular cylinder at the surface, with thickness d, called the skin depth. This ﬁeld problem can be solved from Maxwell’s equations. In a linear homogeneous isotropic medium, Maxwell’s equations take the following form: r Â H ¼ sE r Â E ¼ Àm0 @H @t ð6:1Þ ð6:2Þ For a sinusoidal current, the H ﬁeld takes the form H ¼ Hfejvt. This gives rise to current in the z direction that varies with radius only. Combining Equations 6.1 and 6.2 with sinusoidal excitation yields a general expression for the current density J in a conductor such as copper with relative permeability mr ¼ 1: r Â H ¼ sE ¼ J ð6:3Þ ik iφ I B(r) B(r) ir Eddy currents B(r) ro r Current distribution Figure 6.1 Eddy currents in a circular conductor. High Frequency Effects in the Windings 161 s is the conductivity of the conductor material and: d2J d r2 þ 1 r dJ dr À jvm0sJ ¼ 0 ð6:4Þ This is a modiﬁed Bessel’s equation. The general solution is: JðrÞ ¼ AI0ðmrÞ þ BK0ðmrÞ ð6:5Þ wherepIﬃ0ﬃﬃﬃﬃaﬃﬃnﬃﬃdﬃﬃﬃﬃK0 are modiﬁed Bessel functions of the ﬁrst and second kind, of order 0, and m¼ jvm0s, so that the pﬃﬃ the square root, that is, j argument ¼ jp e4. of I0 and K0 is complex. We take the principal value of The coefﬁcients A and B are determined from the boundary conditions and will be com- kpKleeeilxðvr.ipnItﬃvﬃfﬃiuﬃmsﬃﬃnﬃ0wﬃcﬃsﬃtﬃoiÞor, tnthhsonwuogittihhngirnetaohluaatr rtaghnueamlsyeosnliust,twivoienzs.ﬁnbodefrEiðtrqpmuoaﬃvﬃﬃtrﬃmieﬃoﬃﬃ0ﬃncﬃsﬃoﬃÞ6n,.v4beenaiirðeernpitnﬃvtﬃoﬃfﬃmﬃaﬃuﬃ0cﬃsﬃstﬃeﬃÞc,tohkmeebrmðinropadtiﬃviﬁﬃoﬃﬃemnﬃﬃdﬃs0ﬃﬃsﬃBoﬃÞef satsnhedel functions with complex arguments. MATLAB can calculate these functions directly without resorting to calculation of the real and imaginary parts individually. The quantity m is a function of the frequency and the conductivity of the conductor; this may be related to the skin depth d0 of the conducting material: mro ¼ ð1 þ jÞ ro d0 d0 ¼ pﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃ pf m0s ð6:6Þ ð6:7Þ The solution of Equation 6.5, taking the boundary conditions into account, means that B ¼ 0, since the current is ﬁnite at the centre of the core at r ¼ 0. Taking the current density as J(ro) at the outer surface, the solution is: JðrÞ JðroÞ ¼ I0ðmrÞ I0ðmroÞ ð6:8Þ A plot of the normalized magnitude of J(r) is shown in Figure 6.2 for different values of ro/d0, corresponding to 1 kHz, 10 kHz, 100 kHz and 1 MHz in a 2.5 mm diameter copper conductor. It is clear that, even at 100 kHz, there is a signiﬁcant reduction of current in the centre of the copper wire. The MATLAB program to generate the plots in Figure 6.2 is given at the end of this chapter, and it may be used to generate plots for other conductors or for different frequencies. Assuming cylindrical symmetry, the various components of the electric ﬁeld intensity E and the magnetic ﬁeld intensity H inside the cylinder, in cylindrical coordinates (r, f, z), satisfy the following identities [2]: Er ¼ 0; Ez ¼ 0; @Ef @z ¼ 0 ð6:9Þ Hr ¼ 0; Hf ¼ 0; @Hz @f ¼ 0 ð6:10Þ 162 Transformers and Inductors for Power Electronics rO = 0.599 (f = 1 kHz for 2.5 mm. diam. Cu wire) 1δ 0.9 0.8 0.7 J (r) 0.6 J (ro) 0.5 0.4 rO = 1.89 (f = 10 kHz for 2.5 mm. diam. δ Cu wire) rO = 5.99 (f = 100 kHz for 2.5 mm. δ diam. Cu wire) 0.3 0.2 rO = 18.9 (f = 1 MHz for 2.5 mm. δ diam. Cu wire) 0.1 0 -1 mm Outer radius -0.5 mm 0 Wire axis 0.5 mm 1 mm Outer radius Figure 6.2 Current distribution in a circular conductor. The two Maxwell’s equations then reduce to: Ez ¼ EðroÞ I0ðmrÞ I0ðmroÞ and: Hw ¼ 1 jvm0 dEz dr ¼ sEðroÞ m2 I00ðmrÞ I0ðmroÞ Ampere’s law relates the magnetic ﬁeld intensity to the current (Equation 1.8): þ H Á dl ¼ i C Taking the contour at the outer surface: HwðroÞ ¼ I 2pro Substituting back into Equation 6.12 and noting that: d dr I0ðmrÞ ¼ mI1ðmrÞ yields the internal impedance of the conductor per unit length: Zi ¼ EðroÞ I ¼ Rdc mroI0ðmrÞ 2I1ðmroÞ ð6:11Þ ð6:12Þ ð1:8Þ ð6:13Þ ð6:14Þ ð6:15Þ High Frequency Effects in the Windings 163 6 5 4 Rac Rdc 3 2 1 0 1 2 3 4 5 6 7 8 9 10 ro Ratio of radius to depth of penetration δ Figure 6.3 Rac/Rdc due to skin effect. Rdc is simply the DC resistance of the conductor per unit length. We now have a very compact form of the internal impedance of the conductor. The AC resistance is given by the real part of Zi and internal inductive reactance is given by the imaginary part of Zi. This expression is easily evaluated with MATLAB, but the following approximations are useful, deﬁning ks ¼ Rac/Rdc: ro 4 ks ¼ 1 þ 48 þ d0 0:8 ro 4 d0 ro < 1:7 d0 ð6:16Þ ¼ 0:25 þ 0:5 ro þ 3 d0 d0 32 ro ro > 1:7 d0 ð6:17Þ At very high frequencies, d0 << ro and: ks % 0:5 ro d0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:5ro pf m0s ð6:18Þ showing that the AC resistance is proportional to the square root of frequency. A plot of Rac/Rdc calculated with Equation 6.15 is shown in Figure 6.3. 6.2 Proximity Effect Factor The proximity effect arises when the distribution of current in one layer of a winding inﬂuences the distribution in another layer. Consider the transformer depicted in Figure 6.4, which has a two-layer primary winding, each layer carrying current I. Assuming ideal magnetic material (mr ! 1, s ! 0), the magnetic ﬁeld intensity H goes to zero inside the core. With the aid of Ampere’s law, we can plot the ﬁeld intensity across the window of the transformer. 164 C1 Transformers and Inductors for Power Electronics Eddy currents C2 a 1 bc2d w mmf H1w H0w H0 H1 Primary Secondary b Figure 6.4 mmf in a transformer winding. Consider N turns carrying current I within a layer. For contour C1 in Figure 6.4: H0w ¼ NI ð6:19Þ and: H0 ¼ NI w ð6:20Þ For contour C2 in Figure 6.4: H1w ¼ 2NI ð6:21Þ and H1 ¼ 2NI w ¼ 2H0 ð6:22Þ The situation is best illustrated by an mmf plot. Recall from Chapter 4 that the mmf in the secondary winding will cancel the mmf in the primary, restoring the ﬁeld intensity to zero at the far side of the window. The ﬁeld intensity is alternating at the operating frequency of the transformer. The presence of the ﬁeld intensity adjacent to the winding layers will induce eddy currents in the conductors, thus opposing the magnetic ﬁeld in accordance with Faraday’s law, as shown in Figure 6.4. The ﬁeld intensity to the right of layer 1 in Figure 6.4 is H0 and it is zero to the left. The direction of the induced eddy currents is such that the overall current distribution is increased on the right hand side of the current layer and reduced on the left. At sufﬁciently high frequency (where the skin depth is less than the thickness d of the layer), the current distribution High Frequency Effects in the Windings 165 will be zero on the left hand side of the layer and the current may be considered to ﬂow in a layer of thickness d; the overall resistance is increased because there is a smaller conduction area. It must be remembered that the net current ﬂowing is not changed from its DC value. In the next section, the proximity effect factor is derived from ﬁrst principles. The derivation is based on foil layers which extend the full height of the window. Reﬁnements for layers that consist of round conductors or foils that do not extend the full length of the window can be accommodated by the concept of ‘porosity’, in which an equivalent area of copper is spread along the length w. The general formula for the proximity effect factor can be derived by solving Maxwell’s equations for an annular cylindrical layer of thickness d and with ﬁeld intensities Hþ and HÀ on either side. 6.2.1 AC Resistance in a Cylindrical Conductor Maxwell’s equations (Equations 6.1 and 6.2) apply to the annular cylindrical conducting layer, shown in Figure 6.5, which carries a sinusoidal current if(t) ¼ Ifejvt. The conductivity of the conducting medium is s and the physical dimensions are shown in Figure 6.5. HÀ and Hþ are the magnetic ﬁelds parallel to the inside and outside surfaces of the cylinder, respectively. We shall see shortly that HÀ and Hþ are independent of z. Assuming cylindrical symmetry, the various components of the electric ﬁeld intensity E and the magnetic ﬁeld intensity H inside the cylinder, in cylindrical co-ordinates (r, f, z), satisfy the identities in Equations 6.9 and 6.10. The two equations (6.1 and 6.2) then reduce to: À @Hz @r ¼ sEf 1 r @ @r ðrEfÞ ¼ Àjvm0Hz ð6:23Þ ð6:24Þ Z ik iφ ir H- H+ w r ro Figure 6.5 Conducting cylinder. 166 Transformers and Inductors for Power Electronics C1 C2 H0 Primary Secondary mmf H1w H0w J 2J0 J0 0 -J0 -2J0 (a) (n-1)H0 w rni E(rni) E(rn0) rn0 nthlayer (b) Figure 6.6 Transformer cross-section with (a) associated mmf diagram and current density at high frequency, (b) generalized nth layer. Since H has only a z-component and E has only a f-component, we drop the subscripts without ambiguity. Furthermore, the electric and magnetic ﬁeld intensities are divergence-free, so it follows that E and H are functions of r only. Substituting the expression for E given by Equation 6.23 into Equation 6.24 then yields the ordinary differential equation: d2H d r2 þ 1 r dH dr À jvm0sH ¼ 0 ð6:25Þ This is another modiﬁed Bessel’s equation, in the same format as Equation 6.4, and the general solution is: HðrÞ ¼ AI0ðmrÞ þ BK0ðmrÞ ð6:26Þ A typical transformer cross-section is shown in Figure 6.6(a), with associated mmf diagram and current density distribution for a two-turn primary and a three-turn secondary winding. The physical dimensions of a generalized nth layer are shown in Figure 6.6(b) (the innermost layer is counted as layer 1). We assume that the magnetic material in the core is ideal (mr ! 1, s ! 0), so that the magnetic ﬁeld intensity goes to zero inside the core. We also assume that the dimension w is much greater than the radial dimensions, so that end effects are taken as negligible. Invoking Ampere’s law for the closed loops C1 and C2: H0 ¼ NI w ð6:27Þ where N is the number of turns in layer n, each carrying constant current I. n here refers to the layer number and should not be confused with the harmonic number n to be used later. High Frequency Effects in the Windings 167 Applying the inner and outer boundary conditions for layer n, that is: Hðrni Þ ¼ ðn À 1ÞH0 and Hðrno Þ ¼ nH0 to the general solution (Equation 6.26), we obtain the coefﬁcients: ð6:28Þ ð6:29Þ A ¼ ½ðnK0ðmrni Þ À ðn À 1ÞK0ðmrno ÞH0 I0ðmrno ÞK0ðmrni Þ À K0ðmrno ÞI0ðmrni Þ B ¼ ½ðÀnI0ðmrni Þ þ ðn À 1ÞI0ðmrno ÞH0 I0ðmrno ÞK0ðmrni Þ À K0ðmrno ÞI0ðmrni Þ ð6:30Þ ð6:31Þ The corresponding value of E(r) is found from Equation 6.23, that is: EðrÞ ¼ À 1 dHðrÞ s dr Using the modiﬁed Bessel function identity (Equation 6.14) and: ð6:32Þ d dr K0ðmrÞ ¼ Àm K 1 ðmrÞ the electric ﬁeld intensity is then given by: ð6:33Þ EðrÞ ¼ À m s ½AI1ðmrÞ À BK1ðmrÞ ð6:34Þ The Poynting vector E Â H represents the energy ﬂux density per unit area crossing the surface per unit time. In the cylindrical coordinate system in Figure 6.5, the power per unit area into the cylinder is given by E Â H on the inside surface and –E Â H on the outside surface. Since E and H are orthogonal, the magnitude of the Poynting vector is simply the product E(r)H(r), and its direction is radially outwards. The power per unit length (around the core) of the inside surface of layer n is: Pni ¼ Eðrni ÞHðrni Þw ¼ À m s ðn À 1ÞH0w½AI1ðmrni Þ À BK1ðmrni Þ ð6:35Þ A and B are given by Equations 6.30 and 6.31 respectively, H0 is given by Equation 6.27, so: Pni ¼ N2I2m swC fðn À 1Þ2 ½I 0 ðmrno ÞK 1 ðmrni Þ þ K 0 ðmrno ÞI 1 ðmrni Þ Ànðn À 1Þ½I0ðmrni ÞK1ðmrni Þ þ K0ðmrni ÞI1ðmrni Þg ð6:36Þ 168 Transformers and Inductors for Power Electronics where we deﬁne: C ¼ I0ðmrno ÞK0ðmrni Þ À K0ðmrno ÞI0ðmrni Þ ð6:37Þ In a similar fashion, we ﬁnd the power per unit length (around the core) of the outside surface of layer n is: Pno ¼ ÀEðrno ÞHðrno Þw ¼ m s nHow½AI1ðmrno Þ À BKðmrno Þ ¼ N2I2m swC fn2½I0ðmrni ÞK1ðmrno Þ þ K0ðmrni ÞI1ðmrno Þ Ànðn À 1Þ½I0ðmrno ÞK1ðmrno Þ þ K0ðmrno ÞI1ðmrno Þg ð6:38Þ The minus sign is required to ﬁnd the power into the outer surface. We now assume that mr >> 1 and use the leading terms in the asymptotic approxima- tions for the modiﬁed Bessel functions I0, I1, K0, K1 (noting for purposes of validity argðmrÞ ¼ p < p ): 42 I0ðmrÞ % I1ðmrÞ % pﬃeﬃﬃmﬃﬃﬃrﬃﬃﬃﬃﬃ 2pmr ; rﬃﬃﬃﬃﬃﬃﬃﬃ K0ðmrÞ % K1ðmrÞ % p eÀmr 2mr ð6:39Þ Substituting these into Equations 6.36 and 6.38 and rearranging, yields the total power dissi- pation for layer n as: Pni þ Pno ¼ N2I2m sw ð2n2 À 2n þ 1Þ cothðmd n Þ À n2 À n sinhðmd n Þ rﬃﬃﬃﬃﬃ rno rni þ rﬃﬃﬃﬃﬃ! rni rno ð6:40Þ where dn rno À rni is the thickness of layer n. This result was obtained from the Poynting vector for the complex ﬁeld intensities, so the real part represents the actual power dissipation. We now assume that each layer has constant thickness d, so that dn ¼ d (independent of n). Furthermore, we assume that d << rni . Then, using the Taylor expansion: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þe þ pﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2 þ e2 þ Oðe3Þ 1þe 4 ð6:41Þ it follows that if d < 10%, the error incurred by approximating the sum of the square roots rni in Equation 6.40 by 2 is in the order of 0.1%. The total power dissipation in layer n then becomes: <ðPni þ Pno Þ % 2.5. Terms up to the order of D3 in Equations 6.48 and 6.49 are sufﬁciently accurate for our purposes. Thus, Equation 6.47, with the aid of Equations 6.48 and 6.49, becomes: Rac ¼ 1 þ C D4 Rdc 3 ð6:54Þ where: C ¼ 5p2 À 1 15 ð6:55Þ We now have a simple and straightforward formula to calculate the AC resistance for sinusoidal excitation in a winding with p layers. 6.3 Proximity Effect Factor for an Arbitrary Waveform So far, we have the proximity effect factor for sinusoidal excitation. For power electronic applications, we need to extend the analysis to include higher frequencies as they appear in the Fourier series of a non-sinusoidal waveform. An arbitrary periodic current waveform may be represented by its Fourier series: X 1 iðtÞ ¼ Idc þ an cos nvt þ bn sin nvt: n¼1 ð6:56Þ The sine and cosine terms may be combined to give an alternative form X 1 iðtÞ ¼ Idc þ cn cosðnvt þ wnÞ n¼1 ð6:57Þ 172 Transformers and Inductors for Power Electronics where Idc is the DC value of i(t) and cn is the amplitude of the nth harmonic with corresponding phase wn. The rms value of the nth harmonic is In ¼ pcnﬃﬃ. 2 The total power loss due to all the harmonics is: X 1 P ¼ RdcI2dc þ Rdc kpn I 2 n n¼1 ð6:58Þ where kpn is the AC resistance factor at the nth harmonic frequency. The skin depth at the nth harmonic is: dn ¼ pﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pnf mrmos ¼ pdoﬃﬃ n ð6:59Þ and deﬁning Dn as: Dn ¼ d dn ¼ pﬃﬃ n d do ¼ pﬃﬃ nD ð6:60Þ kpn may be found from Equation 6.47. kpn ¼ pﬃnﬃD pﬃﬃ sinhð2pnﬃﬃ DÞ þ pﬃﬃ sinð2 pnﬃﬃDÞ þ 2ðp2 À 1Þ pﬃﬃ sinhðpﬃnﬃ DÞ À pﬃﬃ ! sinð pnﬃﬃDÞ : coshð2 nDÞ À cosð2 nDÞ 3 coshð nDÞ þ cosð nDÞ ¼ Dn sinhð2Dn coshð2Dn Þ Þ þ À sinð2DnÞ cosð2DnÞ þ 2ðp2 À 3 1Þ sinhðDnÞ coshðDnÞ À þ ! sinðDnÞ cosðDnÞ ð6:61Þ Reff is the AC resistance due to i(t) so that the total power loss is P ¼ ReffI2rms, Irms being the rms value of i(t). Thus, the ratio of effective AC resistance to DC resistance is: Reff Rdc ¼ I 2 dc þ P1 n¼1 I rms 2 kpn I 2 n : ð6:62Þ Rdc is the DC resistance of a foil of thickness d. Deﬁne Rd as the DC resistance of a foil of thickness d0, recalling that d0 is the skin depth at the fundamental frequency of the periodic waveform. Therefore: Rd ¼ d ¼ D Rdc d0 ð6:63Þ So: Reff ¼ Reff Rd ¼ Reff D Rdc Rd Rdc Rd ð6:64Þ High Frequency Effects in the Windings 173 and: Reff Reff Rd ¼ Rdc D ð6:65Þ Example 6.1 Calculate the ratio of effective resistance to DC resistance Reff/Rdc in a copper foil of thickness d carrying the pulsed current waveform shown in Figure 6.9. This waveform is often encountered in switch mode power supplies. The Fourier series of i(t) is: ! iðtÞ ¼ Idc þ ^I1 sin vt þ ^I3 sin 3vt þ . . . ¼ Io 2 þ 2Io p sin vt þ 1 sin 3vt þ . . . 3 The rms value of the current is Irms ¼ pI oﬃﬃ . 2 The proximity effect factor due to i(t) is found from Equation 6.62: X 1 I 2 dc þ kpn I 2 n Reff ¼ Rdc n¼1; odd I2rms ¼ Io!2 þ 2 pﬃﬃ !2 ! 2Io p 12 kp1 pI oﬃﬃ !2 þ 1 32 kp3 þ . . . 2 ¼ 1 þ 4 X 1 kpn 2 p2 n¼1; odd n2 kpn is the proximity effect factor due to the nth harmonic found from Equation 6.47. The ratio Reff/Rd is found from Equation 6.65: Reff ¼ 1 2 þ 4 p2 X 1 n¼1; odd kpn n2 ¼ 0:5 þ 4 X 1 kpn Rd D D p2 n¼1; odd n2D i I0 0 T/2 T 3T/2 2T 5T/2 t Figure 6.9 Pulsed current waveform. 174 Transformers and Inductors for Power Electronics 16 p = 10 14 12 p=9 10 Reff Rδ 8 6 4 2 0 p=8 p=7 Δopt 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Δ p=6 p=5 p=4 p=3 p=2 p=1 1 Figure 6.10 Plot of Reff/Rd versus D for various numbers of layers. Taking Equation 6.47 for kpn : Reff Rd ¼ 0:5 D þ 4 p2 XN 1 n¼1; odd n3 2 sinh cosh 2Dn 2Dn þ À sin 2Dn cos 2Dn þ 2ðp2 À 3 1Þ sinh cosh Dn Dn À þ ! sin Dn cos Dn The expression is plotted in Figure 6.10 for N ¼ 13 harmonics. The plots in Figure 6.10 may be interpreted for a number of situations. Consider a case where the frequency of the waveform is ﬁxed. Both Rd and d0 are ﬁxed and the plots may be interpreted as the effective resistance as a function of the layer thickness. Beginning with small values as the layer thickness increases, there is a reduction in resistance. However, as the thickness approaches values comparable to the skin depth at the fundamental frequency, proximity effects become dominant and the resistance increases accordingly. For values of D > 3, the current distribution at the centre approaches zero and the resistance levels off at a value based on the current being conﬁned to region at each side of the layer and with a thickness d0. In a practical design, values of D < 2 would normally be considered. The horizontal axis in Figure 6.10 is related to foil thickness. The plots show that, for any given number of layers, there is an optimum value of D, labelled Dopt, which minimizes the loss as a function of layer thickness. For 0 < D < Dopt, the DC resistance decreases as the thickness of the foil increases. However, for D > Dopt, the AC effects on resistance are greater than the mitigating effect of increased thickness on DC resistance. 6.3.1 The Optimum Thickness The plot in Figure 6.10, for the pulsed waveform in Figure 6.9, is redrawn in Figure 6.11 in 3-D and the locus of the minimum value of AC resistance is shown. High Frequency Effects in the Windings 175 p = 10 Minima p=9 20 18 16 14 Reff 12 Rδ 10 8 6 4 2 0 0.5 Δ 1 p=8 p=7 p=6 p=5 p=4 10 p=3 8 6 2 4p 1.5 0 Figure 6.11 Plot of Reff/Rd versus D for various numbers of layers. The plot in Figure 6.9 is for the pulsed waveform in Figure 6.8. This type of plot can be generated for any waveform, once its Fourier coefﬁcients are known, by following the procedure in Example 6.1. The Fourier series for various waveforms encountered in power electronics are given in Table 6.1. For each value of p in Figure 6.11, there is an optimum value of D where the AC resistance of the winding is minimum. These optimum points lie on the line marked ‘minima’ in Figure 6.11, and the corresponding value of the optimum layer thickness is: dopt ¼ Doptd0 ð6:66Þ Recognizing that we would have to construct a plot similar to Figure 6.11 for each waveform, a more direct approach is desirable. The approximation given by Equation 6.54 determines the AC resistance factor at the nth harmonic frequency and may be stated as: kpn ¼ 1 þ C n2D4: 3 on the basis that the skin depth at the nth harmonic is dn ¼ pd0ﬃﬃ . n Substituting Equation 6.67 into Equation 6.62 yields: Reff Rdc ¼ I 2 dc þ P1 n¼1 I 2 n þ C D4 X1 3 n¼1 I 2 rms n2I2n ð6:67Þ ð6:68Þ 178 Transformers and Inductors for Power Electronics The rms value of the current in terms of its harmonics is: X 1 I 2 rms ¼ I2dc þ I2n n¼1 The derivative of i(t) in Equation 6.57 is: di dt ¼ Àv X 1 n¼1 ncn sinðnvt þ wnÞ and the rms value of the derivative of the current is: I 0r2ms ¼ v2 X 1 n2c2n n¼1 2 ¼ v2 X 1 n¼1 n2 I 2 n ð6:69Þ ð6:70Þ ð6:71Þ which, upon substitution into Equation 6.68 using Equation 6.69, yields: Reff ¼ 1 þ C D4 I0rms !2 Rdc 3 vIrms ð6:72Þ This is a straightforward expression for the effective resistance of a winding with an arbitrary current waveform, and it may be evaluated without knowledge of the Fourier coefﬁcients of the waveform. Taking Equation 6.72 with Equation 6.64: Reff Rd ¼ 1 D þ C 3 D3 I 0 rms !2 vIrms ð6:73Þ Setting the derivative to zero yields the optimum value of D: d Reff dD Rd ¼ À 1 D2 þ CD2 I0rms !2 ¼ 0 vIrms ð6:74Þ The optimum value of D is then: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Dopt ¼ p1ﬃﬃﬃﬃ 4C vIrms I 0rms ð6:75Þ Substituting this result back into Equation 6.72 produces a very simple expression for the optimum value of the effective AC resistance with an arbitrary periodic current waveform: Reff ¼4 Rdc opt 3 ð6:76Þ High Frequency Effects in the Windings 179 i 0 T/2 T t tr i’ 2I0 tr 0 t 2I0 tr Figure 6.12 Pulsed current waveform and its derivative. Snelling [4] has already established this result for sinusoidal excitation. The corresponding value for wire conductors with sinusoidal excitation is 3/2 [4]. We may also write Equation 6.72 in terms of Dopt: Reff Rdc ¼ 1 þ 1 D 4 3 Dopt ð6:77Þ We now have a set of simple formulae with which to ﬁnd the optimum value of the foil or layer thickness of a winding and its effective AC resistance, where these formulae are based on the rms value of the current waveform and the rms value of its derivative. The formulae have been applied to each of the waveforms in Table 6.1 and the results are presented therein. Example 6.2 Calculate the ratio of effective resistance to DC resistance Reff/Rdc in a copper foil of thickness d carrying the pulsed current waveform shown in Figure 6.12. This waveform is a more realistic version of the waveform in Example 6.1, with a rise and fall time along with a slower characteristic at the start and end of the rising and falling fronts. The derivative is also shown. This waveform is typical of the input in a forward converter. Find the optimum value of D for six layers and tr/T ¼ 4%. This waveform has a Fourier series: iðtÞ ¼ 1 Io 2 À tr T þ X 1 n¼1 4Io n3 p3 tr 2 h 1 À cosnptrisinnp1 T 2 À tr cos n vt T À p 2 T The rms value of i(t) is: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Irms ¼ Io 0:5 À 37tr 30T 180 Transformers and Inductors for Power Electronics 16 14 12 Reff 10 Rδ 8 6 p=6 4 2 Δopt 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Δ Figure 6.13 Optimum value of foil thickness. Following the procedure outlined in Example 6.1, Reff/Rdc is plotted in Figure 6.13 for p ¼ 6 and tr/T ¼ 4%. The optimum value of D is 0.418. The MATLAB program for this example is listed at the end of the chapter. Example 6.3 Repeat Example 6.2 using the approximation in Equation 6.75 to ﬁnd the optimum value of D. The rms value of i(t) is: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Irms ¼ Io 0:5 À 37tr 30T ¼ Io 0:5 À 37 30 0:04 ¼ 0:6713Io The rms value of the derivative of i(t) is: I 0 rms v ¼ Io p rﬃﬃﬃﬃﬃﬃ 2T 3tr ¼ Io p sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ð3Þð0:04Þ ¼ 1:2995Io The optimum value of D is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Dopt ¼ p1ﬃﬃﬃﬃ 4C vIrms I0rms ¼ sﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 ð5Þð6Þ2 À 1 0:6713 1:2995 ¼ 0:387 15 The value obtained by Fourier analysis is 0.418, and this represents an error of 7.4%. However, the Fourier series of the waveform in Figure 6.12 is not readily available. If waveform 5 in Table 6.1 were High Frequency Effects in the Windings 181 used to approximate it, the optimum value of D would be: Dopt ¼ v u u u t4 ﬃﬃﬃDﬃﬃﬃﬃﬃÀﬃﬃﬃﬃ43ﬃﬃﬃtTﬃrﬃﬃ!ﬃﬃ2ﬃﬃﬃpﬃﬃﬃ2ﬃﬃtTﬃﬃrﬃ¼ C v u u u u u t4 ﬃﬃﬃ0ﬃﬃ:ﬃﬃ5ﬃﬃﬃÀﬃﬃﬃﬃﬃðﬃﬃð4ﬃﬃ5ÞﬃﬃÞðﬃﬃð03ﬃﬃ6:ﬃﬃ0Þﬃﬃ24ﬃﬃÞﬃÀﬃ!ﬃﬃﬃ2ﬃ1ﬃpﬃﬃﬃ2ﬃﬃðﬃﬃ0ﬃﬃ:ﬃﬃ0ﬃﬃ4ﬃﬃÞﬃﬃ ¼ 0:414 15 On the other hand, the optimum value of D obtained by Fourier analysis of waveform 5 is 0.448, which would represent an error of 7.2% when compared to the Fourier analysis of the waveform in Figure 6.12. Waveforms with known Fourier series are often approximations to the actual waveform and can give rise to errors which are of the same order as the approximation, but the approximation is simpler to evaluate. Example 6.4 Push-pull Converter Calculate the optimum thickness for the winding in the push-pull converter in Example 5.3. In Example 5.3, there are six layers, p ¼ 6, the duty cycle is D ¼ 0.67, the rise time is tr ¼ 2.5% and the frequency is f ¼ 50 kHz. The closest waveform to the input voltage shown in Figure 6.12 is number 6 in Table 6.1, and the optimum layer thickness is: Dopt ¼ v u u u t4 ﬃﬃﬃDﬃﬃﬃﬃﬃÀﬃﬃﬃﬃ83ﬃﬃﬃtTﬃrﬃﬃ!ﬃﬃpﬃﬃﬃ2ﬃﬃﬃtTﬃﬃrﬃ¼ ð5p2 À 1Þ15 v u u u tﬃﬃﬃ0ﬃﬃ:ﬃﬃ6ﬃﬃ7ﬃﬃﬃﬃÀﬃﬃﬃﬃðﬃﬃ8ﬃﬃﬃÞﬃðﬃﬃ0ﬃﬃ3ﬃ:ﬃ0ﬃﬃ2ﬃﬃﬃ5ﬃﬃÞﬃﬃ!ﬃﬃﬃpﬃﬃ2ﬃﬃðﬃﬃ0ﬃﬃﬃ:ﬃ0ﬃﬃ2ﬃﬃﬃ5ﬃﬃÞﬃﬃﬃ ½ð5Þð6Þ2 À 1=15 ¼ 0:3342 For copper at 50 kHz, the skin depth is: d0 ¼ p66ﬃﬃ f ¼ qﬃﬃﬃﬃﬃﬃ6ﬃﬃﬃ6ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð50 Â 103Þ ¼ 0:295 mm The optimum foil thickness is equal to: dopt ¼ Doptd0 ¼ ð0:3342Þð0:295Þ ¼ 0:1 mm and the AC resistance at the optimum layer thickness is: Reff ¼ 4 3 Rdc The DC resistance of the primary and secondary windings made of 0.1 mm Â 30 mm foil is 3.29 mV, and therefore the AC effective resistance is 4.40 mV. A 2 mm diameter bare copper wire has the same copper area as a in Figure 6.14. The skin effect factor of the round conductor, as given by Equation 6.17, is: ks ¼ 0:25 þ ð0:5Þ ro d0 ¼ 0:25 þ ð0:5Þ 1:0 0:295 ¼ 1:945 The AC resistance of this round conductor due to skin effect is now 3.3 Â 1.945 ¼ 6.42 mV. 182 Transformers and Inductors for Power Electronics 0.1 mm × 30 mm 2 mm Diameter Figure 6.14 Round versus foil conductor. This is for the fundamental frequency only. Adding the loss for the higher frequencies would increase this value further, compared with the proximity effect factor of 4/3 ¼ 1.33. Clearly, in this case, the choice of a foil conductor is vastly superior. 6.4 Reducing Proximity Effects by Interleaving the Windings Consider a transformer with a turns ratio of 2:3, as shown in Figure 6.15. This is the same transformer that was discussed in Section 6.2. The mmf distribution given by Equations 6.19 and 6.21 is also shown. The current density at the surface of the layers in Figure 6.15 is related to the electric ﬁeld intensity E (J ¼ sE) which, in turn, is related to the magnitude of the magnetic ﬁeld intensity H. The current density in the windings before interleaving is illustrated in Figure 6.16. The loss is proportional to J2 and, taking the sum of the squares of the current density in each layer as an indication of the total loss, the loss is 14.4, taking 1 as the base unit H0 H1 w mmf H1w H0w Primary Secondary b Figure 6.15 Transformer windings before interleaving. High Frequency Effects in the Windings 183 mmf H1w H0w J 2J0 J0 0 -J0 -2J0 2 2 ͚ J2 = 2 12 + 22 + 4 + 2 33 = 14.4 Figure 6.16 Current density distribution before interleaving. corresponding to J0. It makes sense, therefore, that the surface value of H should be kept to a minimum. The current density distribution determines the shape of the mmf distribution, and this is shown. The dotted line represents the low frequency distribution. We will return to this point later. The proximity effect loss may be reduced by interleaving the windings, alternating the primary and secondary layers. This is shown in Figure 6.17 with the current density shown in Figure 6.18. Adding up the contributions of J2 in this case yields a total of 4.4, showing an indicative reduction by a factor of more than 3 in the proximity effect loss. Interleaving also reduces the leakage inductance and interwinding capacitance. H0 w Primary Secondary Primary mmf b H0w -H0w Figure 6.17 Transformer windings after interleaving. 184 Transformers and Inductors for Power Electronics mmf H0w 0 -H0w J J0 0 -J0 2 2 ͚ J 2 = 2 12 + 1 + 1 + 12 = 4.4 33 Figure 6.18 Current density distribution after interleaving. 6.5 Leakage Inductance in Transformer Windings Consider the windings shown in Figure 6.4, the magnetic ﬁeld is directed along the plane of the layers. The energy per unit volume associated with the H ﬁeld in Figure 6.4 is given by Equation 2.31. dWm ¼ 1 2 m0H2 ð6:78Þ The energy associated with the winding in Figure 6.4 is: ð Wm ¼ 1 2 m0 H2dV volume ð6:79Þ For Np turns carrying current Ip in the primary of the transformer, the maximum value of H would be NpIp. w In order to evaluate the integral in Equation 6.79, we assume a triangular shape for the H ﬁeld distribution. Performing the integration in Equation 6.79 yields: Wm ¼ 1 2 m0N2pMLTb 3w I2p ð6:80Þ where b is the difference between the inside and outside radii of the windings, MLT is the mean length of a turn and b Â MLT is the volume of the winding. All the ﬂux in the window of the transformer in Figure 6.4 is leakage ﬂux, so that: Ll1 þ Ll2 ¼ m0N2pMLTb 3w ð6:81Þ High Frequency Effects in the Windings 185 ϕ17.70 ϕ 5.56 ϕ 42.40 29.60 17.6 ϕ 35.60 20.40 Primary Secondary Figure 6.19 Pot core, all dimensions in mm. Ll1 and Ll2 are the components of the leakage ﬂux associated with the primary and secondary windings, as described in Chapter 4. Evidently, the total leakage is directly related to the total volume occupied by the windings, and the greater spread of windings along a core (increased w) reduces leakage effects. Equation 6.81 shows that the leakage inductance may be reduced by using fewer turns; the most direct approach is to interleave the windings. The ratio b/w suggests that the winding should be placed in a long, narrow window. This is the ideal case and it takes no account of fringing effects or magnetic energy stored in the windings themselves; however, it does give a reasonable estimate of the leakage inductance in a typical transformer. A more accurate estimate may be obtained by ﬁnite element analysis. Example 6.5 Calculate the leakage inductance for the windings in the pot core shown in Figure 6.19 with the dimensions given in Table 6.2. The MLT is based on the average radius of the coils: MLT ¼ 2 Â p Â ð9:86 þ 13:04Þ=2 ¼ 71:94 mm The breadth of the winding is b ¼ 13.04 À 9.86 ¼ 3.18 mm Table 6.2 Coil dimensions Coil 1: Coil 2: Number of turns Inside radius Outside radius Width Number of turns Inside radius Outside radius Width 20 9.86 mm 11.51 mm 17.6 mm 20 11.39 mm 13.04 mm 17.6 mm 186 Transformers and Inductors for Power Electronics The leakage inductance is: Ll1 þ Ll2 ¼ m0N 2 p MLT b 3w ¼ ð4 Â 10À7Þð20Þ2ð71:94 Â ð3Þð17:6 Â 10À3Þð3:18 10À3Þ Â 10À3Þ Â 106 ¼ 2:2 mH The result in Equation 6.81 was based on the low frequency distribution of the H ﬁeld in Figure 6.4 and that, in turn, was based on the low frequency current distribution, i.e. constant current density across each layer. However, at high frequency, the proximity effect redistributes the current, as illustrated in Figure 6.15. Performing the integration in Equation 6.79 on the high-frequency distribution on Figure 6.15 results in lower energy and, consequently, lower leakage inductance. Remember, the real part of the expression in Equation 6.47 represents the loss as predicted by Dowell’s formula. The imaginary part of the expression represents the leakage reactance, and this can be easily deduced: Ll ac Ll ¼ 3 2p2D2 Im Dð1 þ jÞ coth ðDð1 þ jÞ þ 2ðp2 À 1Þ tanh D ð1 þ ! jÞ 3 2 ¼ 3 2p2D sinh cosh 2D 2D À À sin 2D cos 2D þ 2ðp2 À 3 1Þ sinh cosh D D þ þ sin D ! cos D ð6:82Þ This is plotted in Figure 6.20. There is a fall off in leakage inductance for D > 1. It is common practice to depend on leakage ﬂux to provide the resonant inductor in reso- nant power supplies. The dramatic fall-off when the skin depth is comparable to the layer thickness must be considered in the design. 1 0.9 0.8 0.7 Llac 0.6 Ll 0.5 0.4 0.3 0.2 0.1 0 0.1 1 10 100 Δ Figure 6.20 Leakage inductance at high frequency. High Frequency Effects in the Windings 187 6.6 Problems 6.1 Derive the formula for the optimum value of D for waveform 2 in Table 6.1. 6.2 Verify that the Fourier series for the waveform 3 in Table 6.1 is correct. 6.3 Verify that the Fourier series for the waveform 4 in Table 6.1 is correct. 6.4 Compare the results for the optimum value of D for waveform 5 in Table 6.1 obtained by Fourier analysis, with the result given by Dopt for p ¼ 6. D ¼ 0.5 and tr/T ¼ 0.4%. 6.5 Derive the formula for the optimum value of D for waveform 6 in Table 6.1. 6.6 Compare the results for the optimum value of D for waveform 7 in Table 6.1 obtained by Fourier analysis with the result given by Dopt for p ¼ 6. D ¼ 0.4 and tr/T ¼ 0.4%. 6.7 Write a MATLAB program to generate the 3-D plot of Figure 6.11 for waveform 8 in Table 6.1. 6.8 Derive Equation 6.82 from Dowell’s formula. 6.9 Calculate the leakage inductance in the push-pull transformer in Example 5.3. MATLAB Program for Figure 6.2 % Figure 6.2 Current distribution in a circular conductor for f = [1e3,10e3,100e3,1000e3]; %Hz r0 = 1.25e-3; %m mu0 = 4*pi*10^-7; sigma = 1/(1.72*10^-8); delta0=1/sqrt(pi*f*mu0*sigma); %skin depth w = 2*pi*f; m = sqrt(1j*w*mu0*sigma); % besseli (Modiﬁed Bessel function of ﬁrst kind) r = [-1.25e-3:0.001e-3:1.25e-3]; I1 = besseli(0,m*r); I0 = besseli(0,m*r0); I = abs(I1/I0); % plot plot(r,I,’k’,’LineWidth’,2) title(’High Frequency Effects in the Windings’) xlabel(’Wire radius’) ylabel(’|J(r)/J(r0)|’) axis([-1.25e-3 1.25e-3 0 1]) grid off hold on end 188 Transformers and Inductors for Power Electronics MATLAB Program for Figure 6.3 % Figure 6.3 Rac/Rdc due to skin effect n = 0; for delta = [0:0.01:10]; mr0 =(1+1j)*delta; I0 = besseli(0,mr0); I1 = besseli(1,mr0); R = real((mr0.*(I0/(2*I1)))); n = n+1; V(n)= R; D(n)= delta; end plot(D,V,’k’,’LineWidth’,2) title(’Rac/Rdc due to skin effect’) xlabel(’Ratio of Radius to Depht of Penetration’) ylabel(’Rac/Rdc’) axis([0 10 0 6]) grid off MATLAB Program for Figure 6.7 % Figure 6.7 Proximity effect factor for sinusoidal excitation for p = [1:10] n = 0; for delta = [0.1:0.01:10]; R1 = real(delta*(1+1j)*(coth(delta*(1+1j))+((2*(p^2-1))/3) *tanh((delta/2)*(1+1j)))); R2 = delta*[(sinh(2*delta)+sin(2*delta))/(cosh(2*delta)-cos (2*delta))+((2*(p^2-1))/3)*((sinh(delta)-sin(delta))/(cosh(delta)+ cos(delta)))]; n = n+1; V(n) = R2; D(n)= delta; end loglog(D,V,’k’,’LineWidth’,2) High Frequency Effects in the Windings 189 title(’Figure 6.7 Proximity effect factor for sinusoidal excitation’) xlabel(’Delta’) ylabel(’kp’) axis([0.1 10 1 1000]) grid off hold on end MATLAB Program for Figure 6.10 % Figure 6.10 Plot of Reff/Rdelta versus Delta for various numbers of % layers for p = [1:10] u=0; for delta = [0:0.001:1]; sum = 0; y = 0; for n = [1,3,5,7,9,11,13] y= (1/n^(3/2))*((sinh(2*sqrt(n)*delta)+sin(2*sqrt(n)*delta))/(cosh (2*sqrt(n)*delta)-cos(2*sqrt(n)*delta))+((2*(p^2-1))/3)*((sinh (sqrt(n)*delta)-sin(sqrt(n)*delta))/(cosh(sqrt(n)*delta)+cos(sqrt (n)*delta)))); sum = sum+y; end R = (0.5/delta)+(4/pi^2)*sum; u = u+1; V(u) = R; D(u)= delta; end plot(D,V,’k’,’LineWidth’,1) title(’Figure 6.10 Plot of Reff/Rdelta versus Delta for various numbers of layers’) xlabel(’Delta’) ylabel(’kr’) axis([0 1 0 16]) grid off hold on end 190 Transformers and Inductors for Power Electronics MATLAB Program for Figure 6.11 % Figure 6.11 Plot of Reff/Rdelta versus Delta for various numbers of % layers for p = [1:10] u=1; for delta = [0.01:0.04:1.5]; sum = 0; y = 0; for n = [1,3,5,7,9,11,13] y= (1/n^(3/2))*((sinh(2*sqrt(n)*delta)+sin(2*sqrt(n)*delta))/(cosh (2*sqrt(n)*delta)-cos(2*sqrt(n)*delta))+((2*(p^2-1))/3)*((sinh (sqrt(n)*delta)-sin(sqrt(n)*delta))/(cosh(sqrt(n)*delta)+cos(sqrt (n)*delta)))); sum = sum+y; end R = (0.5/delta)+(4/pi^2)*sum; V(p,u) = R; D(u)= delta; u=u+1; end end mesh(D,1:10,V) title(’Figure 6.11 Plot of Reff/Rdelta versus Delta for various numbers of layers’) xlabel(’D’) ylabel(’p’) zlabel(’V’) axis([0 1.5 1 10 0 20]) grid off hold on for p = [0.1:0.1:10] u = 1; for delta = [0.01:0.01:1.5]; sum = 0; y = 0; High Frequency Effects in the Windings 191 for n = [1,3,5,7,9,11,13] y= (1/n^(3/2))*((sinh(2*sqrt(n)*delta)+sin(2*sqrt(n)*delta))/(cosh (2*sqrt(n)*delta)-cos(2*sqrt(n)*delta))+((2*(p^2-1))/3)*((sinh (sqrt(n)*delta)-sin(sqrt(n)*delta))/(cosh(sqrt(n)*delta)+cos(sqrt (n)*delta)))); sum = sum+y; end R = (0.5/delta)+(4/pi^2)*sum; V(round(p*10),u) = R; D(u)= delta; u=u+1; end end for p=0.1:0.1:10 [krmin,delopt] = min(V(round(p*10),:)); A(round(p*10)) = delopt/100; B(round(p*10)) = p; C(round(p*10)) = krmin+0.1; end plot3(A,B,C,’k’,’LineWidth’,2) axis([0 1.5 0 10 0 20]) hold on MATLAB Program for Figure 6.13 % Figure 6.13 Optimum value of foil thickness p = 6; a = 0.04; % a = tr/T I0 = 1; u = 0; for delta = [0:0.001:1]; sum = 0; y = 0; Irms = I0*sqrt(0.5-37*a/30); Idc = I0*(0.5-a); for n = [1:19] deltan = sqrt(n)*delta; In = ((2*sqrt(2)*I0)/(n^3*pi^3*a^2))*[1-cos(n*pi*a)]*sin (n*pi* (0.5-a)); 192 Transformers and Inductors for Power Electronics Kpn = deltan*((sinh(2*deltan)+sin(2*deltan))/... (cosh(2*deltan)-cos(2*deltan))+... ((2*(p^2-1))/3)*((sinh(deltan)-sin(deltan))/... (cosh(deltan)+cos(deltan)))); y = Kpn*In^2; sum = sum+y; end R = ((Idc^2+sum)/Irms^2)/delta; R = R/1; u = u+1; V(u) = R; D(u)= delta; end plot(D,V,’k’,’LineWidth’,2) title(’Figure 6.13 Optimum value of foil thickness’) xlabel(’Delta’) ylabel(’kr’) axis([0 1 0 16]) grid off hold on MATLAB Program for Figure 6.20 % Figure 6.20 Leakage inductance at high frequency p = 6; n=0; for delta = [0:0.1:100]; R1 = (3/(2*p^2*delta^2))*imag(delta*(1+1j)*(coth(delta*(1+1j))+ ((2*(p^2-1))/3)*tanh((delta/2)*(1+1j)))); R2 = 1-(1/30)*((p^2-1)/p^2)-((2^8)/factorial(7))*delta^4; n = n+1; V(n) = R1; W(n) = R2; D(n)= delta; end semilogx(D,V,’b’,’LineWidth’,2) title(’Leakage inductance’) High Frequency Effects in the Windings 193 xlabel(’Delta’) ylabel(’FL’) axis([0 100 0 1]) grid off hold on References 1. Dowell, P.L. (1966) Effects of eddy currents in transformer windings. Proceedings of the Institution of Electrical Engineers 113 (8), 1387–1394. 2. Hurley, W.G., Gath, E., and Breslin, J.G. (2000) Optimizing the AC resistance of multilayer transformer windings with arbitrary current waveforms. IEEE Transactions on Power Electronics 15 (2), 369–376. 3. Ferreira, J.A. (1994) Improved analytical modeling of conductive losses in magnetic components. IEEE Transactions on Power Electronics 9 (1), 127–131. 4. Snelling, E.C. (1988) Soft Ferrites: Properties and Applications, 2nd edn, Butterworths, London. Further Reading 1. Bartoli, M., Reatti, A., and Kazimierczuk, M.K. (1994) High-frequency models of ferrite core inductors. Proceedings of the IEEE Industrial Electronics, Control and Instrumentation, IECON, pp. 1670–1675. 2. Bartoli, M., Reatti, A., and Kazimierczuk, M.K. (1994) Modelling iron-powder inductors at high frequencies. Proceedings of the IEEE Industry Applications Conference, IAS, pp. 1225–1232. 3. Bennett, E. and Larson, S.C. (1940) Effective resistance to alternating currents of multilayer windings. Transactions of the American Institute of Electrical Engineers 59 (12), 1010–1017. 4. Carsten, B. (1986) High frequency conductor losses in switchmode magnetics. Proceedings of the High Frequency Power Converter Conference, pp. 155–176. 5. Cheng, K.W.E. and Evans, P.D. (1994) Calculation of winding losses in high-frequency toroidal inductors using single strand conductors. IEE Proceedings-Electric Power Applications B 141 (2), 52–62. 6. Cheng, K.W.E. and Evans, P.D. (1995) Calculation of winding losses in high frequency toroidal inductors using multistrand conductors. IEE Proceedings-Electric Power Applications B 142 (5), 313–322. 7. Dale, M.E. and Sullivan, C.R. (2006) Comparison of single-layer and multi-layer windings with physical constraints or strong harmonics. Proceedings of the IEEE International Symposium on Industrial Electronics, pp. 1467–1473. 8. Dauhajre, A. and Middlebrook, R.D. (1986) Modelling and estimation of leakage phenomenon in magnetic circuits. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 213– 216. 9. Erickson, R.W. (2001) Fundamentals of Power Electronics, 2nd edn, Springer, Norwell, MA. 10. Evans, P.D. and Chew, W.M. (1991) Reduction of proximity losses in coupled inductors. IEE Proceedings- Electric Power Applications B 138 (2), 51–58. 11. Ferreira, J.A. (2010) Electromagnetic Modelling of Power Electronic Converters (Power Electronics and Power Systems), 1st edn, Springer, Norwell, MA. 12. Hanselman, D.C. and Peake, W.H. (1995) Eddy-current effects in slot-bound conductors. IEE Proceedings- Electric Power Applications B 142 (2), 131–136. 13. Hui, S.Y.R., Zhu, J.G., and Ramsden, V.S. (1996) A generalized dynamic circuit model of magnetic cores for low- and high-frequency applications. II. Circuit model formulation and implementation. IEEE Transactions on Power Electronics 11 (2), 251–259. 14. Hurley, W.G. and Wilcox, D.J. (1994) Calculation of leakage inductance in transformer windings. IEEE Transactions on Power Electronics 9 (1), 121–126. 15. Hurley, W.G., Wilcox, D.J., and McNamara, P.S. (1991) Calculation of short circuit impedance and leakage impedance in transformer windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 651–658. 194 Transformers and Inductors for Power Electronics 16. Hurley, W.G., Wolﬂe, W.H., and Breslin, J.G. (1998) Optimized transformer design: inclusive of high-frequency effects. IEEE Transactions on Power Electronics 13 (4), 651–659. 17. Jiankun, H. and Sullivan, C.R. (2001) AC resistance of planar power inductors and the quasidistributed gap technique. IEEE Transactions on Power Electronics 16 (4), 558–567. 18. Judd, F. and Kressler, D. (1977) Design optimization of small low-frequency power transformers. IEEE Transactions on Magnetics 13 (4), 1058–1069. 19. Kassakian, J.G. and Schlecht, M.F. (1988) High-frequency high-density converters for distributed power supply systems. Proceedings of the IEEE 76 (4), 362–376. 20. Kazimierczuk, M.K. (2009) High-Frequency Magnetic Components, John Wiley & Sons, Chichester. 21. McAdams, W.H. (1954) Heat Transmission, 3rd edn, McGraw-Hill, New York. 22. McLyman, C.W.T. (2004) Transformer and Inductor Design Handbook, 3rd edn, Marcel Dekker Inc., New York. 23. Muldoon, W.J. (1978) Analytical design optimization of electronic power transformers. Proceedings of Power Electronics Specialists Conference, PESC, pp. 216–225. 24. Pentz, D.C. and Hofsajer, I.W. (2008) Improved AC-resistance of multiple foil windings by varying foil thickness of successive layers. The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 27 (1), 181–195. 25. Perry, M.P. (1979) Multiple layer series connected winding design for minimum losses. IEEE Transactions on Power Apparatus and Systems PAS-98 (1), 116–123. 26. Petkov, R. (1996) Optimum design of a high-power, high-frequency transformer. IEEE Transactions on Power Electronics 11 (1), 33–42. 27. Ramo, S., Whinnery, J.R., and Van Duzer, T. (1984) Fields and Waves in Communication Electronics, 2nd edn, John Wiley & Sons, New York. 28. Sagneri, A.D., Anderson, D.I., and Perreault, D.J. (2010) Transformer synthesis for VHF converters. Proceedings of the International Power Electronics Conference, IPEC, pp. 2347–2353. 29. Schaef, C. and Sullivan, C.R. (2012) Inductor design for low loss with complex waveforms. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 1010–1016. 30. Sullivan, C.R. (1999) Optimal choice for number of strands in a litz-wire transformer winding. IEEE Transactions on Power Electronics 14 (2), 283–291. 31. Sullivan, C.R. (2001) Cost-constrained selection of strand diameter and number in a litz-wire transformer winding. IEEE Transactions on Power Electronics 16 (2), 281–288. 32. Sullivan, C.R. (2007) Aluminum windings and other strategies for high-frequency magnetics design in an era of high copper and energy costs. Proceedings of the IEEE Applied Power Electronics Conference, APEC, pp. 78–84. 33. Sullivan, C.R., McCurdy, J.D., and Jensen, R.A. (2001) Analysis of minimum cost in shape-optimized Litz-wire inductor windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 1473–1478. 34. Sullivan, C.R. and Sanders, S.R. (1996) Design of microfabricated transformers and inductors for high-frequency power conversion. IEEE Transactions on Power Electronics 11 (2), 228–238. 35. Urling, A.M., Niemela, V.A., Skutt, G.R., and Wilson, T.G. (1989) Characterizing high-frequency effects in transformer windings-a guide to several signiﬁcant articles. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 373–385. 36. Van den Bossche, A. (2005) Inductors and Transformers for Power Electronics, 1st edn, CRC Press, New York. 37. Vandelac, J.P. and Ziogas, P.D. (1988) A novel approach for minimizing high-frequency transformer copper losses. IEEE Transactions on Power Electronics 3 (3), 266–277. 38. Venkatraman, P.S. (1984) Winding eddy current losses in switch mode power transformers due to rectangular wave currents. Proceedings of the 11th National Solid-State Power Conversion Conference, Powercon 11, pp. A1.1–A1.11. 39. Williams, R., Grant, D.A., and Gowar, J. (1993) Multielement transformers for switched-mode power supplies: toroidal designs. IEE Proceedings-Electric Power Applications B 140 (2), 152–160. 40. Xi, N. and Sullivan, C.R. (2004) Simpliﬁed high-accuracy calculation of eddy-current loss in round-wire windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 873–879. 41. Xi, N. and Sullivan, C.R. (2009) An equivalent complex permeability model for litz-wire windings. IEEE Transactions on Industry Applications 45 (2), 854–860. High Frequency Effects in the Windings 195 42. Xu, T. and Sullivan, C.R. (2004) Optimization of stranded-wire windings and comparison with litz wire on the basis of cost and loss. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 854–860. 43. Zhu, J.G., Hui, S.Y.R., and Ramsden, V.S. (1996) A generalized dynamic circuit model of magnetic cores for low- and high-frequency applications. I. Theoretical calculation of the equivalent core loss resistance. IEEE Transactions on Power Electronics 11 (2), 246–250. 44. Zimmanck, D.R. and Sullivan, C.R. (2010) Efﬁcient calculation of winding-loss resistance matrices for magnetic components. Proceedings of the IEEE Workshop on Control and Modeling for Power Electronics, COMPEL, pp. 1–5. 45. Ziwei, O., Thomsen, O.C., and Andersen, M. (2009) The analysis and comparison of leakage inductance in different winding arrangements for planar transformer. Proceedings of the IEEE Power Electronics and Drive Systems, PEDS, pp. 1143–1148. 7 High Frequency Effects in the Core1 We saw in Chapter 6 that skin and proximity effects have a major inﬂuence on the operation of a transformer or inductor at high frequencies due to the circulation of eddy currents, the main effect being to increase the winding loss. Eddy currents can also ﬂow in the core material, since it has ﬁnite resistivity. In the past, laminations were used at power frequencies to overcome the eddy current effects of the low resistivity of silicon steel. In power electronics, the operating frequencies are constantly being increased, with new materials; the high but ﬁnite resistivity of materials such as ferrites means that upper limits on operating frequency apply. In this chapter, analysis is provided to establish the frequency limitations due to eddy currents in the core material. 7.1 Eddy Current Loss in Toroidal Cores Eddy current loss arises in the same way that skin effect loss arises in conductors, namely due to alternating ﬂux in a conducting medium. Consider the toroidal core in Figure 7.1. When an alternating current ﬂows in the coil, eddy currents ﬂow as shown in accordance with the principle of Lenz’s law. The ﬂux level is reduced at the centre of the core and bunched to the surface. This has two effects: the net ﬂux is reduced, with a consequential reduction in inductance, and eddy current loss appears. Both of these effects can be represented as impedance in the coil, with the real part representing eddy current loss and the imaginary part representing the inductance. These quantities are clearly frequencydependent. Consider a coil fully wound on the core of Figure 7.1. For the purposes of calculating self inductance and eddy current loss, it is reasonable to neglect leakage effects. The winding 1 Part of this chapter is reproduced with permission from [1] Hurley, W.G. Wilcox, D.J. and McNamara, P.S. (1991) Calculation of short circuit impedance and leakage impedance in transformer windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 651–658. Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 198 Transformers and Inductors for Power Electronics ac flux Eddy current Coil +_ Magnetic material with ferrite conductivity σ and relative permeability μr Figure 7.1 Eddy current loss in a toroidal core. may be replaced by a current sheet, as shown in Figure 7.2, with a surface current density Kf A/m. In the absence of leakage, the magnetic ﬁeld intensity H is z directed only and the electric ﬁeld E has a w component only. Taking the general form of Maxwell’s equations in a linear homogeneous isotropic medium (Equation 6.3): r Â H ¼ sE ¼ J so that: and using Equation 6.2: À @Hz @r ¼ sEf r Â E ¼ Àm0 @H @t ð7:1Þ Kf r b z 1 unit Figure 7.2 Current sheet. High Frequency Effects in the Core 199 yielding: 1 r @ðrEfÞ @r ¼ jvmrm0Hz Eliminating H gives an expression for E: ð7:2Þ d2E d r2 þ 1 r dE dr À E r2 þ jvmrm0sE ¼ 0 ð7:3Þ This Bessel’s equation has a solution of the form: EðrÞ ¼ AI1ðG0rÞ þ BK1ðG0rÞ ð7:4Þ where: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G0 ¼ jvmrm0s ¼ ð1 þ jÞd ð7:5Þ and d is the skin depth in the core. d ¼ pﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pf mrm0s ð7:6Þ I1 and K1 are modiﬁed Bessel functions of the ﬁrst and second kind respectively. The constant B ¼ 0 in Equation 7.4 since E(r) ¼ 0 at r ¼ 0. The constant A is found by applying Ampere’s law to a contour just above and below the current sheet in Figure 7.2, noting that H ¼ 0 outside the current sheet means that: HðbÞ ¼ Kf ð7:7Þ From Equations 7.4 and 7.7: HðbÞ ¼ À G0 jvmrm0 AI0ðG0bÞ ¼ Kf and A is then: A ¼ Àjvmrm0 Kf G0I0ðG0bÞ ð7:8Þ ð7:9Þ The electric ﬁeld is found from Equation 7.4: EðrÞ ¼ Àjvmrm0 I1ðG0rÞ G0I0ðG0bÞ ð7:10Þ At the surface of the core at r ¼ b, the electric ﬁeld is: EðbÞ ¼ Àjvmrm0b I1ðG0bÞ G0bI0ðG0bÞ Kf ð7:11Þ The induced voltage at the surface at r ¼ b due to the current in a segment of length ‘ is: ð2p V ¼ À EðbÞbdw ¼ À2prEðrÞjr¼b 0 ð7:12Þ 200 Transformers and Inductors for Power Electronics and the impedance is: Z ¼ V I ¼ À2pKrfE‘ ðrÞr¼b ð7:13Þ Substituting for E(b), given by Equation 7.11, gives the impedance for a one turn coil: Z ¼ jvmrm02pb2 Kf ‘ Kf I1ðG0bÞ G0bI0ðG0bÞ ¼ jvmrm0 Ac ‘c 2I1ðG0bÞ G0bI0ðG0bÞ ð7:14Þ where Ac ¼ pb2 is the cross-sectional area of the core and lc is the length of the core. For N turns, the self impedance of the coil is: Z ¼ j vmr m0 N 2 Ac ‘c 2I1ðG0bÞ G0bI0ðG0bÞ ð7:15Þ This result may be written down in terms of the low frequency asymptotic value of the inductance of the toroid L0: L0 ¼ mr m0 N 2 Ac ‘c ð7:16Þ Thus: Z ¼ jvL0 2I1ðG0bÞ G0bI0ðG0bÞ ð7:17Þ The self impedance of the coil given by Equation 7.15 has two components: Z ¼ Rs þ jvLs ð7:18Þ where Rs represents the eddy current loss in the core and Ls is the self inductance. Both Rs and Ls are frequency dependent. This dependence is best illustrated by numerical approximations to the quantities given by Equation 7.17. 7.1.1 Numerical Approximations The term G0b may be rewritten in terms of the skin depth d: G0b ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jvmrm0sb ¼ ð1 þ jÞ b d ¼ pﬃﬃ pﬃﬃ 2 jD ð7:19Þ D is a dimensionless parameter that is the ratio of the radius of the toroidal core cross-section to the skin depth. High Frequency Effects in the Core 201 The function 2I1ðG0bÞ G0bI0ðG0bÞ may be approximated as follows by asymptotic expansions of the Bessel functions: 8 ( pﬃﬃpﬃﬃ ) Re pﬃﬃ2pIﬃ1ﬃ ð 2pﬃjﬃDpÞﬃﬃ 2 jDI0ð 2 jDÞ ¼ >>>< 1 À 12 D4 þ 1:43D4 >>>: 1 D þ 1 16D3 þ 1 16D4 D D < > 2:1 2:1 8 ( pﬃﬃ pﬃﬃ ) Im pﬃﬃ2pIﬃ1ﬃ ð 2pﬃjﬃDpÞﬃﬃ 2 jDI0ð 2 jDÞ ¼ >>>>>< À D2 4 þ 384 D6 þ 4:16D4 >>>>>: À 1 D þ 11 1 2D2 þ 1 16D3 D D < > 2:8 2:8 ð7:20Þ ð7:21Þ The resistance and inductance terms of the core impedance may be evaluated by these approximations: 80 1 Rs ¼ >>>>>>< >>>>>>: vL0 B@D2 4 À 384 D6 þ 4:16D4 CA 11 vL0 1 D À 1 2D2 À 1 16D3 D D < > 2:8 2:8 Ls ¼ 8 >>>< >>>: D4 L0 1 À 12 þ 1:43D4 L0 1 D þ 1 16D3 þ 1 16D4 D D < > 2:1 2:1 ð7:22Þ ð7:23Þ 7.1.2 Equivalent Core Inductance L0 in Equation 7.16 is recognized as the classical expression for the inductance of a toroid at low frequencies. L0 was derived in Example 2.7; the derivation was based on the assumption of uniform ﬂux density in the core, i.e. no eddy currents. Ls from Equation 7.18 is plotted in Figure 7.3 as a function of D. The program is listed at the end of the chapter. The inductance is reduced to 50% of its low frequency value at D ﬃ 2; that is, when the diameter of the core is approximately four skin depths. Example 7.1 Calculate the maximum operating frequency of an inductor, made of a toroidal ferrite core with a circular cross-sectional diameter of 1.0 cm, that will ensure that the inductance of a toroidal inductor does 202 Transformers and Inductors for Power Electronics 1 0.9 0.8 0.7 LS 0.6 L0 0.5 0.4 0.3 0.2 0.1 0 0 5 1 1.5 2 2.5 3 3.5 4 4.5 5 Δ Figure 7.3 Self inductance in a toroidal core as a function of frequency. not drop below 80% of its low frequency value. The ferrite material is Mn-Zn with relative permeability mr ¼ 2500 and electrical resistivity of 1 Vm. The approximation in Equation 7.23 means that Ls/Lo ¼ 0.8 and: D4 1 À 12 þ 1:43D4 ¼ 0:8 Solve to give D ¼ 1.35. The diameter of the core is 2b and this means that 2b ¼ 2 Â D Â d ¼ 1.7d ¼ 1.0 cm. The maximum skin depth is then 1.0/1.7 ¼ 0.588 cm. The conductivity is s ¼ 1(Vm)À1. The skin depth is: d ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðpÞf maxð2500Þð4p Â 10À7Þð1Þ Â 102 ¼ p10ﬃﬃ0ﬃﬃﬃ6ﬃﬃﬃ f max ¼ 0:588 cm Taking this value of skin depth means that the maximum frequency is 2.93 MHz in this example. This is well above the recommended operating frequency for a core of this material and crosssection. 7.1.3 Equivalent Core Resistance The approximation given by Equation 7.22 for Rs gives us further insight into the nature of the core loss in terms of operating frequency and maximum ﬂux density. Invoking the approximation in Equation 7.22 for D < 2.8 and taking the ﬁrst term yields: Rs ¼ vL0 D2 4 ¼ pf 2 s mr m0 N 2 lc lcpb2 2 ð7:24Þ High Frequency Effects in the Core 203 Noting that the volume of the core is simply lcpb2 ¼ lcAc, the instantaneous power loss is i2Rs and the instantaneous power loss per unit volume is: p ¼ pf 2s mrm0 Ni2 2 lc ð7:25Þ Applying Ampere’s law, using Equation 7.7 with the current Ni ¼ Kf lc, means that the term inside the bracket in Equation 7.25 is the magnetic ﬂux density. We have assumed that the outside diameter of the toroid is much greater than the diameter of the core cross-section to take the length of the winding as the mean core length. Assuming the magnetic ﬂux density B ¼ Bmaxsinvt at the surface of the core, the average value of (Bmaxsinvt)2 is Bmax/2 and the average power loss in the core due to eddy currents is: p ¼ pf 2sB2maxpb2 4 ð7:26Þ This result shows that the core loss may be reduced by increasing the electrical resistivity (reducing the electrical conductivity) of the core material. The total loss due to eddy currents is proportional to the square of the radius of the cross-section. The use of a smaller core cross-section to reduce eddy current loss suggests the use of laminated layers of core material for high-frequency operation. Example 7.2 Calculate the equivalent series resistance and the average eddy current loss in the core of Example 7.1 that operates at a maximum ﬂux density of 0.4 T at 1 MHz with 100 turns. The outside diameter of the toroid is 8 cm. The inside diameter of the toroid is (8 – 2) cm ¼ 6 cm and the average diameter is 7 cm. The mean length of the magnetic path is p Â 7 cm. L0 ¼ ð2500Þð4p Â 10À7Þð1002ÞðpÞð0:5 pð7 Â 10À2Þ Â 10À2Þ2 ¼ 0:0112 H d ¼ qﬃﬃﬃ1ﬃﬃ0ﬃﬃ0ﬃﬃﬃ6ﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1:006 cm ð1 Â 106Þ D ¼ 0:5 1:006 ¼ 0:497 Rs ¼ ð2pÞð1 Â 106Þð0:0112Þ0:4972 4 À 384=11 0:4976 þ ð4:16Þð0:4974Þ ¼ 4315:45 V p ¼ pð1 Â 106Þ2ð1Þð0:4Þ2pð0:5 Â 10À2Þ2 ¼ 9:87 W=cm3 4 204 Transformers and Inductors for Power Electronics Example 7.3 Estimate the total core loss for the core in Example 7.2 based on the manufacturer’s data for N87 Mn-Zn. Pfe ¼ Kcf aBbmax For N87 Mn-Zn the parameters of the Steinmetz equation may be taken from Table 1.1, the total core loss per unit volume is p ¼ ð16:9Þð1 Â 106Þ1:25ð0:4Þ2:35 ¼ 62:05 W=cm3 This suggests that the hysteresis loss is 62.05À9.87 ¼ 52.18 W/cm3. 7.2 Core Loss The celebrated general Steinmetz equation [2] for core loss is commonly used to describe the total core loss under sinusoidal excitation (Equation 1.29): Pfe ¼ Kcf aBbmax ð1:29Þ Pfe is the time-average core loss per unit volume; Bmax is the peak value of the ﬂux density with sinusoidal excitation at the frequency f; Kc, a and b are constants that may be found from manufacturers’ data. Non-sinusoidal excitation is normal for typical power electronics applications. The non- linear nature of ferromagnetic materials means that it is not a simple case of adding the individual frequency components of a Fourier series. Simply using the peak value of the ﬂux density would underestimate the total core loss. The improved General Steinmetz Equation (iGSE) [3] addresses this issue by modifying the original equations while retaining the origi- nal coefﬁcients, Kc, a and b. The time-average power loss using the iGSE is: Pv ¼ 1 T ðT 0 ki d BdðttÞa jDBjbÀa d t ¼ ki jDBjbÀa 1 T ðT 0 ki d BdðttÞa d t ¼ ki jDBjbÀad BdðttÞa ð7:27Þ Here, DB is the peak-to-peak ﬂux density of the excitation. ki is given by: ki ¼ ð K 0 c 2bÀ1paÀ1 jcosðuÞjadu 2p A useful approximation for ki is: ki ¼ Kc 2bÀ1paÀ1 1:1044 þ 6:8244 a þ 1:354 ð7:28Þ ð7:29Þ High Frequency Effects in the Core 205 Example 7.4 Calculate the core loss in the forward converter of Example 5.2. Recall that the frequency is 25 kHz and the duty cycle is 0.75. The core material speciﬁcations are given in Table 5.5 and the core speciﬁcations are given in Table 5.6. These tables are repeated below for convenience: The ﬂux waveform is shown in Figure 7.4. Normally, iGSE should be used together with a piecewise linear model (PWL) applied to the ﬂux waveform in Figure 7.4, and the power loss per unit volume is: Pv ¼ ki jDBjbÀa 1 T ðDT 0 DDBT a d t þ ðT DT ð1 DB À DÞT a d ! t Rearranging: Pv ¼ ki jDBjbÀa 1 T h jDBja ðDT Þ1Àa þ jDBja½ð1 À i DÞT 1Àa Table 5.5 Material speciﬁcations Kc 37.2 a 1.13 b 2.07 Bsat 0.4T Table 5.6 Core and winding speciﬁcations Ac Wa Ap Vc kf ku MLT r20 a20 1.25 cm2 1.78 cm2 2.225 cm4 11.5 cm3 1.0 0.4 6.9 cm 1.72 mV-cm 0.00393 B(T ) ΔB 0 DT T t Figure 7.4 Flux waveform in the transformer of a forward converter. 206 Transformers and Inductors for Power Electronics With further simpliﬁcation: Pv ¼ ki jDBjb 1 Ta h D1Àa þ ð1 À i DÞ1Àa ki can be calculated by using Equation 7.29: ki ¼ Kc 2bÀ1paÀ1 1:1044 þ 6:8244 a þ 1:354 ¼ 37:2 22:07À1p1:13À1 1:1044 þ 6:8244 1:13 þ 1:354 ¼ 3:964 DB can be calculated by using Faraday’s law: DB ¼ V P DT N P Ac ¼ ð12Þð0:75Þ ð9Þð2:125 Â 10À4Þð25 000Þ ¼ 0:32 T The core loss per unit volume is: Pv ¼ ki jDBjb 1 Ta h D1Àa þ ð1 À i DÞ1Àa ¼ 5:56ð0:198Þ2:07ð25 h 000Þ1:13 0:751À1:13 þ ð1 À i 0:75Þ1À1:13 ¼ 7:815 Â 104 W=m3 The total core loss is then 11.5 Â 10À6 Â 7.815 Â 104 ¼ 0.899 W. The result using GSE was 0.898 W. Example 7.5 The ripple on the output inductor of the forward converter of Example 5.3 is 0.1 T (peak to peak) and the DC current is 7.5 A. Calculate the core loss per unit volume; assume the same material characteristics as given in Table 5.5. The voltage waveform on the output inductor is shown in Figure 7.5; the core loss is mainly due to the AC current ripple. The effect of DC bias on the core loss is not taken into account; this effect is described in [4]. The iGSE, together with a PWL model, may be used to calculate the core loss. Since the waveform shape and the corresponding time periods are the same as the transformer in Example 7.4, the same expressions apply. Pv ¼ ki jDBjb 1 Ta h D1Àa þ ð1 À i DÞ1Àa ¼ ð3:964Þð0:1Þ2:07ð25 h 000Þ1:13 0:751À1:13 þ ð1 À i 0:75Þ1À1:13 ¼ 7 034 W=m3 With GSE Pf e ¼ Kcf aBmaxb ¼ ð37:2Þð25 000Þ1:3ð0:1=2Þ2:1 ¼ 7 032 W=m3 The error is less than 1%. High Frequency Effects in the Core 207 vL Vin - Vo 0 DT T t iL Δi 0 DT T t Figure 7.5 Flux waveform for the output inductor in a forward converter. Example 7.6 Calculate the core loss in a push-pull converter of Example 5.3. Recall that the frequency is 50 kHz and the duty cycle is 0.67 with input voltage of 36 V. The voltage waveform factor is Kv ¼ 4.88. The core material speciﬁcations are given in Table 5.8 and the core speciﬁcations are given in Table 5.9. These tables are repeated below for convenience: The peak value of the ﬂux density in the core is: pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Bmax ¼ DV d c Kvf NpAc ¼ ð4:88Þð50 0:67ð36Þ 000Þð6Þð1:73 Â 10À4Þ ¼ 0:116 T The ﬂux waveform is shown in Figure 7.6. Table 5.8 Material speciﬁcations Kc a b Bsat 9.12 1.24 2.0 0.4 T Table 5.9 Core and winding speciﬁcations Ac Wa Ap Vc kf ku MLT r20 a20 1.73 cm2 2.78 cm2 4.81 cm4 17.70 cm3 1.0 0.4 7.77 cm 1.72 mV-cm 0.00393 208 vp Transformers and Inductors for Power Electronics Bmax ΔB t -Bmax 0 DT/2 T/2 (1+D)T/2 T Figure 7.6 Flux waveform for the push-pull converter. Taking iGSE with a piecewise linear model (PWL) applied to the ﬂux waveform in Figure 7.6, the power loss per unit volume is: Pv ¼ ki jDBjbÀa 1 T "ð DT =2 0 DDTB=2a d t þ ð ð1þDÞT =2 T =2 DDTB=2a d # t Rearranging and simplifying: Pv ¼ ki jDBjbÀa 1 T h i j2DBja ðDT Þ1Àa ki is found from Equation 7.29: ki ¼ Kc 2bÀ1paÀ1 1:1044 þ 6:8244 a þ 1:354 ¼ 9:12 22:0À1p1:24À1 1:1044 þ 6:8244 1:24 þ 1:354 ¼ 0:9275 The core loss per unit volume is: Pv ¼ ki jDBjbÀa 1 T h i j2DBja ðDT Þ1Àa h i ¼ ð0:9275Þð0:232Þ2:0À1:24ð50 000Þ ð2 Â 0:232Þ1:24 Â ð0:67=ð50 000ÞÞ1À1:24 ¼ 0:871 Â 105 W=m3 The total core loss is then (17.71 Â 10À6)(0.871 Â 105) ¼ 1.543 W. Using GSE, taking equation : Pfe ¼ VcKcf aBbmax ¼ ð17:70 Â 10À6Þð9:12Þð50 000Þ1:24ð0:116Þ2:0 ¼ 1:458 W The difference is approximately 6% when compared to iGSE. High Frequency Effects in the Core 209 7.3 Complex Permeability Returning to Equation 7.15, the equivalent series impedance may be written in terms of this equation, with the approximations in Equations 7.20 and 7.21 for D < 2: Z ¼ Rs þ jvLs ¼ vmr D2 4 L1 þ jvmr 1 À 12 D4 À 1:43D4 L1 ð7:30Þ Or, in terms of complex permeability: Z ¼ jvL1ðm0rs À jm0r0sÞ ð7:31Þ Here, L1 is the low frequency inductance of the toroid with a relative permeability of 1 (air): L1 ¼ m0 N 2 Ac ‘c ð7:32Þ and the series complex relative permeability is deﬁned as: mrs ¼ m0rs À jm0r0s ð7:33Þ Based on the analysis provided above, we can ﬁnd the components of the relative permeability as: m0rs ¼ mr Re 2I1ðG0bÞ G0bI0ðG0bÞ ¼ mr 1 À 12 D4 À 1:43D4 m0r0s ¼ Àmr Im 2I1ðG0bÞ G0bI0ðG0bÞ ¼ mr D2 4 ð7:34Þ ð7:35Þ mr is simply the relative permeability of the core material as before. The loss tangent is deﬁned in terms of the loss angle d: tan d ¼ m0r0s m0rs ¼ Rs vLs The Q factor is: Q ¼ vLs Rs ¼ m0rs m0r0s ð7:36Þ ð7:37Þ Manufacturers provide plots of mr vs. frequency so that the inductance is calculated by using the appropriate value of mr in the low frequency toroidal equation (Equation 7.16). This achieves the same results as Figure 7.3. However, D is a function of the core diameter and measurements on one core are not entirely appropriate for a different-sized core of the same material. 210 Transformers and Inductors for Power Electronics Equation 7.16 is based on a homogeneous isotropic core. In particular, it is assumed that the relative permeability and electrical resistivity are constant. It turns out that the electrical resistivity of a ferrite is frequency-dependent. Snelling [5] gives typical values for Mn-Zn ferrites as 1 Vm at low frequencies and 0.001 Vm at high frequencies. There are two factors which contribute to resistivity in polycrystalline ferrites such as Mn-Zn. There is a granular structure, in which the resistance across the grain boundaries is on the order of one million times greater than that of the ferrite material inside the grains. At low frequencies, the grain boundaries dominate the resistivity, while at high frequencies, capacitive effects shunt the grain boundaries and the ferrite material within the grains’ resis- tivity dominates the overall resistivity. The resistivity was measured on a sample of P type Mn-Zn ferrite and the variation with frequency is illustrated in Figure 7.7. Manufacturers present the imaginary part of the complex relative permeability m0r0s to allow the designer to ﬁnd the inductance by inserting the frequency-dependent value of rela- tive permeability in the classical toroidal inductance formula given by Equation 7.16. In effect, the manufacturer generates the complex permeability by taking measurements on a toroidal core of known dimensions, and deduces the complex permeability from the meas- ured impedance as given by Equation 7.30. The ratio of Ls/L0 is given by: 8 Ls L0 ¼ >>>< >>>: 1 1 D À þ D4 12 þ 1:43D4 1 16D3 þ 1 16D4 D < 2:1 D > 2:1 ð7:38Þ Recall that L0, given by Equation 7.16, is L0 ¼ mrL1, and L1 is given by Equation 7.32. 10 1 Resistivity (Ωm) 0.1 0.01 1 kHz 10 kHz 100 kHz 1 MHz Frequency 10 MHz 100 MHz Figure 7.7 Resistivity of P type ferrite. High Frequency Effects in the Core 211 1.5 1 LS L0 0.5 0 10 kHz LS , ρ from Fig. 7.7 L0 LS , ρ = 0.1 Ωm L0 μi μi 0 (manufacturer) 100 kHz 1 MHz Frequency 10 MHz 100 MHz Figure 7.8 Initial permeability of P type ferrite. Equation 7.38 is plotted in Figure 7.8 for a 10 mm diameter P type ferrite core (b ¼ 5 mm). One calculation was performed using a single low-frequency value of resistivity, 0.1 Vm, and another calculation was performed with values taken from Figure 7.7. The manufacturer supplied initial permeability as a function of frequency for P type Mn-Zn ferrite, and this is normalized to the low-frequency value (mi0 ¼ 2500). Clearly, Equation 7.38 is a very good ﬁt for the manufacturer’s measurements, even in the case where a single value of resistivity is used. The peak in the manufacturer’s data at 600 kHz is due to a complex capacitive effect at domain walls in the ferrite. It is important to stress that only one value of relative permeability was used in the calculations in Figure 7.8. The permeability is constant, and the reduction in self inductance at high frequency is due to the electrical resistivity of the core material giving rise to eddy currents. The manufacturer accounted for the reduction in inductance at high frequency by supplying a frequency-dependent permeability for use in the classical toroidal formula. In fact, the variation is entirely predictable and is calculable with Equation 7.38, knowing the electrical resistivity and the relative permeability of the core material and the physical dimensions of the core. Example 7.7 Calculate the series impedance of a toroidal inductor wound with 100 turns on a toroid with the dimensions shown in Figure 7.9 at 1 MHz. Take r ¼ 0.1 Vm and mr ¼ 2000. d ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Â 102 ¼ 0:356 cm pð106Þð2000Þð4p Â 10À7Þð10Þ D ¼ 0:5 0:356 ¼ 1:40 212 Transformers and Inductors for Power Electronics 1cm 8cm 10cm Figure 7.9 Toroidal core. m0rs ¼ mr 1 À 12 D4 À 1:43D4 ¼ 2000 ! 1 À 12 À ð1:40Þ4 ð1:43Þð1:40Þ4 ¼ 819 m0r0s ¼ mr D2 4 ¼ 2000 ð1:4Þ2 4 ¼ 980 L1 ¼ m0 N 2 Ac ‘c ¼ ð4p Â 10À7Þð100Þ2pð0:5 pð9 Â 10À2Þ Â 10À2 Þ2 ¼ 3:49 Â 10À6 H Z ¼ jvL1ðm0rs À jm0r0sÞ ¼ jð2p Â 106Þð3:49 Â 10À6Þð819 À j980Þ ¼ 21; 489 þ j17; 962 V 7.4 Laminations The factors which inﬂuence the choice of core lamination material and thickness may be deduced by considering a laminated plate in a uniform magnetic ﬁeld, as shown in Figure 7.10. t a b 1 1 Jx dx dx d c B xx y z x Figure 7.10 Lamination in a uniform magnetic ﬁeld. High Frequency Effects in the Core 213 Consider a volume of magnetic lamination material of unit height and depth, with a uniform sinusoidal ﬂux density B ¼ Bmaxsinvt at right angles to the face of the lamination. The alternating ﬂux induces eddy currents, as represented by the current density Jx. Applying Faraday’s law in the form of Equation 1.14 around the loop abcd, with contributions from the sides ab and cd at a distance x from the centre line, yields an expression for the induced emf Ex: 2Ex ¼ À2 dB dt ð7:39Þ where Ex has the same direction as current density Jx and is related to it by the microscopic form of Ohm’s law: Jx ¼ sEx ð7:40Þ where s is the conductivity of the material in the lamination. The instantaneous power loss per unit volume due to Jx is: pðtÞ ¼ 1 s J2x ¼ sE2x ¼ sx2 d B2 dt ð7:41Þ The instantaneous power loss in a tranche of thickness dx with unit height and depth is: pðtÞ ¼ sx2 d B2 d x dt ð7:42Þ Integrating from x ¼ 0 to x ¼ t/2 and taking both sides of the centre plane into account gives the instantaneous power loss in a volume of unit height and depth, in a lamination of thickness t, as: pðtÞ ¼ d 2s B2 ð t 2 x2 d x ¼ t3 d B2 s dt 0 12 dt ð7:43Þ In a transformer core, there are 1/t laminations in a unit thickness so that, in a unit cube, the instantaneous eddy current loss per unit volume is: pðtÞ ¼ t2 d B2 s 12 dt ð7:44Þ The average value of d B2 dt is the average value of (vBmaxcosvt)2 which is v2B2max=2, so that the average power loss per unit volume of the core is: hpi ¼ t2 12 sv2B2max 1 2 ¼ p2f 2st2B2max 6 ð7:45Þ Example 7.8 A power transformer, operating at 50 Hz would have laminations of thickness 0.3 mm, with Bmax ¼ 1.5 T and s ¼ 2 Â 106 (Vm)À1, giving: hpi ¼ p2ð50Þ2ð2 Â 106ÞÀ0:3 Â 10À3Á2ð1:5Þ2 ¼ 1:66 kW=m3 6 214 Transformers and Inductors for Power Electronics With a density of 7650 kg/m3, this corresponds to 0.22 W/kg. The core loss for silicon steel in Table 1.1 is 5.66 kW/m3, which includes the hysteresis loss. This simpliﬁed analysis assumes that the presence of the eddy currents does not disturb the magnetic ﬁeld, and a suitable choice of lamination would ensure this. The skin depth in the material used in the above example would be 0.5 mm for mr ¼ 10 000. In general, the thickness of the lamination should be less than the skin depth given by Equation 1.21 at the operating frequency in order to ensure a uniform distribution of the ﬂux density. The result in Equation 7.45 shows that the eddy current loss under AC operating conditions is proportional to the square of frequency and the square of the maximum ﬂux density. While this is somewhat simpliﬁed, it shows that eddy current loss is an important design consideration for iron cores. For ferrites, the conductivity s would be in the order of 2 (V-m)À1 and, therefore, eddy current loss would not arise until frequencies into the MHz range are encountered. 7.5 Problems 7.1 Write a MATLAB program to compare the exact value of Z given by Equation 7.15, with the approximations given by Equations 7.22 and 7.23. 7.2 Repeat Example 7.1 for Ni-Zn ferrite with a relative permeability of 400 and electrical resistivity of 10 000 Vm. 7.3 Repeat Example 7.2 for Ni-Zn ferrite with a relative permeability of 400 and electrical resistivity of 10 000 Vm. 7.4 Evaluate the integral in Equation 7.28 for a ¼ 1 and a ¼ 2. 7.5 Calculate the core loss in Example 3.3 using iGSE. 7.6 Calculate the eddy current loss per unit volume for amorphous alloy tape at 25 kHz. The tape thickness is 40 mm, the maximum ﬂux density is 0.2 T and the resistivity of the material is 1.3 mVm. MATLAB Program for Figure 7.3 % Figure 7.3 : inductance in a toroidal core as a function of frequency n=0; Lo=1; for delta = [0:0.01:10]; I0 = besseli(0,sqrt(2)*sqrt(j)*delta); I1 = besseli(1,sqrt(2)*sqrt(j)*delta); Ls = Lo*real(2*I1/(sqrt(2)*sqrt(j)*delta*I0)); n = n+1; V(n)= Ls/Lo; D(n)= delta; end plot(D,V,’k’,’LineWidth’,2) title(’Self’) High Frequency Effects in the Core 215 xlabel(’Delta’) ylabel(’Ls/Lo’) axis([0 5 0 1]) grid off MATLAB Program for Figure 7.8 % ﬁgure 7.8 : Initial permeability of P type ferrite n = 0; Lo = 1; b = 0.005;%m ro = 0.1;%omega-m mur = 2500; mu0 = 4*pi*10^-7; for f1 = [10000:100000:100000000]; delta = b*sqrt(pi*f1*mu0*mur/ro); I0 = besseli(0,sqrt(2)*sqrt(1j)*delta); I1 = besseli(1,sqrt(2)*sqrt(1j)*delta); Ls = real(2*I1/(sqrt(2)*sqrt(1j)*delta*I0)); n = n+1; V(n)= Ls; D(n)= f1; end semilogx(D,V,’k’,’LineWidth’,2) title(’Self’) xlabel(’frequency’) ylabel(’Ls/Lo’) axis([10000 100000000 0 1.5]) grid off hold on n = 0; f2 =[100,200,400,700,1000,2000,4000,7000,10000,20000,40000,70000, 100000,200000,400000,700000,1000000,2000000,4000000,7000000, 10000000,20000000,40000000]; for delta = [0.00263,0.00374,0.00530,0.00701,0.00838,0.01186,0.01704, 0.02291,0.02812,0.04340,0.07188,0.11353,0.15553,0.29685,0.57167, 216 Transformers and Inductors for Power Electronics 0.98783,1.38301,2.58237,4.53450,6.65482,8.16618,12.22863, 19.11912]; I0 = besseli(0,sqrt(2)*sqrt(1j)*delta); I1 = besseli(1,sqrt(2)*sqrt(1j)*delta); L = real(2*I1/(sqrt(2)*sqrt(1j)*delta*I0)); n = n+1; U(n)= L; end semilogx(f2,U,’b’,’LineWidth’,2) f3 = [10000,100000,150000,200000,250000,300000,350000,400000, 450000,500000,550000,600000,700000,800000,900000,1000000,1500000, 1750000]; mu = [1,1,1,1.04,1.10,1.18,1.28,1.35,1.38,1.40,1.38,1.36,1.30,1.20, 1.10,1,0.40,0]; semilogx(f3,mu,’r’,’LineWidth’,2) References 1. Hurley, W.G., Wilcox, D.J., and McNamara, P.S. (1991) Calculation of short circuit impedance and leakage impedance in transformer windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC pp. 651–658. 2. Steinmetz, C.P. (1984) On the law of hysteresis. P. Proceedings of the IEEE, 72 (2), 197–221. 3. Venkatachalam, K., Sullivan, C.R., Abdallah, T., and Tacca, H. (2002) Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters. Proceedings of IEEE Workshop on Computers in Power Electronics, COMPEL, pp. 36–41. 4. Muhlethaler, J., Biela, J., Kolar, J.W., and Ecklebe, A. (2012) Core losses under the DC bias condition based on Steinmetz parameters. IEEE Transactions on Power Electronics 27 (2), 953–963. 5. Snelling, E.C. (1988) Soft Ferrites: Properties and Applications, 2nd edn, Butterworths, London. Further Reading 1. Del Vecchio, R.M., Poulin, B., Feghali, P.T. et al. (2001) Transformer Design Principles: With Applications to Core-Form Power Transformers, 1st edn, CRC Press, Boca Raton, FL. 2. Di, Y. and Sullivan, C.R. (2009) Effect of capacitance on eddy-current loss in multi-layer magnetic ﬁlms for MHz magnetic components. Proceedings of the IEEE Energy Conversion Congress and Exposition, ECCE, pp. 1025–1031. 3. Dowell, P.L. (1966) Effects of eddy currents in transformer windings. Proceedings of the Institution of Electrical Engineers 113 (8), 1387–1394. 4. Dwight, H.B. (1961) Tables of Integrals and Other Mathematical Data, 4th edn, Macmillan, London. 5. Erickson, R.W. (2001) Fundamentals of Power Electronics, 2nd edn, Springer, Norwell, MA. 6. Goldberg, A.F., Kassakian, J.G., and Schlecht, M.F. (1989) Issues related to 1–10-MHz transformer design. IEEE Transactions on Power Electronics 4 (1), 113–123. 7. Han, Y., Cheung, G., Li, A. et al. (2012) Evaluation of magnetic materials for very high frequency power appli- cations. IEEE Transactions on Power Electronics 27 (1), 425–435. 8. Hurley, W.G. and Wilcox, D.J. (1994) Calculation of leakage inductance in transformer windings. IEEE Trans- actions on Power Electronics 9 (1), 121–126. 9. McLachlan, N.W. (1955) Bessel Functions for Engineers, 2nd edn, Clarendon Press, Oxford. High Frequency Effects in the Core 217 10. McLyman, C.W.T. (1997) Magnetic Core Selection for Transformers and Inductors, 2nd edn, Marcel Dekker Inc., New York. 11. McLyman, C.W.T. (2002) High Reliability Magnetic Devices, 1st edn, Marcel Dekker Inc., New York. 12. McLyman, C.W.T. (2004) Transformer and Inductor Design Handbook, 3rd edn, Marcel Dekker Inc., New York. 13. E.S. MIT (1943) Magnetic Circuits and Transformers (MIT Electrical Engineering and Computer Science), The MIT Press, Cambridge, MA. 14. Muhlethaler, J., Biela, J., Kolar, J.W., and Ecklebe, A. (2012) Improved core-loss calculation for magnetic components employed in power electronic systems. IEEE Transactions on Power Electronics 27 (2), 964–973. Section Three Advanced Topics 8 Measurements Traditionally, measurements for transformers and inductors have been focused on power frequency operation. These approaches remain relevant today, but they must be modiﬁed or replaced to take account of the high frequencies encountered in power electronics. Measurement of inductance provides several challenges, since it is not always a single value, particularly when saturation is involved. Two methods for inductance measurement are treated – one involving DC current and another involving AC signals. Traditionally, the measurements of losses in transformers have been carried out by the established open-circuit and short-circuit tests. However, these tests must be adapted to the ﬁeld of power electronics in a manner that allows us to measure the core loss over a wide frequency range and to present the data in a format compatible with manufacturers’ data sheets. The measurement of the B-H loop is described for core materials. Knowledge of winding capacitance is important, particularly when there are resonant frequencies involved. Winding capacitance also plays a role in the dynamic response when a step change in voltage is shorted by the winding capacitance. A suitable measurement of capacitance in transformer windings is presented, along with some basic formulas to estimate the capacitance in typical winding conﬁgurations. 8.1 Measurement of Inductance The measurement of inductance has traditionally been carried out using the Anderson bridge, named after Alexander Anderson, who was president and professor of physics at the National University of Ireland, Galway (formerly Queen’s College Galway) in the late nineteenth century. The Anderson Bridge operated in much the same way as the Wheatstone bridge. However, the advent of modern network analyzers has consigned the Anderson bridge to the history museum. Inductance can be highly variable as saturation takes effect. We may divide inductance measurement into two categories: the quiescent or DC value; and the incremental or AC value. The DC measurement of inductance is easily obtained from the response of an applied step function of voltage. The incremental value may be found from the application of an AC signal. Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 222 Transformers and Inductors for Power Electronics + DC source _ S R + t=0 i C v L _ Figure 8.1 Inductance measurement by step voltage method. 8.1.1 Step Voltage Method The inductor may be modelled by a series combination of resistance and inductance, as shown in Figure 8.1. The simplest way to apply the step voltage is to charge a large capacitor – an ultracapacitor is ideal. The capacitor is initially charged from a power supply and the switch S is closed at t ¼ 0. The capacitor provides a constant voltage, and the current rises in the inductor and may be measured by a small non-inductive precision sampling resistor. The general solution for the setup in Figure 8.1 is: V ¼ Ri þ dl dt ð8:1Þ In discretized form, this becomes, for constant V: V ¼ RiðkÞ þ lðkÞ À lðk Dt À 1Þ k ¼ 1; 2; 3; . . . ð8:2Þ Samples of current are taken at intervals of time Dt, so Equation 8.2 may be rewritten: lðkÞ ¼ VDt À RiðkÞDt þ lðk À 1Þ ð8:3Þ l(k) is the ﬂux linkage at time tk ¼ k Dt that is after k time steps and the corresponding value of current is i(k). This is best illustrated by an example. Example 8.1 Table 8.1 shows the data for the current in an inductor following a step change of voltage. V ¼ 70 V, the time step is 0.24 ms and R ¼ 2.5 V. Calculate the value of the inductance and estimate the value of current at the onset of saturation. Inserting the values given above into Equation 8.3 yields: lðtkÞ ¼ ð70Þð0:24 Â 10À3Þ À ð2:5Þð0:24 Â 10À3ÞiðtkÞ þ lðtkÀ1Þ Measurements 223 Table 8.1 Inductance measurements k tk(ms) i(k) (A) 1 0.24 0.32 2 0.48 0.60 3 0.72 0.92 4 0.96 1.20 5 1.20 1.44 6 1.44 1.84 7 1.68 2.40 8 1.92 3.40 9 2.16 4.44 l(k) (Wb-T) 16.608Ã10À3 33.048Ã10À3 49.296Ã10À3 65.376Ã10À3 81.312Ã10À3 92.008Ã10À3 112.368Ã10À3 127.128Ã10À3 141.264Ã10À3 L(ik) (mH) 51.9 55.5 53.5 54.5 56.5 52.3 46.8 32.4 31.8 or lðtkÞ ¼ ð16:8 Â 10À3Þ À ð0:60 Â 10À3ÞiðtkÞ þ lðtkÀ1Þ The inductance at each value of current is: Lðik Þ ¼ lðkÞ iðkÞ L(ik) is shown in Figure 8.2. Above 2 A, the inductance falls off to indicate that the onset of saturation was reached in the core at that point. 8.1.2 Incremental Impedance Method This method measures the AC or incremental inductance as a function of its DC bias. For this purpose a DC current is fed through the inductor, while an AC voltage source is applied to 70 60 50 Inductance (mH) 40 30 20 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Current (A) Figure 8.2 Inductance and incremental inductance measured by the step voltage method. 224 Transformers and Inductors for Power Electronics Inductor under test Decoupling inductor Decoupling capacitor I dc DC + source _ + L vac R _ i ac + AC source _ Figure 8.3 Inductance measurement by incremental impedance. the inductor driving a small AC current through the inductor as shown in Figure 8.3. The incremental inductance may be calculated from the AC voltage and current. In this circuit, the AC and the DC circuits must be totally decoupled from each other, otherwise the DC supply would short-circuit the AC voltage on the device under test with its large internal capacitance. The DC current through the inductor can be set independently from the AC values. The DC supply is decoupled from the AC circuit with a large decoupling inductor Lc (typically 1 H). The AC source is decoupled from the DC circuit by a large capacitor Cc (typically 1500 mF). The impedance can be calculated from the measured AC voltage vac and current iac on the inductor under test. Z ¼ vac iac ð8:4Þ The impedance of the inductor is given by the effective inductance and the resistance of the coil. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z ¼ R2 þ ðvLacÞ2 ð8:5Þ From which we obtain: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Lac ¼ 1 v v2ac i2ac À R2 ð8:6Þ R may be measured separately. Example 8.2 Table 8.2 shows the measurement data for an inductor obtained from the incremental impedance method. Calculate the AC inductance as a function of DC bias. The resistance of the coil was 4.0 V and the measurements were carried out at 50 Hz. The impedance and inductance values are given by Equations 8.4 and 8.6 and the results are summarized in Table 8.2. The inductance is plotted in Figure 8.4. Measurements Table 8.2 Inductance measurements Idc(A) 0.000 0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.250 3.500 3.750 4.000 vac(V) 1.105 1.103 1.112 1.115 1.128 1.115 1.184 1.193 1.182 1.147 1.105 1.080 1.040 1.010 0.984 0.966 0.944 iac(mA) 52.70 51.40 53.90 58.10 67.50 83.70 109.00 131.00 161.00 179.00 187.00 191.00 195.00 199.00 197.00 201.00 210.00 Z(V) 20.97 21.46 20.63 19.19 16.71 13.32 10.86 9.11 7.34 6.41 5.91 5.65 5.33 5.08 4.99 4.81 4.50 225 Lac(mH) 65.5 67.1 64.5 59.8 51.7 40.5 32.2 26.1 19.6 15.9 13.9 12.7 11.2 9.9 9.5 8.5 6.5 8.2 Measurement of the B-H Loop The B-H loop is of interest because we need knowledge of the magnetic parameters such as Bsat, the saturation ﬂux density, the coercive force Hc and the residual ﬂux Br. These terms are explained in Chapter 2. The simplest set up is a coil of N turns on a toroidal core, as illustrated in Figure 8.5. 80 70 60 50 Inductance (mH) 40 30 20 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Current (A) Figure 8.4 AC inductance measured by the incremental impedance method. 226 Transformers and Inductors for Power Electronics lc Ac i _ e+ φ =BAc Figure 8.5 Toroidal core for B-H measurement. Recall from Chapter 2, Equation 2.1þ: ¼ H Á dl ¼ Ni C The magnetic ﬁeld intensity may be obtained directly from the current measurement. Hc ¼ N lc i ð8:7Þ where lc is the mean length of the magnetic path in the test specimen. The ﬂux density in the coil is found from Faraday’s law, assuming negligible winding resistance: ð B ¼ 1 vdt NAc ð8:8Þ where v is the terminal voltage of the coil. The number of turns must be selected to ensure that the correct values of Bsat and Hc are correctly included in the measurement. For a sinusoidal voltage input at frequency f, the maximum ﬂux density is related to the peak value of the applied voltage, from Equation 8.8: Bmax ¼ V peak 2pf NAc ð8:9Þ It is a straightforward matter to implement the integration in Equation 8.8 with the set-up shown in Figure 8.6. The gain-phase analyzer may be programmed in a LabVIEW environment to integrate the input voltage to the coil, and the current may be measured by the use of a sampling resistor Rref. The signal generator sets the frequency for the power ampliﬁer. The normal magnetization curve may be obtained by taking the B/H ratio at the tips of the hysteresis loops for different values of Bmax. Measurements 227 Control Data v acquisition i Signal generator Inductor under test + + Power vi v amplifier _ R ref _ i Figure 8.6 Test set-up for measurement of the B-H loop. Example 8.3 Select the number of turns for a closed core made of Mn-Zn ferrite, with mean magnetic path length of 75.5 mm and a cross-sectional area of 81.4 mm2, that yields a value of maximum ﬂux density Bmax of 400 mT and coercive force Hc of 155 A/m at a current of 1.15 A. Calculate the rms value of the applied voltage for 50 Hz excitation. Using Equation 8.7, the number of turns is: N ¼ ð155Þð75:5 Â 1:15 10À3Þ ¼ 10 turns The rms value of the applied voltage at 400 mT and 50 Hz excitation is: Vrms ¼ 4:44f NBmaxAc ¼ ð4:44Þð50Þð10Þð0:4Þð81:4 Â 10À6Þ ¼ 72:3 mV pﬃﬃ Vmax ¼ 2ð72:3Þ ¼ 102:2 mV So the amplitude of v is in the order of 100 mV for Bsat ¼ 0.4 T at 50 Hz excitation. The B-H loop was generated for the PC40 material by winding ten turns on a core with the dimen- sions given above. The B-H loop is shown in Figure 8.7. We may also obtain the hysteresis loss per unit volume by integrating the B-H characteristic to ﬁnd the area enclosed by the B-H loop. 8.3 Measurement of Losses in a Transformer The main parameters in a transformer may be measured by two simple tests, namely the short-circuit test that forces rated current through the windings at a low voltage, and the open-circuit test that is carried out at rated voltage to include the magnetizing current. 228 Transformers and Inductors for Power Electronics Figure 8.7 Hysteresis loop for PC40 material at 25 C. The tests may be carried out on either the low voltage or high voltage side of the transformer. These tests are traditionally associated with 50 Hz or 60 Hz power transformers, with the test power readily supplied at line frequency. However, in power electronics applications, the transformers operate at hundreds of kHz. In this case, we can measure the core loss at high frequency by using a power ampliﬁer and a gain-phase analyzer. It is also possible to apply a DC step voltage to the transformer, with one winding short-circuited to infer the leakage reactance from the step response. 8.3.1 Short-Circuit Test (Winding/ Copper Loss) With one winding short-circuited, typically 10% of rated voltage on the other winding is sufﬁcient to establish rated full load current. For convenience, we will short the secondary winding, take measurements in the primary winding and refer the secondary quantities as appropriate. Measure: Vsc short-circuit primary voltage Isc short-circuit primary (rated) current Psc short-circuit power (measured with a wattmeter). The core loss is negligible, since the input voltage is very low. For that reason, the core circuit parameters are shown dotted in Figure 8.8. Measurements 229 R1 Xl1 + AC source_ + Isc Vsc _ Xl21 R 1 2 Figure 8.8 Transformer short-circuit test. Invoking the transformer analysis of Chapter 4 for the equivalent circuit of Figure 8.8, the equivalent impedance Zeq looking into the terminals of the transformer is given by the shortcircuit impedance Zsc: Z eq ¼ Z sc ¼ V sc Isc ð8:10Þ By examination of Figure 8.8, the real part of Zeq is the equivalent resistance Req of the windings. This is made up of the resistance of the primary winding and the resistance of the secondary winding reﬂected into the primary. The primary to secondary turns ratio is a: Req ¼ Rsc ¼ Psc I 2 sc Req ¼ R1 þ a2R2 ð8:11Þ ð8:12Þ The imaginary part of Zeq is the equivalent leakage reactance of the windings. This consists of the leakage reactance of the primary winding and the leakage reactance of the secondary winding reﬂected into the primary winding: qÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÁﬃﬃ Xeq ¼ Xsc ¼ Z2sc À R2sc ð8:13Þ Xeq ¼ Xl1 þ a2Xl2 ð8:14Þ As a ﬁrst approximation, it is reasonable to assume that R1 ¼ a2R2, Xl1 ¼ a2Xl2 in a welldesigned transformer. A more realistic approach is to take the ratios of R1 and Xl1 as the ratio R2 Xl2 of the DC resistance of the individual windings, which may be easily measured. 8.3.2 Open-Circuit Test (Core/ Iron Loss) With rated voltage on the primary winding and with the secondary winding open-circuited, the magnetizing current ﬂows in the primary winding. The voltage drops in Req and Xeq are very small due to small magnetizing current and the power input is very nearly equal to the core loss. 230 Transformers and Inductors for Power Electronics Ioc + + AC source _ Voc Rc Xc _ Figure 8.9 Transformer open-circuit test. Measure: Voc open-circuit primary voltage Ioc open-circuit primary current Poc open-circuit power (use a wattmeter) The equivalent circuit for these measurements is shown in Figure 8.9. Again, by invoking the analysis of Chapter 4, we may deduce the core parameters. The core reactance is given by: 1¼1þ 1 Zf Rc jXc ð8:15Þ where Rc represents the core loss and Xc represents the magnetizing reactance of the core. 1 ¼ Ioc Zf Voc ð8:16Þ The core resistance is: and the core reactance is: Rc ¼ V 2 oc Poc Xc ¼ sﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 Z 2 f À 1 R2c ð8:17Þ ð8:18Þ These two tests provide sufﬁcient data to characterize the transformer in terms of its equivalent circuit. Measurements 231 Example 8.4 In a 1000 VA, 220: 110 V transformer, the following readings were obtained on the low voltage side of the transformer: Short-circuit test:Vsc ¼ 8:18 V Isc ¼ 9:1 A Psc ¼ 27:4 W Open-circuit test:Voc ¼ 110 V I1 ¼ 0:53 A P1 ¼ 18:5 W Determine the equivalent resistance and reactance of the transformer. The equivalent impedance is: Z eq ¼ 8:18 9:1 ¼ 0:9 V The equivalent resistance is: Req ¼ Psc=Isc2 ¼ 27:4=ð9:1Þ2 ¼ 0:33 V and the equivalent leakage reactance is: qÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÁﬃ qÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÁﬃﬃ Xeq ¼ Z2sc À R2sc ¼ 0:92 À 0:332 ¼ 0:84 V The measurements were taken on the low voltage side, so we must convert these to the high voltage side by multiplying by the square of the turns ratio: Turns ratio ¼ 220/110 ¼ 2 On the high voltage side: RHeq ¼ ð2Þ2ð0:33Þ ¼ 1:32 V XHeq ¼ ð2Þ2ð0:84Þ ¼ 3:36 V Taking the results of the open-circuit test, the core impedance is: Zf ¼ V oc Ioc ¼ 110 0:53 ¼ 207:5 V and the core equivalent resistance is: Rc ¼ V 2 oc Poc ¼ 1102 18:5 ¼ 654 V Determine the core parameters: Xc ¼ v u u tﬃﬃﬃﬃZﬃﬃ1ﬃ2fﬃ1ﬃﬃÀﬃﬃﬃﬃﬃRﬃﬃ1ﬃ2cﬃﬃ!ﬃﬃﬃ ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 207:52 À 1 6542 ¼ 218:8 V 232 Transformers and Inductors for Power Electronics On the high voltage side: RHc ¼ ð2Þ2ð654Þ ¼ 2616 V XHc ¼ ð2Þ2ð218:8Þ ¼ 875:2 V The efﬁciency of the transformer may be obtained from the measurements taken in the open-circuit test. Example 8.5 Find the efﬁciency of the transformer in Example 8.4 when it is connected to a load with a power factor of 0.8. The rated current in the primary (high voltage) winding is: I1 ¼ VA V1 ¼ 1000 220 ¼ 4:55 A The total copper loss is: I21RHeq ¼ ð4:55Þ2ð1:32Þ ¼ 27:3 W The core loss is the power loss measured in the open-circuit test ¼ 18.5 W. The total losses are 27.3 W þ 18.5 W ¼ 45.80 W. The load is connected to the low voltage side and the output current is 4.55 A Â 2 ¼ 9.1 A at 110 V with a power factor of 0.8. The output power is: VIcos w ¼ ð110Þð9:1Þð0:8Þ ¼ 801 W The input supplies this power as well as the losses. The input power is 801 W þ 45.8 W ¼ 846.8 W. The efﬁciency is: Efficiency ¼ Power Out ¼ Power In 801 846:8 ð100Þ ¼ 94:6% 8.3.3 Core Loss at High Frequencies Normally, a wattmeter is used for the transformer tests at line frequency but, at the frequencies encountered in power electronics, a gain-phase meter must replace the wattmeter. The gain-phase analyzer can be readily programmed to measure voltage, current and phase angle and the set-up is shown in Figure 8.10. The core is made from the material under test (the shape of the core is not important). The core should not contain air gaps; recall from Chapter 2 that the air gap determines the level of magnetic ﬁeld intensity in the core. It is generally desirable to measure the core loss over a wide frequency range while maintaining a constant ﬂux level in the core. Recall from Chapter 4, for sine wave excitation, Measurements 233 Control Signal generator Data acquisition Gain-phase v meter i Power amplifier + + vi _ Rref v1 _ Transformer under test + v2 _ vref Figure 8.10 Test set-up for measurement of core loss at high frequency. the transformer equation is: Erms ¼ 4:44f NBmaxAc ð8:19Þ Constant ﬂux can be maintained over the desired range of frequencies by keeping the ratio Erms / f constant. In the case of core loss, the open-circuit test informs our approach. The signal generator in Figure 8.10 sets the frequency for the power ampliﬁer, which is programmed to give the correct output open-circuit voltage for each frequency, based on Equation 8.19. For the purpose of this calculation, Erms in Equation 8.19 is interpreted as the correct open-circuit voltage for each frequency. The input voltage adjustment for frequency is easily implemented in the LabVIEW environment. The voltage is measured in the opencircuit winding to avoid errors associated with voltage drops due to current ﬂow in the test winding. The gain-phase analyzer measures the input current by sampling the voltage vref across the non-inductive resistor Rref, the output voltage v2 and the corresponding phase u between vref and v2. The core loss is now: Pfe ¼ av2 vref Rref cos u ð8:20Þ The primary to secondary turns ratio a is included because the voltage is measured in the secondary winding and the current is measured in the primary winding. The LabVIEW controller may be programmed to present the data in the format shown in Figure 8.11. The core loss is usually expressed in watts per unit volume (kW/m3), as shown in Figure 8.11 for PC40 Mn-Zn ferrite core material at three different ﬂux levels. Since the core heats up as a result of the test, it is important to place the transformer in a controlled oven to stabilize its temperature, and also to ensure that the subsequent test is carried out very quickly. This is easily achieved in a programmable test set-up. 234 1000 100 Core losses ( kW m3 10 ( Transformers and Inductors for Power Electronics 200mT 150mT 100mT 50mT 1 10 100 1000 Frequency (kHz) Figure 8.11 Core loss as a function of frequency at 25 C. The core loss in Figure 8.11 may be expressed in the empirical Steinmetz equation, as described in Chapter 4 using Equation 1.29: Pfe ¼ Kcf aBbmax Example 8.6 Determine the parameters a, b and Kc for the material in whose core loss data is shown in Figure 8.11. Essentially we have three unknowns, so we need three data points. Pick two points at 20 kHz, corre- sponding to maximum ﬂux densities of 50 mT and at 200 mT respectively, and pick the third point at 100 kHz and 200 mT: At point A: f ¼ 20 kHz, Bmax ¼ 50 mT and Pfe ¼ 4.5 kW/m3. At point B: f ¼ 20 kHz, Bmax ¼ 200 mT and Pfe ¼ 90 kW/m3. At point C: f ¼ 100 kHz, Bmax ¼ 200 mT and Pfe ¼ 700 kW/m3. Taking logarithmic values of ratios given by Equation , the following identities apply: 700 ln a¼ 90 100 ¼ 1:275 ln 20 90 b ¼ ln 4:5 ¼ 2:161 200 ln 50 Measurements 10000 1000 Core losses ( kW m3 100 ( 235 100 kHz 40 kHz 20 kHz 10 10 100 B (mT) 1000 Figure 8.12 Core loss as a function of induction level at 25 C. and: Kc ¼ ð100 Â 700 Â 103 103Þ1:275ð0:2Þ2:161 ¼ 9:563 Core loss may also be expressed as a function of induction level for different frequencies. We can generate the core loss versus induction level as shown in Figure 8.12, using the parameters we have deduced above. 8.3.4 Leakage Impedance at High Frequencies The leakage reactance in the short-circuit test can be found by applying a step voltage to the short-circuited transformer through a known inductance Lm (normally a toroid), as illustrated in Figure 8.13. + DC source _ S t=0 Vdc Lm + vm _ 0 Tmax L eq + veq _ Short circuit Transformer under test Figure 8.13 Transformer short-circuit test at high frequency. 236 Transformers and Inductors for Power Electronics Taking the known inductance as Lm and the unknown leakage inductance as Leq, then: V dc ¼ V m þ V eq ð8:21Þ By measuring the voltage across the known inductor, we can deduce the voltage across the equivalent leakage inductance of the transformer. The voltage across the leakage inductance is: V eq ¼ Leq di dt ð8:22Þ The input DC voltage appears across the known inductance and the unknown leakage induc- tance: V in ¼ ðLm þ LeqÞ di dt ð8:23Þ We have neglected the effect of the winding resistance because, at high frequencies, the leakage inductance will dominate the voltage, in which case the ratio of the inductances is directly related to the ratio of the voltages, since di/dt is common. In that case, the leakage inductance is simply: Leq ¼ Veq ¼ Vdc À Vm ¼ Vdc À 1 Lm Vm Vm Vm ð8:24Þ We must ensure that the core of the known toroidal inductor is not saturated. From the analy- sis of Section 2.4.2, in a toroid of N turns and cross-sectional area Ac: Vm ¼ NAc dB dt ð8:25Þ The maximum ﬂux density is reached at Tmax. Therefore, the voltage Vin should be applied for a maximum pulse width of Tmax that satisﬁes: Bmax ¼ Vm NAc T max ð8:26Þ and: T max ¼ NAc Vm Bmax ð8:27Þ As a general rule, the maximum current allowed in the test should be comparable to the nominal current of the transformer. From Ampere’s law of Section 1.2.1, the maximum current in the known toroid is: Imax ¼ Bmaxlc meff m0N ð8:28Þ where meff is the effective permeability of the core. Measurements 237 Example 8.7 The following readings were obtained from a leakage test on a 50 kHz transformer. The input voltage of 5 V was applied to a known inductance of 3 mH, connected in series with a transformer with a shorted secondary. The known inductor was built with 500 turns on a powder iron core (Bmax ¼ 0.3 T, meff ¼ 75) with a cross-sectional area of 1 cm2 and a magnetic path length of 9.8 cm. The voltage measured across the known inductor was 2.3 V. Calculate the width of the applied voltage pulse and the leakage inductance of the transformer. The maximum pulse width is: T max ¼ ð500Þð1:0 Â 2:3 10À4Þ ð0:3Þ Â 103 ¼ 6:5 ms The leakage inductance is: Leq ¼ 3 5 2:3 À 1 ¼ 3:52 mH As a general rule, the maximum current allowed in the test should be comparable to the nominal current of the transformer. From Ampere’s law in Section 1.2.1, the maximum current is: Imax ¼ Bmaxlc meff m0N ¼ ð0:3Þð9:8 Â 10À2Þ ð75Þð4p Â 10À7Þð500Þ ¼ 0:62 A 8.4 Capacitance in Transformer Windings The capacitance of a transformer or inductor winding is of interest in power electronic switching circuits, because the capacitance may provide a short circuit to a step change in voltage, causing over-current circuitry to trip. Flyback circuits are particularly prone to this problem. The second issue relates to resonances in the windings, so the designer needs to know the effective capacitance of the windings. Figure 8.14 shows the classical lumped-parameter representation of a transformer winding. The mutual impedance between winding sections i and j is represented by Zij. Z ij C1 C1 C1 C1 C0 2 C0 C0 C0 C0 C0 2 Figure 8.14 Lumped parameter transformer model. 238 Transformers and Inductors for Power Electronics C0 represents the distributed shunt capacitance between each section and the core, and C1 represents the series capacitance between sections of the winding. The winding is continuous, but it is normal to discretize the winding for analysis. 10–20 sections are normally sufﬁcient to represent the winding accurately, i.e. as regards natural resonant frequencies. The model is similar to that of a transmission line, except that mutual coupling exists between sections and there is also series capacitance. The presence of mutual coupling makes transformer analysis much more complex than transmission line analysis. 8.4.1 Transformer Effective Capacitance For n sections of winding, the total shunt capacitance is: Cg ¼ nC0 ð8:29Þ and the total series capacitance is: Cs ¼ C1 n ð8:30Þ The network of capacitances C0 and C1 in Figure 8.14 has an equivalent capacitance Ceff [1]. This is the equivalent capacitance placed across the transformer terminals that would draw the same charging current as the capacitance distribution in the transformer winding shown in Figure 8.14. The effective capacitance is given by: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ceff ¼ CgCs ð8:31Þ Ceff cannot be measured directly due to the presence of the coil inductance. However, it can be measured by suddenly discharging a known capacitance Ck into the winding and recording the instantaneous voltage drop, as shown in Figure 8.15. By conservation of charge: Ceff ¼ Ck DE Eo À DE ð8:32Þ transformer Rs t=0 winding Eo Ck Figure 8.15 Circuit to measure effective capacitance. Measurements 239 E = 1.78 V ΔE = 0.83 V 1 200 mV/div 100 μs/div Figure 8.16 Measurement of effective capacitance. where Eo is the initial voltage on Ck and DE is the drop in voltage at the instant of discharge [1]. Rs is chosen so that the time constant for charging Ck is much longer than the transient time in the winding after the switch is closed. Due to the inductance of the winding, the circuit will oscillate after the initial discharge and, eventually, the capacitances will recharge to Eo through Rs. Ck is normally chosen to be of the same order of magnitude as Ceff. Example 8.8 Calculate the effective capacitance of the transformer for which the test results of Figure 8.16 were obtained. The value of the known capacitance is 500 pF. Ceff ¼ Ck DE Eo À DE ¼ ð500Þ 0:83 1:78 À 0:83 ¼ 437 pF 8.4.2 Admittance in the Transformer Model The admittance terms in the transformer model of Figure 8.14 are made up of the shunt capacitance C0 and the series capacitance C1. The individual capacitances that make up the shunt capacitance C0 and the series capacitance C1 may be comprised of many different combinations of sections of windings. However, the capacitance values in most conﬁgurations can be calculated using three formulae for the geometries shown in Figure 8.17. Co-Axial Cylinders This conﬁguration consists of a cylinder of radius r1 inside a cylinder of radius r2, and the radius of the boundary between the two dielectric mediums is R. The capacitance between the cylinders is: 240 Transformers and Inductors for Power Electronics R r1 r2 D r2 r1 Medium 1 Medium 2 (a) (b) Figure 8.17 Geometries for capacitance calculations (a) co-axial cylinders, (b) external cylinders. C ¼ lnerrR112pþe0llcnerrR22 ð8:33Þ lc is the cylinder length and er1 and er2 are the relative permittivities of the dielectrics in mediums 1 and 2 respectively. Typically, one medium is air and the other medium is insulation. External Cylinders This conﬁguration consists of two cylinders of radius r1 and r2 respectively, at a distance D apart. The capacitance between the cylinders is: C ¼ coshÀ1&2Dpe2rÀe0lrc21 À r22' 2r1r2 ð8:34Þ Parallel Plates In many cases, the radius of curvature of a winding is much greater than the separation of the windings under consideration. In this case, it is perfectly reasonable to assume parallel plates. This conﬁguration consists of two parallel plates of area A with a com- posite dielectric of thicknesses d1 and d2. The corresponding dielectric constants are er1 and er2 respectively. C ¼ e0A d1 er1 þ d2 er2 ð8:35Þ The approach is best illustrated by the following examples. Measurements C1 C1 241 C1 47 71 91 30 Figure 8.18 Disc winding dimensions. Example 8.9 A high-voltage disc winding is illustrated in Figure 8.18. The winding consists of 16 discs in series and each disc winding has 12 turns tightly wound in two layers as shown. The disc dimensions are: HV winding inside radius ¼ 71.0 mm HV winding outside radius ¼ 91.0 mm HV winding conductor ¼ 5 Â 3.15 mm Width of disc winding ¼ 12.06 mm Insulation thickness ¼ 0.08 mm Dielectric constant insulation ¼ 3.81. Calculate the series capacitance of the winding. The individual capacitances that make up the series capacitance C1 of the disc are shown in Figure 8.19. Cd is the capacitance between each of the adjacent conductors in a disc winding pair and Ct is the capacitance between conductors within a disc. The conductors in Figure 8.19 have rectangular cross-sections; it is equally likely that they may be circular conductors. In general, both Cd and Ct will depend on the position of the conductor within the disc but, for convenience, it is reasonable to assume average dimensions without a signiﬁcant loss of accuracy. The shunt capacitance C1 is then the equivalent capacitance of the network above. Both capacitances in Figure 8.19 may be found from the parallel plate formula as follows: The average radius of the disc is 81 mm and the width of the conductors that make up Ct is 5 mm. Therefore, the area of the capacitor plate is: A ¼ 2pRd ¼ ð2pÞð81Þð5Þ ¼ 2545 mm2 There are two layers of insulation so the plate separation is 0.16 mm and the capacitance is obtained from Equation 8.35 with one medium: Ct ¼ ð3:81Þð8:854 Â 10À12Þð2545 0:16 Â 10À3 Â 10À6Þ Â 1012 ¼ 536 pF 242 Transformers and Inductors for Power Electronics Ct Ct Ct Ct Ct Cd Cd Cd Cd Cd Cd Ct Ct Ct Ct Ct Figure 8.19 Geometry for series capacitance. The capacitance Cd between the parallel conductors that make up the disc may be modelled as parallel plates. The total width of the disc is 12.06 mm and the conductors and insulation make up 10.16 mm, leaving 0.16 mm of insulation (dielectric constant 3.81) and 1.74 mm of air. The area of the plate is: A ¼ 2pRt ¼ ð2pÞð81Þð3:15Þ ¼ 1603 mm2 The capacitance is then given by Equation 8.35: Cd ¼ ð8:854 Â 10À12Þð1603 Â 10À6Þ 0:16 Â 10À3 3:81 þ 1:74 Â 10À3 Â 1012 ¼ 8:0 pF The series capacitance C1 across the winding in Figure 8.19 is now: C1 ¼ Cd þ 5Cd Ct 2Cd þ Ct ¼ 8:0 þ ð5Þð8:0Þð536Þ ð2Þð8:0Þ þ 536 ¼ 46:84 pF Example 8.10 Figure 8.20 shows a transformer layout consisting of a primary winding with 39 turns in two layers, and a seven-turn secondary with a reset winding of three turns. The primary winding has two parallel 0.5 mm wires. The screens are made of 0.035 mm copper foil. The main dimensions are shown. Estimate the effective capacitance of the primary winding when measured between the input to the primary winding and the screen with the other end of the winding in open circuit. The individual capacitances that make up the shunt capacitance C0 of the primary winding is may be deﬁned as follows: C0g is the capacitance between the primary winding and the screen W1; C0g0 is the capacitance between the primary winding and screen W3; C1 is the series capacitance of the primary winding; The shunt capacitance C0 of Figure 8.14 is the parallel combination of C0g and C0g0. C0 in this case is the same as C0 in Equation 8.29. The series capacitance Cs in this case is the same as C1. Measurements 243 W6 reset W5 secondary W4 screen W3 screen W2 primary W1 screen Reset (3 turns) Insulation Secondary (7 turns) Insulation Cg'' … Primary (19 turns) C1 … Insulation Cg' Insulation Core Figure 8.20 Transformer dimensions. 10.235 10.344 11.514 11.648 C0g is made of two concentric cylinders with r1 ¼ 10.235 mm, the insulation thickness is 0.075 mm with a relative permittivity of 3.8, and there is an extra 0.034 mm layer of wire insulation, giving r2 ¼ 10.344 mm. The length of the winding is approximately 20 mm, based on 19 turns of two 0.5 mm wires in parallel. C0g ¼ ð2pÞð3:8Þð8:854 Â 10À12Þð20 10:344 Â 10À3Þ Â 1012 ¼ 399 pF ln 10:235 C0g0 is made of two concentric cylinders with r1 ¼ 11.514 mm, the insulation thickness is 0.100 mm with a relative permittivity of 3.8, and there is an extra 0.034 mm layer of wire insulation, giving r2 ¼ 11.648 mm. C0g0 ¼ ð2pÞð3:8Þð8:854 Â 10À12Þð20 11:648 Â 10À3Þ Â 1012 ¼ 365 pF ln 11:514 Cg ¼ C0g þ C0g0 ¼ 399 þ 365 ¼ 764 pF The capacitance between the two layers of the primary winding may be considered a parallel plate capacitor, since the separation between the layer is much smaller than the radius of the windings. The 244 Transformers and Inductors for Power Electronics insulation thickness between the layers is 0.1 mm and there are two extra 0.034 mm layers of wire insulation, giving d ¼ 0.168 mm. The mean diameter is 21.858 mm, but this does not take the air medium created by the wire shape into account. From Equation 8.35: Cs ¼ ð3:8Þð8:854 Â 10À12ÞðpÞð21:858 Â 0:169 Â 10À3 10À3Þð20 Â 10À3Þ Â 1012 ¼ 273 pF Finally the effective capacitance of the winding is obtained form Equation 8.31 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ceff ¼ 764 Â 273 ¼ 457 pF 8.5 Problems 8.1 Calculate the value of the inductance for which the data in Table 8.3 was generated. The coil resistance was 2.5 V and the applied voltage was 72 V. 8.2 Reconstruct the initial permeability versus magnetic ﬁeld intensity curve for the material in the core tested in Example 8.2. 8.3 A single phase, 100 KVA 1000/100 V transformer gave the following test results with the instruments connected to the high voltage side: Open-circuit test: 1000 V, 0.6 A, 400 W Short-circuit test: 50 V, 100 A, 1800 W (a) Determine the rated voltage and rated current for the high-voltage and low-voltage sides. (b) Derive the approximate equivalent circuit referred to the HV side. 8.4 Table 8.4 shows the measurement data for an inductor obtained from the incremental impedance method. Calculate the AC inductance as a function of DC bias. The resistance of the coil was 4.0 V and the measurements were carried out at 50 Hz. 8.5 Devise a method to estimate the number of turns in a transformer winding based on a turns ratio. Table 8.3 Inductance measurements k tk(ms) 1 1 2 2 3 3 4 4 5 5 i(k) (A) 0.5 1.6 6.0 14.6 17.3 Measurements Table 8.4 Inductance measurements Idc(A) 0.000 0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.250 3.500 3.750 4.000 vac(V) 1.090 1.086 1.092 1.119 1.143 1.169 1.181 1.176 1.148 1.119 1.085 1.051 1.024 0.998 0.978 0.960 0.934 iac(mA) 42.90 42.10 47.00 60.40 80.10 104.00 126.00 151.00 175.00 190.00 199.00 204.00 208.00 212.00 219.00 222.00 231.00 Z(V) 25.41 25.80 23.23 18.53 14.27 11.24 9.37 7.79 6.56 5.89 5.45 5.15 4.92 4.71 4.47 4.32 4.04 245 Lac(mH) 79.9 81.2 72.9 57.6 43.6 33.5 27.0 21.3 16.6 13.8 11.8 10.3 9.1 7.9 6.3 5.2 1.9 Reference 1. Heller, B. and Veverka, A. (1968) Surge Phenomena in Electrical Machines, Iliffe Books, London. Further Reading 1. Bi, S., Sutor, A., and Yue, J. (2008) Optimization of a measurement system for the hysteretic characterization of high permeable materials. Proceedings of the Virtuelle Instrumente in der Praxis VIP 2008, pp. 22–30. 2. Dalessandro, L., da Silveira Cavalcante, F., and Kolar, J.W. (2007) Self-capacitance of high-voltage transformers. IEEE Transactions on Power Electronics 22 (5), 2081–2092. 3. Dauhajre, A. and Middlebrook, R.D. (1986) Modelling and estimation of leakage phenomenon in magnetic circuits. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 213–226. 4. Gradzki, P.M. and Lee, F.C. (1991) High-frequency core loss characterization technique based on impedance measurement. Proceedings of the Virginia Power Electronic Center Seminar, VPEC, pp. 1–8. 5. Hai Yan, L., Jian Guo, Z., and Hui, S.Y.R. (2007) Measurement and modeling of thermal effects on magnetic hysteresis of soft ferrites. IEEE Transactions on Magnetics 43 (11), 3952–3960. 6. Hui, S.Y.R. and Zhu, J. (1995) Numerical modelling and simulation of hysteresis effects in magnetic cores using transmission-line modelling and the Preisach theory. IEE Proceedings-Electric Power Applications B 142 (1), 57–62. 7. Hurley, W.G. and Wilcox, D.J. (1994) Calculation of leakage inductance in transformer windings. IEEE Transactions on Power Electronics 9 (1), 121–126. 8. Hurley, W.G., Wilcox, D.J., and McNamara, P.S. (1991) Calculation of short circuit impedance and leakage impedance in transformer windings. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 651–658. 9. Jiles, D.C. and Atherton, D.L. (1984) Theory of ferromagnetic hysteresis (invited). Journal of Applied Physics 55 (6), 2115–2120. 10. Kazimierczuk, M.K. (2009) High-Frequency Magnetic Components, John Wiley & Sons, Chichester. 246 Transformers and Inductors for Power Electronics 11. Kis, P., Kuczmann, M., F€uzi, J., and Ivanyi, A. (2004) Hysteresis measurement in LabView. Physica B: Condensed Matter, 343 (1–4), 357–363. 12. Kuczmann, M. and Sarospataki, E. (2006) Realization of the Jiles-Atherton hysteresis model applying the LabVIEW and MATLAB software package. Journal of Electrical Engineering 57 (8), 40–43. 13. Kulkarni, S.V. (2004) Transformer Engineering: Design and Practice, 1st edn, CRC Press, New York. 14. McLyman, C.W.T. (1997) Magnetic Core Selection for Transformers and Inductors, 2nd edn, Marcel Dekker Inc., New York. 15. McLyman, C.W.T. (2002) High Reliability Magnetic Devices, 1st edn, Marcel Dekker Inc., New York. 16. McLyman, C.W.T. (2004) Transformer and Inductor Design Handbook, 3rd edn, Marcel Dekker Inc., New York. 17. E.S. MIT (1943) Magnetic Circuits and Transformers (MIT Electrical Engineering and Computer Science), The MIT Press, Cambridge, MA. 18. Muhlethaler, J., Biela, J., Kolar, J.W., and Ecklebe, A. (2012) Improved core-loss calculation for magnetic com- ponents employed in power electronic systems. IEEE Transactions on Power Electronics 27 (2), 964–973. 19. Muhlethaler, J., Biela, J., Kolar, J.W., and Ecklebe, A. (2012) Core losses under the DC bias condition based on steinmetz parameters. IEEE Transactions on Power Electronics 27 (2), 953–963. 20. Prabhakaran, S. and Sullivan, C.R. (2002) Impedance-analyzer measurements of high-frequency power pas- sives: techniques for high power and low impedance. Conference Record of the Industry Applications Conference, IAS, pp. 1360–1367. 21. Snelling, E.C. (1988) Soft Ferrites: Properties and Applications, 2nd edn, Butterworths, London. 22. Steinmetz, C.P. (1984) On the law of hysteresis. Proceedings of the IEEE 72 (2), 197–221. 23. Van den Bossche, A. (2005) Inductors and Transformers for Power Electronics, 1st edn, CRC Press, New York. 24. Venkatachalam, K., Sullivan, C.R., Abdallah, T., and Tacca, H. (2002) Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters. Proceedings of IEEE Workshop on Computers in Power Electronics, COMPEL, pp. 36–41. 9 Planar Magnetics1 The relentless drive towards high-density electronic circuits has been a feature of microelectronics for several decades and, typically, improvements in power efﬁciency have paralleled the shrinkage of electronic circuits. The effects are obvious in IC design; component densities continue to increase, with no sign of letting up. The latest trends in microelectronic techniques and nanotechnology, such as thick ﬁlm and thin ﬁlm technologies, are being pushed by the requirements of microelectronics to reduce size and cost and to improve reliability. By extension, low-proﬁle planar magnetic components can be incorporated into the production processes that are already used to fabricate resistors and capacitors, and magnetic materials that are suitable to meet these requirements are constantly under development. One of the major drawbacks in establishing planar magnetic technology is the lack of accurate analytical models for typical structures. Prototypes are expensive to fabricate and test and, while they offer useful insights into speciﬁc designs, they do not always extend to the general case. Analytical models that take account of frequency-dependent losses in the magnetic materials are needed to enhance our understanding of planar magnetics [1–3]. The reduction in the size of magnetic devices is essential for further overall miniaturisation and increased functionality of power conversion systems. Conventional magnetics using bobbins are bulky and the assembly process is difﬁcult to automate. Planar magnetics fabrication and assembly processes have several advantages over conventional magnetics: Low proﬁle: due to the fabrication process, planar magnetic components have a lower proﬁle than their wire-wound counterparts. 1 Parts of this chapter are reproduced with permission from [1] Hurley, W.G. and Duffy, M.C. (1995) Calculation of self and mutual impedances in planar magnetic structures. IEEE Transactions on Magnetics 31 (4), 2416–2422; [2] Hurley, W.G. and Duffy, M.C. (1997) Calculation of self- and mutual impedances in planar sandwich inductors. IEEE Transactions on Magnetics 33 (3), 2282–2290; [3] Hurley, W.G., Duffy, M.C., O’Reilly, S, and O’Mathuna, S.C. (1999) Impedance formulas for planar magnetic structures with spiral windings. IEEE Transactions on Industrial Electronics 46 (2), 271–278; [4] Wang, N., O’Donnell, T., Meere, R. et al. (2008) Thin-ﬁlm-integrated power inductor on Si and its performance in an 8-MHz buck converter. IEEE Transactions on Magnetics 44 (11), 4096– 4099; [5] Mathuna, S.C.O., O’Donnell, T., Wang, N., and Rinne, K. (2005) Magnetics on silicon: an enabling technology for power supply on chip. IEEE Transactions on Magnetics 20 (3), 585–592. Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 248 Transformers and Inductors for Power Electronics Automation: it is difﬁcult to automate the winding of conventional inductors and transformers, whereas the processes used in planar magnetics are based on advanced computeraided manufacturing techniques. High power densities: planar inductors and transformers are spread out, which gives them a bigger surface-to-volume ratio than conventional components. The increased surface area enhances the thermal performance and this, in turn, means that the power density is increased. Increased heat convection also means that higher operating frequencies are achievable. Predictable parasitics: in wire-wound components, it is very difﬁcult to control the winding layout, which can mean more leakage effects and winding capacitance. As a result, signiﬁcant variations in these electrical parameters appear in devices manufactured at the same time. With planar magnetics, the windings are precise and consistent, yielding magnetic designs with highly controllable and predictable characteristic parameters. The size of magnetic components may be reduced by operating at high frequency. Planar magnetic components use this principle to reduce component size while taking advantage of microelectronic processing. The magnetic materials used in planar magnetics have ﬁnite conductivity and, at sufﬁciently high frequency, unwanted eddy current loss appears. In general, the number of turns in a planar device tends to be limited by the manufacturing process. The low proﬁle tends to lead to a larger footprint compared with its conventional counterpart. The spreading effect leads to high capacitance between layers and between the windings and the core. In multilayer devices, the interlayer capacitance introduces resonance at high frequencies. When considering conventional versus planar devices, several issues must be addressed: the shape; the trade-off between magnetic core area and winding window area; the magnetic path length versus the mean length of a turn; the surface area. Planar magnetics have opened up new applications such as coreless transformers for gate drives, radio frequency (rf) inductors and a Power Supply on Chip (PwrSoC). The circuit models have not necessarily changed, but the new layouts of windings and cores require new models for inductance and loss mechanisms. 9.1 Inductance Modelling The mutual inductance between two ﬁlaments in air was treated in Chapter 2. This was extended to formulae for mutual inductance between coils with ﬁnite cross-sections by integrating the ﬁlamentary formula over the cross-section; the current density was taken as uniform over the section. Accurate formulae emerged, but the lack of computing power meant that results were often presented in look-up tables. Another approach was to develop approximations based on the ﬁlament formula, with judicious choice of ﬁlament placement. In the case of planar magnetic components, the current density is not uniform because the aspect ratio of width to height of a section is typically very large. This means that the length of the conducting path is much shorter on the inside of a ﬂat wide coil and, consequently, the Planar Magnetics 249 current density is higher. The accuracy of the classic formulae is much improved when the proper current density is included in the analysis. The starting-point for a planar magnetic component is the air-cored spiral inductor. Despite its physical simplicity, this forms the basis for more advanced conﬁgurations, such as coils on magnetic substrates [1] and sandwich inductors [2], where the coil is placed between two magnetic substrates. The analysis begins with a formula for the mutual inductance between two planar spiral coils in air that takes full account of the current density distribution across the planar section. The result can be extended to a component with several turns per layer and with several layers. The next step is to add a magnetic substrate. A ferromagnetic substrate with ﬁnite conductivity introduces eddy current loss and hysteresis loss that add to the winding resistance loss. A frequency-dependent mutual impedance formula for this case is derived, which takes the eddy current loss into account. Finally, we will add a second substrate layer in a sandwich conﬁguration. 9.1.1 Spiral Coil in Air The mutual inductance between two ﬁlaments shown in Figure 9.1 is the basis for establish- ing the general mutual inductance formula for planar structures [1]. The formula has the form: ð1 M ¼ m0par J1ðkrÞJ1ðkaÞeÀkjzjdk 0 ð9:1Þ where J1 is a Bessel function of the ﬁrst kind, a and r are the ﬁlament radius and m0 is the permeability of free space. An alternative to Equation 9.1 can be written in terms of elliptic integrals: M ¼ pﬃﬃﬃﬃ m0 ar 2 f 1 À f 2 K ðf Þ À Eðf ! Þ 2 ð9:2Þ where K(f) and E(f) are complete elliptic integrals of the ﬁrst and second kind, respectively, and where: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f¼ 4ar z2 þ ða þ rÞ2 ð9:3Þ a z r Figure 9.1 Circular concentric ﬁlaments in air. 250 Transformers and Inductors for Power Electronics Coil 2 w2 a1 a2 z h2 r1 r2 Coil 1 h1 w1 Figure 9.2 Planar coils of rectangular cross-section. The next step is to consider two circular and concentric planar sections, as depicted in Figure 9.2, with dimensions as shown. In practice, a spiral coil would connect two sections in series, which can be accurately modelled by the concentric circular coils as illustrated. In Chapter 2, the mutual inductance between sections was found by integrating the ﬁlamentary formula (Equation 9.1) over each cross-section, with the assumption that the current density is constant over each section. The approach is adequate when the width and height of the coil section are approximately equal. In a planar structure, however, the ratio of width to height could be as big as 50 : 1. The shorter path on the inside edge of the conducting section means that the resistance to current ﬂow is lower, and therefore the current density is higher on the inside than on the outside. On the basis of this observation, it is reasonable to assume that there is an inverse relation- ship between the current density J(r) and the radius r. By the same token, current density in the z direction may be taken as constant because of the low proﬁle. Integrating the current density over the cross-section yields the total current I: ðr2 h ¼ JðrÞdr ¼ I ð9:4Þ r1 The current density has the form: JðrÞ ¼ K r ð9:5Þ with K a constant. Solving Equations 9.4 and 9.5 over a cross-section with an inside radius of r1, outside radius r2 and height h yields: JðrÞ ¼ I hrln r2 r1 ð9:6Þ In the following analysis, the current is sinusoidal: Jfðr; tÞ ¼ JðrÞejvt where v is the angular frequency. ð9:7Þ Planar Magnetics 251 The induced voltage in the ﬁlament at (r, t1) in coil 1 due to the current in an annular section da Â dt2 at radius a in coil 2 is: dV ¼ jvMJðaÞ da dt2 ð9:8Þ where M is the mutual inductance between the ﬁlaments at (r, t1) and (a, z þ t2). Integrating Equation 9.8 over the cross-section of coil 2 yields the voltage at (r, t1) due to all the current in coil 2: V ðrÞ ¼ jvm0pr ð1 0 ðh2 2 Àh2 2 ða2 a1 aJ ðaÞJ 1 ðkrÞJ 1 ðkaÞeÀkjzþt2 Àt1 j d a dt2 dk ð9:9Þ The power transferred to the annular segment at (r, t1) due to coil 2 is: dP ¼ VðrÞJðrÞdr dt1 ð9:10Þ The total power transferred to coil 1 may be obtained by integrating Equation 9.10 over its cross-section: P ¼ jvm0p ð1 0 ð h1 2 Àh1 2 ðh2 2 Àh2 2 ðr2 r1 ða2 a1 rJ ðrÞJ 1 ðkrÞaJ ðaÞJ 1 ðkaÞeÀkjzþt2 Àt1 j d a dr dt1 dt2 dk ð9:11Þ The internal integrals are readily solved, with J(r) given by Equation 9.6, to yield: P ¼ jvm0p I1I2 h1ln r2 r1 h2ln a2 a1 ð1 0 Sðkr2; kr1ÞSðka2; ka1ÞQðkh1; kh2 ÞeÀkjzj d k ð9:12Þ where: Qðkx; kyÞ ¼ 2 k2 h cosh k x þ 2 y À cosh k x À 2 yi ¼ 2 h þ eÀkh À 1 z ¼ 0; x ¼ y ¼ h k k z > h1 þ h2 2 ð9:13Þ and: Sðkx; kyÞ ¼ J0ðkxÞ À J0ðkyÞ k ð9:14Þ but: P ¼ v2i2 ¼ jvM12I1I2 ð9:15Þ 252 Transformers and Inductors for Power Electronics where M12 is the mutual inductance between the two coils. Substituting Equation 9.15 establishes M12: M12 ¼ m0p h1h2ln r2 r1 ln a2 a1 ð1 0 Sðkr2; kr1ÞSðka2; ka1ÞQðkh1; kh2 ÞeÀkjzj d k ð9:16Þ This is the ﬁnal result and, despite its complex appearance, it is easily solved with numerical integration using MATLAB. Combined with a simple formula for the DC resistance, the equivalent circuit model includes a resistance in series with an inductance given by Equation 9.16. Example 9.1 Calculate the self inductance of the four-turn planar coil shown in Figure 9.3. This conﬁguration is commonly used for inductive charging in electric vehicles. The device consists of four planar coils in series. We need to calculate the self inductance of coil 1 (same as the self inductance of coil 2) and the self inductance of coil 3 (same as the self inductance of coil 4). Next, we need the mutual inductances M12, M13 (same as M24), M14 (same as M23) and M34. The required dimensions for the self inductance calculations are: Coil 1 : r1 ¼ a1 ¼ 1:15 mm; r2 ¼ a2 ¼ 1:75 mm; h1 ¼ h2 ¼ 15 mm; z ¼ 0 Coil 2 : r1 ¼ a1 ¼ 1:15 mm; r2 ¼ a2 ¼ 1:75 mm; h1 ¼ h2 ¼ 15 mm; z ¼ 0 Coil 3 : r1 ¼ a1 ¼ 2:00 mm; r2 ¼ a2 ¼ 2:60 mm; h1 ¼ h2 ¼ 15 mm; z ¼ 0 Coil 4 : r1 ¼ a1 ¼ 2:00 mm; r2 ¼ a2 ¼ 2:60 mm; h1 ¼ h2 ¼ 15 mm; z ¼ 0 5.2 mm 2.3 mm 2 3 1 4 600μ z 250μ 40μ 15μ r Figure 9.3 Layout of a planar device. Planar Magnetics 253 The required dimensions for the mutual inductance calculations are: Coils 1 and 2 : r1 ¼ a1 ¼ 1:15 mm; r2 ¼ a2 ¼ 1:75 mm; h1 ¼ h2 ¼ 15 mm; z ¼ 55 mm Coils 1 and 3 : r1 ¼ 1:15 mm; r2 ¼ 1:75 mm; h1 ¼ 15 mm a1 ¼ 2:00 mm; a2 ¼ 2:60 mm; h2 ¼ 15 mm z ¼ 55 mm Coils 1 and 4 : r1 ¼ 1:15 mm; r2 ¼ 1:75 mm; h1 ¼ 15 mm a1 ¼ 2:00 mm; a2 ¼ 2:60 mm; h2 ¼ 15 mm z¼0 Coils 2 and 3 : same as coils 1 and 4 Coils 2 and 4 : same as coils 1 and 3 Coils 3 and 4 : r1 ¼ a1 ¼ 2:00 mm; r2 ¼ a2 ¼ 2:60 mm; h1 ¼ h2 ¼ 15 mm; z ¼ 55 mm The total inductance is: L ¼ 2L11 þ 2L33 þ 2M12 þ 4M13 þ 4M14 þ 2M34: The MATLAB program to carry out the calculations is listed at the end of this chapter. The program yielded the following results: L11 ¼ 4.366 nH; L33 ¼ 8.320 nH; M12 ¼ 3.956 nH; M13 ¼ 2.223 nH; M14 ¼ 2.229 nH and M34 ¼ 9.327 nH. The total inductance of the four-turn coil is 69.75 nH. 9.1.2 Spiral Coil on a Ferromagnetic Substrate The self inductance of the coil in Figure 9.2 will be enhanced by adding a magnetic substrate. Placing the coil on an ideal magnetic substrate of inﬁnite thickness would double the overall inductance of the coil. Currents ﬂowing in the coil induce eddy currents in a core material with ﬁnite conductivity. A general impedance formula that takes frequency-dependent eddy current loss in the substrate into account is required to model the loss for highfrequency operation. The geometry for the mutual impedance between two ﬁlaments placed above a magnetic substrate is shown in Figure 9.4, with appropriate dimensions. The lower ﬁlament in Figure 9.4 is at a height d1 above the substrate of thickness t, so that an insulating layer between a coil and the substrate can be modelled. The mutual impedance between the two circular concentric ﬁlaments in Figure 9.4 is: Z ¼ jvM þ Z f t ð9:17Þ where M is the mutual inductance that would exist in the absence of the substrate, and is the same as Equation 9.1. 254 Transformers and Inductors for Power Electronics z Medium 1 r z = d2 a z = d1 μr,σ t Dielectric layer Magnetic substrate Medium 2 z=0 Figure 9.4 Circular concentric ﬁlaments on a magnetic substrate. The additional impedance due to the presence of the substrate may be found by solving Maxwell’s equations. For a magnetoquasistatic system, the following forms of Maxwell’s equations hold in a linear homogeneous isotropic medium: r Â H ¼ Jf r Â E ¼ À @B @t ð9:18Þ ð9:19Þ The ﬁlamentary turn at z ¼ d1 in Figure 9.4 carries a sinusoidal current if(t) ¼ If ejvt. Medium 1 may be air or a dielectric, and medium 2 is the magnetic substrate. The solution is divided into four regions: region 1 (z ! d1); region 2 (0 z < d1); region 3 (–t z < 0); and region 4 (z < –t). Regions 1, 2 and 4 corresponds to medium 1 and region 3 corresponds to medium 2. The following identities apply to the electric ﬁeld intensity E and the magnetic ﬁeld inten- sity H because there is cylindrical symmetry: Er ¼ 0; Ez ¼ 0; @Ef @f ¼ 0 Hf ¼ 0; @Hr @f ¼ 0; @Hz @f ¼ 0 ð9:20Þ ð9:21Þ Maxwell’s equations in each region may be shown to be: Region 1 (z ! d1): @Hr @z À @Hz @r ¼ Ifdðr À aÞdðz À d1Þ @Ef @z ¼ jvm0Hr ð9:22Þ ð9:23Þ Planar Magnetics 255 1 r @ðrEfÞ @r ¼ Àjvm0Hz ð9:24Þ Eliminating H gives the following result for Ef: @2Ef @z2 þ @2Ef @r2 þ 1 r @Ef @r À Ef r2 ¼ jvm0Ifdðr À aÞdðz À d1Þ Region 2 (0 z < d1): There is no current in this region and the electric ﬁeld is given by: ð9:25Þ @2E @z2 þ @2E @r2 þ 1 r @E @r À E r2 ¼ 0 Region 3 (–t z < 0): In this region, Jf ¼ sEf and the electric ﬁeld is: @2Ef @z2 þ @2Ef @r2 þ 1 r @Ef @r À 1 r2 þ jvmrm0s Ef ¼ 0 ð9:26Þ ð9:27Þ Region 4 (z < –t): This is similar to region 2 and Equation 9.26 again applies. The solution of Ef in regions 1–4 is obtained by invoking the Fourier Bessel integral transformation [6]: ð1 EÃ ¼ EfðrÞrJ1ðkrÞdr 0 ð9:28Þ and noting that: ð1 dðr À aÞJ1ðkrÞr dr ¼ aJ1ðkaÞ 0 ð9:29Þ resulting in the transformed version of Equation 9.27: d2EÃ d z2 ¼ k2E Ã þjvm0If a J1ðkaÞ dðz À d1Þ ð9:30Þ The solution is of the form: EÃ ¼ Iekz þ AeÀkz ð9:31Þ I and A are constants to be determined by the boundary conditions. I ! 0 since EÃ ! 0 at infinity: The solution in EÃ in each region becomes: Region 1 z ! d1 EÃ ¼ AeÀkz ð9:32Þ 256 Transformers and Inductors for Power Electronics Region 2 0 z < d1 EÃ ¼ Bekz þ CeÀkz Region 3 À t z < 0 EÃ ¼ DeLz þ FeÀLz Region 4 z < Àt EÃ ¼ Gekz ð9:33Þ ð9:34Þ ð9:35Þ with: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ L ¼ k2 þ jvmrm0s ð9:36Þ There are six constants to be established on the basis of the boundary conditions. The electric ﬁeld is continuous at the following boundaries: z ¼ d1, z ¼ 0 and z ¼ –t, giving: AeÀkd1 ¼ Bekd1 þ CeÀkd1 BþC ¼ DþF DeÀLt þ FeLt ¼ GeÀkt ð9:37Þ ð9:38Þ ð9:39Þ The boundary condition imposed on the radial component of the magnetic ﬁeld intensity is given by: n Â ðHþ À HÀÞ ¼ Kf ð9:40Þ where n is the unit vector normal to the plane at the boundary and Kf is the surface current density at the boundary. The radial component of the magnetic ﬁeld intensity is given by Maxwell’s equations: @Ef @z ¼ jvmrm0Hr ð9:41Þ At z ¼ 0 and z ¼ –t, there is no surface current, and equating Hr given by Equation 9.41 at either side of the boundary yields: L kðB À CÞ ¼ ðD À FÞ mr ð9:42Þ and: L ðDeÀLt À FeLtÞ ¼ kGeÀkt mr ð9:43Þ At z ¼ d1 the surface current density is: ð d 1þ Kf ¼ Ifdðr À aÞdðz À d1Þdz ¼ Ifdðr À aÞ d 1À ð9:44Þ Planar Magnetics 257 and in terms of the transformed variable: KÃf ¼ IfaJ1ðkaÞ ð9:45Þ Hþ and H– may be found by using Equation 9.42 in Equation 9.33 and 9.34: ÀAeÀkd1 À Bekd1 þ CeÀkd1 ¼ jvm0 k I f aJ1 ðkaÞ ð9:46Þ It now remains to solve six equations in six unknowns. Our primary interest is in the mutual impedance between the ﬁlaments in Region 1, where the electric ﬁeld is: EÃ ¼ Àjvm0If aJ1ðkaÞ 2k h eÀkjzÀd 1 j þ i lðtÞeÀkjzþd1j ð9:47Þ l(t) is deﬁned as: lðtÞ ¼ fðkÞ 1 1 À eÀ2Lt À fðkÞ2eÀ2Lt ð9:48Þ with: fðkÞ ¼ mr À L k L mr þ k ð9:49Þ Applying the inverse transform of the Fourier-Bessel integral deﬁned as: ð1 EðrÞ ¼ EÃðkÞkJ1ðkrÞdk 0 ð9:50Þ results in an expression for E(r,z): Eðr; zÞ ¼ Àjvm0Ifa 1 2 ð1 0 J1ðkrÞJ1ðkaÞ Á ÂeÀkjzÀd1j þ lðtÞeÀkjzþd 1 j Ã d k ð9:51Þ The induced voltage in a circular ﬁlament at (r, d2) due to the source at (a, d1) is V ¼ ZIf and: ð2p V ¼ À Eðr; d2Þ rdf ¼ À2prEðr; d2Þ 0 ð9:52Þ It follows that: Z ¼ À 2pEðr; d2Þ If ð9:53Þ 258 Transformers and Inductors for Power Electronics and: Z ¼ jvM þ Z f t ð1 M ¼ m0par J1ðkrÞJ1ðkaÞeÀkjd2Àd1jdk 0 ð1 Zft ¼ Rs þ jvLs ¼ jvm0par J1ðkrÞJ1ðkaÞlðtÞeÀkðd1þd2Þdk 0 ð9:54Þ ð9:55Þ ð9:56Þ In practice, if the substrate is at least ﬁve skin depths thick, it may be considered inﬁnite in the –z direction. For t ! 1, l(t) ! w(k) and Equation 9.56 becomes: ð1 Z f s ¼ Rs þ jvLs ¼ jvm0par J 1 ðkrÞJ 1 ðkaÞfðkÞeÀkðd 1 þd 2 Þ d k 0 ð9:57Þ The subscript t in Z refers to a substrate of ﬁnite thickness, while subscript s refers to a substrate of inﬁnite thickness. The parameter l(t) in Equation 9.56 contains four variables of interest: thickness t, relative permeability mr, conductivity s and frequency v. At low frequencies, h ! k as v ! 0 and f(k) ! f0 for low frequency operation: f0 ¼ mr mr À þ 1 1 ð9:58Þ This factor describes the increase in inductance in a substrate, when mr ) 1, f ¼ 1; this means that the substrate component is equal to the air component, resulting in a doubling of the overall inductance. As a check, if there is no substrate, then mr ¼ 1, h ¼ k and f(k) ¼ 0, which means that the additional component Ls ¼ 0 as expected. Figure 9.5 shows two circular concentric planar sections on a magnetic substrate with appropriate dimensions. Integrating the ﬁlament formula (Equation 9.56) over the coil cross- z Medium 1 w2 h2 Dielectric layer h1 w1 Magnetic substrate Medium 2 z = d2 a1 a2 z = d1 μr,σ t r1 r2 z=0 Figure 9.5 Planar coils on a magnetic substrate. Planar Magnetics 259 sections yields the impedance formula for coils. The assumption that the current density is inversely proportional to the radius as described in Section 9.0 is also made here. The total impedance is: Z ¼ jvM þ Z p t ð9:59Þ M is the air term given by Equation 9.16 and the additional component due to the substrate is: Z p t ¼ jvm0p h1h2ln r2 r1 ln a2 a1 ð1 0 Sðkr2; kr1ÞSðka2; ka1ÞQðkh1; kh2 ÞlðtÞeÀkðd 1 þd 2 Þ d k ð9:60Þ As before, replace l(t) by f(k) for an inﬁnite substrate. The DC resistance may be combined with the terms given by Equation 9.59 to complete the equivalent circuit model. Example 9.2 Calculate the self inductance of the two-turn coil in Figure 9.6, with the dimensions shown. Plot the results for mr going from 1 to 1000 and for three values of substrate thicknesses t ¼ 0.05 mm, 0.1 mm and 0.5 mm. Assume there are no losses in the substrate (s ¼ 0). The self inductance of the device is given by: L ¼ L11 þ 2M12 þ L22 where L11 and L22 are the self inductances of sections 1 and 2 in Figure 9.6 and M12 is the mutual inductance between sections 1 and 2. The individual terms for the inductance in air are given by Equation 9.16 and the contributions for the substrate are given by Equation 9.60. For a lossless substrate, l becomes f0 in Equation 9.58. 5.2 mm 2.3 mm 1 2 15μ μr ,σ t 600μ z 250μ r Figure 9.6 Planar sections of a two-turn device on a ﬁnite substrate. 260 Transformers and Inductors for Power Electronics The required dimensions for the self inductance calculations are: Coil 1 : r1 ¼ a1 ¼ 1:15 mm; r2 ¼ a2 ¼ 1:75 mm; h1 ¼ h2 ¼ 15 mm; d1 ¼ d2 ¼ 7:5 mm Coil 2 : r1 ¼ a1 ¼ 2:00 mm; r2 ¼ a2 ¼ 2:60 mm; h1 ¼ h2 ¼ 15 mm; d1 ¼ d2 ¼ 7:5 mm The required dimensions for the mutual inductance calculations are: Coils 1 and 2 : r1 ¼ 1:15 mm; r2 ¼ 1:75 mm; a1 ¼ 2:00 mm; a2 ¼ 2:60 mm h1 ¼ h2 ¼ 15 mm; d2 ¼ d2 ¼ 7:5 mm The air terms were calculated in Example 9.1, and the MATLAB program at the end of this chapter gives the calculation for the substrate components. Figure 9.7(a) shows the inductance enhancement as a function of relative permeability for difference values of substrate thickness t. The enhancement is in the range: L 1 2 L1 2.0 1.9 1.8 1.7 1.6 L 1.5 L1 1.4 1.3 1.2 1.1 1.0 1 2.0 1.9 1.8 1.7 1.6 L 1.5 L1 1.4 1.3 1.2 1.1 1.0 0.001 t = 0.05 mm t = 0.1 mm t = 0.5 mm 10 100 Relative permeability μr (a) 1000 0.01 0.1 Thickness (mm) (b) μr = 10 μr = 100 μr = 1000 1 Figure 9.7 Enhancement of inductance with magnetic substrate: (a) as a function of mr, (b) as a function of t. Planar Magnetics 261 where L1 is the inductance in air (without the substrate). L1 for the two-turn coil is 17.14 nH The upper limit is achieved for the product mrt > 0.1 m and the lower limit is approached as mr ! 1. Figure 9.7(b) shows the inductance enhancement as a function of substrate thickness for different values of relative permeability mr. The enhancement is in the range: 1 L L1 1 þ f0 The upper limit is approached for mrt > 0.1 m. The lower limit is approached for mrt < 0.1 mm. The calculations are included in the MATLAB program at the end of this chapter. Example 9.3 Calculate the inductance of the device in Example 9.2 on a ferrite substrate with relative permeability mr ¼ 1000 and electrical conductivity s ¼ 10 (V – m)–1. Take the frequency range 1 MHz to 1000 MHz. The skin depth in the material is given by: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d¼ 1 ¼ 5p03ﬃﬃ3 mm pf mrm0s f where f is the frequency in Hz. The self impedance is shown in Figure 9.8(a) for each value of substrate thickness. The frequency dependence due to the eddy currents in the substrate becomes evident when the skin depth becomes comparable to the thickness of the substrate. This is evident at 200 MHz for t ¼ 0.5 mm, where the skin depth is 0.36 mm. The eddy current loss as represented by the resistance in Figure 9.8(b) is pronounced above 100 MHz. The DC resistance of the device in Figure 9.6 is taken as 0.093 V for copper turns, and this is included in R. The resistance increases by a factor of 7 at 500 MHz in the 0.5 mm substrate, where the skin depth is less than half the thickness of the substrate. This analysis shows that it is very important that the skin depth in the substrate should be greater than the thickness of the substrate, to ensure that inductance enhancement takes place without incurring the penalty of high eddy current loss. The calculations are included in the MATLAB program at the end of this chapter. 9.1.3 Spiral Coil in a Sandwich Structure The addition of a second substrate above the planar coils results in a sandwich structure, as shown in Figure 9.9. Adopting the dimensions shown in Figure 9.9, the solution of Maxwell’s equations proceeds as before. The solution is given by [2]: Z ¼ jvM þ Z p sv ð9:61Þ 262 Transformers and Inductors for Power Electronics 2.00 1.99 1.98 1.97 1.96 L 1.95 L1 1.94 1.93 t = 0.05 mm 1.92 t = 0.1 mm 1.91 t = 0.5 mm 1.90 106 107 108 109 Frequency (Hz) (a) 18 16 14 12 R 10 Rdc 8 6 t = 0.05 mm 4 t = 0.1 mm 2 t = 0.5 mm 106 107 108 109 Frequency (Hz) (b) Figure 9.8 Self impedance of a planar coil on a ﬁnite substrate: (a) inductance, (b) resistance. μr,σ t2 a1 h2 a2 s r1 h1 d2 r2 d1 μr,σ t1 Figure 9.9 Planar coils in a sandwich structure. Planar Magnetics 263 M is the air component given by Equation 9.16 and the enhancement due to the substrate is: Z p sv ¼ jvm0p h1h2ln r2 r1 ln a2 a1 ð1 0 Sðkr2; kr1ÞSðka2; ka1Þ½ f ðlÞ þ gðlÞQðkh1; kh2 Þd k ð9:62Þ where: f ðlÞ ¼ lðt1ÞeÀkðd1þd2Þ þ lðt2 ÞeÀkðd 0 1 þd 0 2 Þ 1 À lðt1Þlðt2ÞeÀ2ks ð9:63Þ and: gðlÞ ¼ 2lðt1 Þlðt2 ÞeÀ2ks cosh½kðd 2 1 À lðt1Þlðt2ÞeÀ2ks À d 1 Þ ð9:64Þ In addition to the four parameters (mr, s, t1, v) discussed in Section 9.1.2, there are additional parameters s and t2 to be taken into account in the sandwich inductor. For the purposes of analysis, substrates of equal thickness will be considered (t1 ¼ t2 ¼ t). Example 9.4 Calculate the inductance and equivalent resistance of the device in Example 9.2, sandwiched between two ferrite substrates with relative permeability mr ¼ 1000 and electrical conductivity s ¼ 10 (V-m)–1. The substrate thickness is 0.5 mm. Plot the inductance for substrate separation ranging from 0.02 mm to 10 mm. Take the frequency range 1–1000 MHz. The calculations are included in the MATLAB program at the end of this chapter, based on Equation 9.63. Figure 9.10 shows the variation of inductance with separation s for low frequency (below the frequency where eddy currents in the substrate make a difference). In this case, the range of inductance is between 1 and mr. 100 90 80 70 60 L L1 50 40 30 20 10 L = 1000 at s = 0 L1 0 0.1 1 10 Separation (mm) Figure 9.10 Inductance as a function of substrate separation; L1 ¼ 17.14 nH. 264 Transformers and Inductors for Power Electronics The upper limit is approached for s ! 0 because the coils are enclosed by a material that has a rela- tive permeability of mr. The lower limit is approached when one of the substrates is removed (s ! 1). In this example, mrt > 0.1 m, so that L L1 ! 1 þ f0, which is too large for mr, as shown by Equation 9.58. The analysis in Figure 9.10 was carried out for low frequencies where eddy current loss is negligible. The most important feature of Figure 9.10 is that the increase in inductance in a sandwich structure is strongly inﬂuenced by the gap between the substrates. The increase drops from a factor of 1000 (mr) at s ¼ 0 down to 100 at s ¼ 15 mm. At s ¼ 1 mm, the inductance is just 50% higher than that for a single substrate. Figure 9.11 shows inductance and resistance as a function of frequency for s ¼ 15 mm. The induc- tance remains relatively ﬂat up to 100 MHz. For comparison purposes, the results of Figure 9.8 with t ¼ 0.5 mm for the single substrate are included and, clearly, the presence of the second substrate greatly increases both the overall inductance and the overall loss. At 2 MHz, the loss in the substrates is equal to the DC loss in the conductors. 100 90 80 70 60 L 50 L1 40 30 20 10 0106 s = 15μm Single substrate 107 108 Frequency (Hz) (a) 1000 100 R Rdc 10 s = 15μm Single substrate 1106 107 108 Frequency (Hz) (b) Figure 9.11 Self impedance of a planar sandwich coil as a function of frequency: (a) inductance, (b) resistance; L1 ¼ 17.14 nH, Rdc ¼ 0.093 V. Planar Magnetics 265 9.2 Fabrication of Spiral Inductors The fabrication of planar magnetic devices may be broadly classiﬁed as printed circuit board (PCB), thick ﬁlm, low temperature co-ﬁred ceramic (LTCC) and thin ﬁlm. The technology tends to be aligned to the application; for example, PCB has found applications in microprocessors and computer loads, where etched conductors can carry the load currents. Thick ﬁlm is normally associated with passive components such as resistors and, to a lesser extent, capacitors [7], and it may be found in biomedical applications, where a ceramic substrate is normally expected to avail of its non-toxic properties. LTCC ﬁnds favour in high temperature applications, and thin ﬁlm has emerged in power supply on a chip (PwrSoC) because of its compatibility with silicon. 9.2.1 PCB Magnetics Printed circuit board (PCB) technology was developed and adapted to magnetic components to overcome the size and cost problems associated with conventional magnetics. PCB magnetics may be categorized as discrete PCB magnetics and integrated PCB magnetics. The difference lies in the method of fabrication or assembly. PCB magnetics have found application in low power DC-DC converters. Discrete PCB Magnetics Discrete PCB magnetics mean that the winding is produced separately from the magnetic core. Low-height magnetic cores are used and the windings do not require winding bobbins. Figure 9.12 shows a typical planar magnetic construction with PCB windings. The windings on printed circuit boards are assembled in layers, along with appropriate insulating layers. Standard ferrite cores are placed around the windings to complete the inductor or transformer in a sandwich structure. Ferrite cores are available in many shapes and sizes due to their ease of manufacture and, therefore, ferrite is the primary core material for planar magnetics designs. Most planar magnetics assemblies have core materials of manganese-zinc (Mn-Zn) or nickel-zinc (Ni-Zn). Mn-Zn is more popular in applications where the operating frequencies are less than 2 MHz; above 2 MHz, Ni-Zn is more suitable because its resistivity is several orders of magnitude larger than Mn-Zn. The only disadvantage of ferrite material is the relatively low value of saturation ﬂux density compared with other core materials. One of the main advantages of this planar type construction is that the surface area for heat transfer tends to be higher relative to volume when compared to conventional wire-wound devices. The larger surface area means that more heat can be removed for the same temperature rise and this, in turn, means that higher power densities are achievable. Another advantage is that the parasitics, such as leakage inductance and capacitance, are more consistent for devices built in the same lot, because there is less variation in the winding manufacture. The nature of the mechanical coil winding process in conventional devices leads to more variation from part to part. New materials are constantly appearing that further enhance the advantages of discrete planar devices, with particular emphasis on extending the operating frequency range with materials that exhibit low core loss. 266 Transformers and Inductors for Power Electronics Figure 9.12 A typical planar transformer with an E-I core. Integrated PCB Magnetics The automated fabrication of magnetic components, with attendant improvement in reliability, may be achieved by integrating the core and winding so that the magnetic core is formed during the PCB process. Figure 9.13 shows a typical integrated PCB transformer. The process used to make PCB windings in discrete planar magnetics is also used to integrate the magnetic core into the manufacturing process. The cores are not limited to conventional or off-the-shelf components, and this gives the designer the opportunity to specify the core shape for optimized operation. Due to the low power range addressed to date, the size of the windings is generally smaller in comparison with discrete planar magnetics. PCB track widths down to 150 mm are readily available using standard PCB processing, allowing multiturn inductors in a small footprint – less than 10 Â 10 mm2. The electroplating of the core is compatible with the process used for fabricating the PCB winding. There are two approaches to integrating magnetic material in PCB. One approach involves laminating commercial magnetic foils within multilayer boards. Foils are available in a range of thicknesses and with relative permeability values as high as 100 000. This means that high inductance values may be achieved in a small footprint area. The other approach involves electroplating magnetic materials (e.g. permalloy) onto laminate layers. The process is Planar Magnetics 267 Cu Core Cu Laminate FR4 Cu Core Cu Through-holes Cu Core Figure 9.13 PCB integrated magnetic toroidal transformer. integral to the PCB process, thereby eliminating expensive foils. In addition, the required thickness of core material is controlled, giving greater ﬂexibility in the design. In many cases, integrated PCB magnetics consist of toroidal structures where the winding is wrapped around the core by the use of etching and conductive vias. However due to the higher resistance of the vias, winding loss is higher. Normally, toroids have a high number of turns, and this adds further to the overall winding loss. 9.2.2 Thick Film Devices A thick ﬁlm integrated passive RLC circuit offers cost and reliability beneﬁts over those requiring the addition of discrete wire-wound inductor components. Printed conductor through-holes in alumina substrates have been developed that allow double-sided solenoidal-type inductors to be manufactured, with the substrate acting as the interlayer dielectric. Traditionally, thick ﬁlm materials have been limited in terms of conductivity, line deﬁnition and dielectric performance. However, new materials are constantly evolving, with higher conductivity and with track widths as low as ten microns. Circular spiral inductors may be manufactured on a magnetic substrate with thick ﬁlm technology [3], thereby introducing a magnetic core. These inductors may consist of multiple layers of several conductor turns, each deposited with thick ﬁlm conductor paste. These conductor layers are separated by layers of dielectric material. The magnetic substrates are usually ferrite material. Thick ﬁlm circuits are produced by a screen printing process, which involves using a mesh screen to produce designs on a suitable substrate. The thickness of ﬁlm is generally of the order of 10 mm. The mesh screen may be made from stainless steel or from synthetic ﬁbres such as nylon or Dacron. A viscous paste is forced through the screen to deposit a pattern onto the substrate. A typical screen printing set-up is shown in Figure 9.14. 268 Transformers and Inductors for Power Electronics Squeegee Screen Printed paste Substrate Paste Figure 9.14 Screen printing process. There are three categories of thick ﬁlm paste, namely conductive, resistive and insulating pastes. The fabrication of the spiral inductor involves both conductive paste and insulating paste. Each paste contains: a functional material that determines the electrical properties of the paste; a solvent/thinner that determines the viscosity of the paste; a temporary binder that holds other ingredients together; a permanent binder that fuses particles of the functional material together. There are four main steps in the fabrication process for each layer of printed paste: Screen generation. Firstly, the desired pattern is laid out using a suitable software layout package. The photoplots for a four-layer inductor are shown in Figure 9.15. The masks for the dielectric layer and a via hole through which the various conductor layers can be interconnected, as shown in Figure 9.16. A photosensitive emulsion is then applied to the entire screen and the mask is placed on the screen and exposed to ultraviolet light. The parts of the emulsion that were not exposed by the mask may be washed away using a spray gun, and a negative image is generated on the screen mesh. This part of the process is illustrated in Figure 9.17. Screen printing. The required pattern is deposited onto a substrate by forcing a viscous paste through the apertures in the patterned screen. Layer 1 Layer 2 Layer 3 Layer 4 Figure 9.15 Photoplots of conducting layers. Planar Magnetics 269 Dielectric 1 Dielectric 2 Dielectric 3 Figure 9.16 Masks for dielectric layers. Expose Develop Positive Artwork Negative Acting Emulsion on Screen Negative Photoresist Mask Figure 9.17 Screen generation. Drying/Curing. The aim of this part of the process is to remove the organic solvents from the screen-printed paste in two steps. At the ﬁrst step, the substrate with the freshly printed paste is placed in the air for 5–10 minutes in order to settle the paste. At the next step, the substrate is put into an oven at 125C for 10–20 minutes and the organic solvents are removed by evaporation. Firing. This is carried out in three steps. First, the temporary organic binder is decomposed by oxidation and removed at temperatures up to 500C. Next, the permanent binder melts and wets both the surface of the substrate and the particles of the functional material between 500C and 700C. In the ﬁnal step, the functional particles are sintered and become interlocked with the permanent binder and the substrate between 700C and 850C. A microsection of an experimental device is shown in Figure 9.18. Note, the individual conductor layers can be easily recognized. The actual physical dimensions will vary slightly from the ideal rectangular cross-sections because, by its nature, the screen printing process results in tapered walls and the individual layers do not line up exactly in the vertical direction. Figure 9.18 Optical photograph of a microsectioned device (scale 30: 1) [3] Reproduced with permission from [3]. Copyright 1999 IEEE. 270 Transformers and Inductors for Power Electronics 9.2.3 LTCC Magnetics Low temperature co-ﬁred ceramic (LTCC) technology is a multilayer technology that has its roots in the microelectronics packaging industry. Unlike the successive build-up of layers in the previously described technologies, each layer is processed independently. The layers are stacked, aligned, laminated and ﬁred together at around 900C. The constituent ceramic materials must be sintered at the same temperature. LTCC is particularly suitable for highcurrent switching devices operating in the MHz range. The key to LTCC is a hybrid multilayer structure that lends itself to integrated magnetics. The layers are prepared with individual ceramic green sheets with punched-through vias and screen-printed conductor patterns. Depending on the type of magnetic components, a partial winding or several windings may be printed on each layer. Ferrite ceramic is typically used to enhance the inductance. Under certain circumstances, a non-ferrite ceramic may be used to form air-core inductors or coreless transformers. LTCC technology presents advantages compared to other packaging technologies, such as high temperature co-ﬁred ceramic (HTCC) technology: the ceramic is generally ﬁred below 1000C, due to a special composition of the material. Firing below 1000C means that highly conductive materials such as silver, copper and gold may be co-ﬁred to form the conductors for low DC resistance and high Q factor. LTCC technology can incorporate other passive elements, such as resistors and capacitors, minimizing the size of the ceramic package, with better manufacturing yields and lower costs. LTCC technology has been employed in rf and wireless applications that have highly integrated multilayer circuits with resistors, capacitors and inductors, along with active components in a single package. There are eight main steps in the LTTC process: Tape preparation: putty-like green sheets are supplied on a roll and are cut by laser or punched. Some tapes are baked for up to 30 minutes at 120C. Blanking: individual pieces are cut and orientation marks and tooling holes are formed with a blanking die. Single sheets are rotated to compensate for shrinkage in different directions. Via punching: vias are formed by punching or laser drilling. The thickness of the tapes determines the via diameter. Via ﬁlling: the vias are ﬁlled with thick ﬁlm screen printer, as described in Section 9.2.2, or with a ﬁller. Printing: a thick ﬁlm screen printer is used to print co-ﬁreable conductors onto the green sheets, using standard emulsions. Following printing, the vias and conductors are dried in an oven with a temperature setting between 80C and 120C, for up to 30 minutes. Collating and laminating: each layer is stacked in turn over tooling pins to line them up correctly for lamination. The stacked tapes are pressed together at 20 MPa and 70C for ten minutes. The stack is rotated through 180 after ﬁve minutes. Alternatively, the stacked tapes may be vacuum-sealed in foil. Pressing takes place in hot water for ten minutes at 21MPa. Co-ﬁring: ﬁring of the laminated stack takes place in two stages. In the ﬁrst stage, the temperatures increase at a rate of 2–5C per minute up to 450C, taking about 1.5–2 hours and causing organic ingredients to burn out. At the next stage, the temperature is increased Planar Magnetics 271 Green tape Blanking Via punching Via filling Conductor print Post-processing Co-firing & Testing Laminating Collating Figure 9.19 LTCC Process ﬂow. steadily to around 875C over a period of about 30 minutes and held there for 20 minutes while sintering takes place. The oven is then cooled down over the next hour. Post-ﬁring and singulation: in post-ﬁring, a paste is applied to the ﬁred parts and ﬁred under conditions dictated by the material used. In singulation, the ﬁred parts are now ready for cutting, trimming with a laser or ultrasonic cutter. These steps are illustrated in Figure 9.19. 9.2.4 Thin Film Devices Recent trends towards power supply on chip (PwrSoC) technology have placed further demands on the size of the inductor, with typical footprints of 5–10 mm2. This may be achieved by thin ﬁlm technology, which involves a microinductor fabricated on silicon using electrochemical deposition techniques. these are sometimes referred to as MEMS (microelectro-mechanical systems) inductors. Various approaches to deposit the magnetic core layer have been demonstrated over the last 20 years, such as screen printing, sputtering and electroplating. The screen print technique has been explained earlier in Section 9.2.2. Sputtering is a widely used microfabrication technique whereby atoms are dislodged from the target material and deposited on a surface at relatively low temperatures; the resulting layers are usually several microns in thickness. Low temperatures rule out ferrites, because they require sintering, which takes place well above the temperatures involved in sputtering. Sputtering has found application in microinductors operating in the GHz frequency range. The main disadvantage of sputtering lies in the fact that it is quite expensive to deposit relatively thick layers up to several microns. 272 Transformers and Inductors for Power Electronics Integrated microinductors operating in the MHz range are needed for power conversion and, typically, a magnetic ﬁlm thickness of several microns is required. Electroplating is inexpensive and has relatively fast deposition rates compared to sputtering. Electroplating is a widely used technique for the deposition of relatively thick (several mm to tens of mm) layers in a microfabrication compatible manner. Electroplating takes place when ions move under the inﬂuence of an electric ﬁeld to form a coated conductive seed layer on an electrode. By their nature, the electroplated materials are conductive, and a thickness of up to one skin depth is typically applied to limit eddy current loss. There are three main steps in the electroplating process: 1. The required pattern is formed by photolithography using a dry ﬁlm photoresist, similar to the photoplots in Figure 9.15. 2. This is followed by electroplating of the conductor pattern to form a seed layer. 3. Finally, the photoresist is stripped and the seed layer is etched to produce the ﬁnal circuit pattern. Inductors formed in this way are limited to the nH range because the thickness of deposited thin ﬁlm has to be limited to control the eddy current loss. However, this limitation can be overcome by implementing core lamination techniques. Eddy current loss in the core can be substantially reduced by laminating the core in successive layers of magnetic material separated by dielectric. The top view and cross-section of a microinductor using laminated magnetic core are shown in Figure 9.20 [4,5]. The two-layer magnetic core is effectively laminated to reduce eddy current loss at high frequencies, allowing the device to operate in the tens of MHz range. As we are dealing with a current-carrying component, low resistance is therefore important, and this can only be achieved by thick conductors. From a performance point of view, it is desirable to increase the conductor thickness to reduce winding loss, provided that the conductor thickness does not exceed the skin depth. In this case, the conductors are deposited by electroplating, which limits the achievable thickness. There are ﬁve layers in a typical microinductor: 1. Electroplated magnetic layer: permalloy (Ni Fe) is electroplated to form the bottom magnetic core layer of the inductor. 2. Dielectric layer: a dielectric material is used to insulate the bottom core from the conductors. 3. Electroplated copper layer: the windings are fabricated, using the electroplating of copper through thick photoresist patterns. 4. Insulation layer: thick photoresist is spun on the conductors to provide an insulation layer between the conductors and the top core layer. 5. Electroplated magnetic layer: ﬁnally, the top layer of permalloy is electroplated and patterned to close the magnetic core by connecting the top core to the bottom core. These are illustrated in Figure 9.21. The process ﬂow is shown in Figure 9.17. Planar Magnetics 273 Figure 9.20 Silicon integrated microinductor: (a) Top view. Reproduced with permission from [4]. Copyright 2008 IEEE, (b) Cross-section. Reproduced with permission from [5]. Copyright 2005 IEEE. Layer 1 : Electroplated bottom core layer Layer 2 : Insulator layer between bottom core and winding Layer 3 : Electroplated copper winding layer Layer 4 : Insulator layer between top core and winding Layer 5 : Electroplating top core layer Figure 9.21 Microfabrication process ﬂow for an inductor. 274 Transformers and Inductors for Power Electronics Table 9.1 A comparison on performance of different magnetic technologies Technology Frequency (typical) Power (typical) Inductance (typical) PCB magnetics Thick ﬁlm LTCC Thin ﬁlm 20 KHz $ 2 MHz < 10 MHz 200 KHz $ 10 MHz > 10 MHz 1 W $ 5 kW < 10 W < 10 W < 1W 10 mH $ 10 mH 1 mH $ 1 mH 1 mH $ 1 mH 10s $ 100s nH Size (typical) 100 mm2 $ 100s cm2 < 1 cm2 < 1 cm2 < 10 mm2 9.2.5 Summary Each of the four technologies described above has its own characteristics and its own advantages and disadvantages. The individual processes are under continual development, and improvements in the relevant technologies mean that the range of applications is increasing in each case. The ongoing research into new materials in pushing out the boundaries of operation. The achievable performance (inductance and power level), target operating frequency, and size for each technology are listed in Table 9.1. The advantage and disadvantage of each technology is summarized in Table 9.2. Table 9.2 Advantages and disadvantages of different magnetic technologies Technology Integration method Advantages Disadvantage PCB Thick ﬁlm Discrete core on laminated structure or integrated core in laminated structure Parallel or sequential process Screen printed on sintered ceramic Sequential build-up of multiple layers Low cost Multilayer structure Thick copper High current High inductance Low cost LTCC Thin ﬁlm Screen printed on green tapes Parallel multilayer and ﬁnal co-ﬁred structure Sequential build-up of lithographically deﬁned layers Parallel layer process High layer counts Module reliability Precision value (line width 5 mm) High tolerance High component density High frequency Low resolution (line width 100 mm) Relatively low frequency Difﬁcult to form Long process time and low yield due to sequential build-up Co-ﬁreability of materials Low inductance Equipment costly Limited selection on ﬁlm materials(material compatibility Planar Magnetics 275 9.3 Problems 9.1 Repeat Example 9.2 for the four-layer device in Figure 9.3. 9.2 Repeat Example 9.3 for the four-layer device in Figure 9.3. 9.3 Repeat Example 9.4 for the four-layer device in Figure 9.3. 9.4 What are the main advantages of planar magnetic as compared to conventional wire- wound magnetic? 9.5 List the main planar magnetic technologies. 9.6 What are the disadvantages and advantages of each planar magnetic technology? MATLAB Program for Example 9.1 %This MATLAB program is used to calculate the self and mutual inductances %in Example 9.1 %The parameters are shown in Figure 9.3 rin =[1.15e-3 1.15e-3 2e-3 2e-3]; rout =[1.75e-3 1.75e-3 2.6e-3 2.6e-3]; height =[15e-6 15e-6 15e-6 15e-6]; d=[7.5e-6 62.5e-6 62.5e-6 7.5e-6]; Inductance=ones(4,4); for i=1:4 for j=1:4 Inductance(i,j)= air_mutual(rin(i),rout(i),rin(j),rout(j),height (i),height(j),d(i),d(j)); end end %Export the inductance matrix Inductance %Get the total inductance L_total = sum(sum(Inductance)) %File to deﬁne the function air_mutual function y = air_mutual(r1,r2,a1,a2,h1,h2,d1,d2) %This function is used to calculate the mutual inductance in air core %condition. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. 276 Transformers and Inductors for Power Electronics %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %z is the axis separation. z=0 for self-inductance calculation; z= |d2-d1| %for mutual inductance calculation. global uo; uo=4*pi*1e-7; g=@(k)aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end function y=aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k) z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; y=S1.*S2.*Q.*exp(-z.*k); end Planar Magnetics 277 MATLAB Program for Example 9.2(a) %This MATLAB program is used to calculate the self inductances and plot the %results in Example 9.2(a) %The parameters are shown in Figure 9.6 rin=[1.15e-3 2e-3]; rout=[1.75e-3 2.6e-3]; height=[15e-6 15e-6]; d=[7.5e-6 7.5e-6]; L1_air=air_mutual(rin(1),rout(1),rin(1),rout(1),height(1),height(1), d(1),d(1)); L2_air=air_mutual(rin(2),rout(2),rin(2),rout(2),height(2),height(2), d(2),d(2)); L12_air=air_mutual(rin(1),rout(1),rin(2),rout(2),height(1),height (2),d(1),d(2)); L1=L1_air+L2_air+2*L12_air; t1=0.05e-3; t2=0.1e-3; t3=0.5e-3; L_t1=L1.*ones(1,1000); L_t2=L1.*ones(1,1000); L_t3=L1.*ones(1,1000); i=2; for ur=2:1000; L1_t1=L1_air+inductance_substrate(rin(1),rout(1),rin(1),rout(1), height(1),height(1),d(1),d(1),t1,ur); L2_t1=L2_air+inductance_substrate(rin(2),rout(2),rin(2),rout(2), height(2),height(2),d(2),d(2),t1,ur); L12_t1=L12_air+inductance_substrate(rin(1),rout(1),rin(2),rout(2), height(1),height(2),d(1),d(2),t1,ur); L_t1(i)=L1_t1+L2_t1+2*L12_t1; L1_t2=L1_air+inductance_substrate(rin(1),rout(1),rin(1),rout(1), height(1),height(1),d(1),d(1),t2,ur); L2_t2=L2_air+inductance_substrate(rin(2),rout(2),rin(2),rout(2), height(2),height(2),d(2),d(2),t2,ur); 278 Transformers and Inductors for Power Electronics L12_t2=L12_air+inductance_substrate(rin(1),rout(1),rin(2),rout(2), height(1),height(2),d(1),d(2),t2,ur); L_t2(i)=L1_t2+L2_t2+2*L12_t2; L1_t3=L1_air+inductance_substrate(rin(1),rout(1),rin(1),rout(1), height(1),height(1),d(1),d(1),t3,ur); L2_t3=L2_air+inductance_substrate(rin(2),rout(2),rin(2),rout(2), height(2),height(2),d(2),d(2),t3,ur); L12_t3=L12_air+inductance_substrate(rin(1),rout(1),rin(2),rout(2), height(1),height(2),d(1),d(2),t3,ur); L_t3(i)=L1_t3+L2_t3+2*L12_t3; i=i+1; end x=1:1:1000; y1=L_t1./L1; y2=L_t2./L1; y3=L_t3./L1; semilogx(x,y1,’-’,x,y2,’-’,x,y3,’-’) axis([1,1000,1,2]) %File to deﬁne the function air_mutual function y=air_mutual(r1,r2,a1,a2,h1,h2,d1,d2) %This function is used to calculate the mutual inductance in air core %condition. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %z is the axis separation. z=0 for self-inductance calculation; z=|d2-d1| %for mutual inductance calculation. global uo; uo=4*pi*1e-7; g=@(k)aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; Planar Magnetics 279 end end y=uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end function y=aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k) z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; y=S1.*S2.*Q.*exp(-z.*k); end %File to deﬁne the function inductance_substrate function y=inductance_substrate(r1,r2,a1,a2,h1,h2,d1,d2,t,ur) %This function is used to calculate the additional mutual impedance to the %presence of the substrate. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %t is thickness of the substrate %ur relative permeability of the magnetic substrate global uo; uo=4*pi*1e-7; g=@(k)integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,k); for upper=1000:1000:1000000 280 Transformers and Inductors for Power Electronics [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end function y=integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,k) global uo; z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; phi=(ur-1)./(ur+1); lambda=phi.*(1-exp(-2*t.*k))./(1-(phi.^2).*exp(-2*t.*k)); y=S1.*S2.*Q.*lambda.*exp(-(d1+d2).*k); end MATLAB Program for Example 9.2(b) %This MATLAB program is used to calculate the self inductances and plot the %results in Example 9.2(b) %The parameters are shown in Figure 9.6 rin=[1.15e-3 2e-3]; rout=[1.75e-3 2.6e-3]; height=[15e-6 15e-6]; d=[7.5e-6 7.5e-6]; Planar Magnetics 281 L1_air=air_mutual(rin(1),rout(1),rin(1),rout(1),height(1),height (1),d(1),d(1)); L2_air=air_mutual(rin(2),rout(2),rin(2),rout(2),height(2),height (2),d(2),d(2)); L12_air=air_mutual(rin(1),rout(1),rin(2),rout(2),height(1),height (2),d(1),d(2)); L1=L1_air+L2_air+2*L12_air; ur1=10; ur2=100; ur3=1000; L_ur1=L1.*ones(1,1000); L_ur2=L1.*ones(1,1000); L_ur3=L1.*ones(1,1000); i=1; for t=0.001e-3:0.001e-3:1e-3; L1_ur1=L1_air+inductance_substrate(rin(1),rout(1),rin(1),rout(1), height(1),height(1),d(1),d(1),t,ur1); L2_ur1=L2_air+inductance_substrate(rin(2),rout(2),rin(2),rout(2), height(2),height(2),d(2),d(2),t,ur1); L12_ur1=L12_air+inductance_substrate(rin(1),rout(1),rin(2),rout (2),height(1),height(2),d(1),d(2),t,ur1); L_ur1(i)=L1_ur1+L2_ur1+2*L12_ur1; L1_ur2=L1_air+inductance_substrate(rin(1),rout(1),rin(1),rout(1), height(1),height(1),d(1),d(1),t,ur2); L2_ur2=L2_air+inductance_substrate(rin(2),rout(2),rin(2),rout(2), height(2),height(2),d(2),d(2),t,ur2); L12_ur2=L12_air+inductance_substrate(rin(1),rout(1),rin(2),rout (2),height(1),height(2),d(1),d(2),t,ur2); L_ur2(i)=L1_ur2+L2_ur2+2*L12_ur2; L1_ur3=L1_air+inductance_substrate(rin(1),rout(1),rin(1),rout(1), height(1),height(1),d(1),d(1),t,ur3); L2_ur3=L2_air+inductance_substrate(rin(2),rout(2),rin(2),rout(2), height(2),height(2),d(2),d(2),t,ur3); L12_ur3=L12_air+inductance_substrate(rin(1),rout(1),rin(2),rout (2),height(1),height(2),d(1),d(2),t,ur3); 282 Transformers and Inductors for Power Electronics L_ur3(i)=L1_ur3+L2_ur3+2*L12_ur3; i=i+1; end x=0.001:0.001:1; y1=L_ur1./L1; y2=L_ur2./L1; y3=L_ur3./L1; semilogx(x,y1,’-’,x,y2,’-’,x,y3,’-’) axis([0.001,1,1,2]) %File to deﬁne the function air_mutual function y=air_mutual(r1,r2,a1,a2,h1,h2,d1,d2) %This function is used to calculate the mutual inductance in air core %condition. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %z is the axis separation. z=0 for self-inductance calculation; z=|d2-d1| %for mutual inductance calculation. global uo; uo=4*pi*1e-7; g=@(k)aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end function y=aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k) z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); Planar Magnetics 283 end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; y=S1.*S2.*Q.*exp(-z.*k); end %File to deﬁne the function inductance_substrate function y=inductance_substrate(r1,r2,a1,a2,h1,h2,d1,d2,t,ur) %This function is used to calculate the additional mutual impedance to the %presence of the substrate. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %t is thickness of the substrate %ur relative permeability of the magnetic substrate global uo; uo=4*pi*1e-7; g=@(k)integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end function y=integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,k) global uo; 284 Transformers and Inductors for Power Electronics z = d2-d1; if z == 0 Q = 2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q = 2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; phi=(ur-1)./(ur+1); lambda=phi.*(1-exp(-2*t.*k))./(1-(phi.^2).*exp(-2*t.*k)); y=S1.*S2.*Q.*lambda.*exp(-(d1+d2).*k); end MATLAB Program for Example 9.3 %This MATLAB program is used to calculate the self impedance and plot the %results in Example 9.3 %The parameters are shown in Figure 9.6 rin=[1.15e-3 2e-3]; rout=[1.75e-3 2.6e-3]; height=[15e-6 15e-6]; d=[7.5e-6 7.5e-6]; L1_air=air_mutual(rin(1),rout(1),rin(1),rout(1),height(1),height (1),d(1),d(1)); L2_air=air_mutual(rin(2),rout(2),rin(2),rout(2),height(2),height (2),d(2),d(2)); L12_air=air_mutual(rin(1),rout(1),rin(2),rout(2),height(1),height (2),d(1),d(2)); L1=L1_air+L2_air+2*L12_air; ur=1000; sigma=10; Rdc=0.093; t1=0.05e-3; Planar Magnetics 285 t2=0.1e-3; t3=0.5e-3; L_t1 = L1.*ones(1,1000); L_t2 = L1.*ones(1,1000); L_t3 = L1.*ones(1,1000); Rac_t1 = Rdc.*ones(1,1000); Rac_t2 = Rdc.*ones(1,1000); Rac_t3 = Rdc.*ones(1,1000); i = 1; for frequency=1e6:1e6:1000e6; L1_t1=impedance_substrate(rin(1),rout(1),rin(1),rout(1),height(1), height(1),d(1),d(1),t1,ur,sigma,frequency); R1_t1=real(L1_t1); L1_t1=(imag(L1_t1))/(2*pi*frequency)+L1_air; L2_t1=impedance_substrate(rin(2),rout(2),rin(2),rout(2),height(2), height(2),d(2),d(2),t1,ur,sigma,frequency); R2_t1=real(L2_t1); L2_t1=(imag(L2_t1))/(2*pi*frequency)+L2_air; L12_t1=impedance_substrate(rin(1),rout(1),rin(2),rout(2),height (1),height(2),d(1),d(2),t1,ur,sigma,frequency); R12_t1=real(L12_t1); L12_t1=(imag(L12_t1))/(2*pi*frequency)+L12_air; L_t1(i)=L1_t1+L2_t1+2*L12_t1; Rac_t1(i)=R1_t1+R2_t1+2*R12_t1; L1_t2=impedance_substrate(rin(1),rout(1),rin(1),rout(1),height(1), height(1),d(1),d(1),t2,ur,sigma,frequency); R1_t2=real(L1_t2); L1_t2=(imag(L1_t2))/(2*pi*frequency)+L1_air; L2_t2=impedance_substrate(rin(2),rout(2),rin(2),rout(2),height(2), height(2),d(2),d(2),t2,ur,sigma,frequency); R2_t2=real(L2_t2); L2_t2=(imag(L2_t2))/(2*pi*frequency)+L2_air; L12_t2=impedance_substrate(rin(1),rout(1),rin(2),rout(2),height (1),height(2),d(1),d(2),t2,ur,sigma,frequency); R12_t2=real(L12_t2); L12_t2=(imag(L12_t2))/(2*pi*frequency)+L12_air; L_t2(i)=L1_t2+L2_t2+2*L12_t2; Rac_t2(i)=R1_t2+R2_t2+2*R12_t2; 286 Transformers and Inductors for Power Electronics L1_t3=impedance_substrate(rin(1),rout(1),rin(1),rout(1),height(1), height(1),d(1),d(1),t3,ur,sigma,frequency); R1_t3=real(L1_t3); L1_t3=(imag(L1_t3))/(2*pi*frequency)+L1_air; L2_t3=impedance_substrate(rin(2),rout(2),rin(2),rout(2),height(2), height(2),d(2),d(2),t3,ur,sigma,frequency); R2_t3=real(L2_t3); L2_t3=(imag(L2_t3))/(2*pi*frequency)+L2_air; L12_t3=impedance_substrate(rin(1),rout(1),rin(2),rout(2),height (1),height(2),d(1),d(2),t3,ur,sigma,frequency); R12_t3=real(L12_t3); L12_t3=(imag(L12_t3))/(2*pi*frequency)+L12_air; L_t3(i)=L1_t3+L2_t3+2*L12_t3; Rac_t3(i)=R1_t3+R2_t3+2*R12_t3; i=i+1; end x = 1e6:1e6:1000e6; y1= L_t1./L1; y2 = L_t2./L1; y3 = L_t3./L1; z1= Rac_t1./Rdc+1; z2 = Rac_t2./Rdc+1; z3 = Rac_t3./Rdc+1; semilogx(x,y1,’-’,x,y2,’-’,x,y3,’-’) axis([1e6,1000e6,1.9,2]) ﬁgure; semilogx(x,z1,’-’,x,z2,’-’,x,z3,’-’) axis([1e6,1000e6,1,19]) %File to deﬁne the function air_mutual function y=air_mutual(r1,r2,a1,a2,h1,h2,d1,d2) %This function is used to calculate the mutual inductance in air core %condition. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %z is the axis separation. z=0 for self-inductance calculation; z=|d2d1| Planar Magnetics 287 %for mutual inductance calculation. global uo; uo = 4*pi*1e-7; g = @(k)aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y = uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end function y = aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k) z = d2-d1; if z == 0 Q = 2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1 =(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2 =(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; y = S1.*S2.*Q.*exp(-z.*k); end %File to deﬁne the function impedance_substrate function y=impedance_substrate(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,sigma, freq) %This function is used to calculate the additional mutual impedance to the %presence of the substrate in sandwich structures. 288 Transformers and Inductors for Power Electronics %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %t is the thickness of the substrate %ur relative permeability of the magnetic substrate %sigma relative permeability of the magnetic substrate %freq operation frequency global uo; uo = 4*pi*1e-7; omega = 2*pi*freq; g = @(k)integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,sigma,freq,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=1j.*omega.*uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/ r1)); end function y=integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,sigma,freq,k) global uo; z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; Planar Magnetics 289 end; eta=sqrt((1j*2*pi.*freq*sigma*ur*4*pi*1e-7)+(k.^2)); phi=(ur.*k-eta)./(ur.*k+eta); lambda=phi.*(1-exp(-2*t.*eta))./(1-(phi.^2).*exp(-2*t.*eta)); y=S1.*S2.*Q.*lambda.*exp(-(d1+d2).*k); end MATLAB Program for Example 9.4(a) %This MATLAB program is used to calculate the self impedance and plot the %results in Example 9.4(a) %The parameters are shown in Example 9.2 sandwiched between two ferrite %substrates rin=[1.15e-3 2e-3]; rout=[1.75e-3 2.6e-3]; height=[15e-6 15e-6]; d=[7.5e-6 7.5e-6]; L1_air=air_mutual(rin(1),rout(1),rin(1),rout(1),height(1),height (1),d(1),d(1)); L2_air=air_mutual(rin(2),rout(2),rin(2),rout(2),height(2),height (2),d(2),d(2)); L12_air=air_mutual(rin(1),rout(1),rin(2),rout(2),height(1),height (2),d(1),d(2)); L1=L1_air+L2_air+2*L12_air; ur=1000; sigma=10; t1=0.5e-3; t2=0.5e-3; frequency=1e6; L_total=L1.*ones(1,1000); i=1; for s=0.01e-3:0.01e-3:10e-3; L1=impedance_sandwich(rin(1),rout(1),rin(1),rout(1),height(1), height(1),d(1),d(1),t1,t2,s,ur,sigma,frequency); L1=(imag(L1))/(2*pi*frequency)+L1_air; L2=impedance_sandwich(rin(2),rout(2),rin(2),rout(2),height(2), height(2),d(2),d(2),t1,t2,s,ur,sigma,frequency); 290 Transformers and Inductors for Power Electronics L2=(imag(L2))/(2*pi*frequency)+L2_air; L12=impedance_sandwich(rin(1),rout(1),rin(2),rout(2),height(1), height(2),d(1),d(2),t1,t2,s,ur,sigma,frequency); L12=(imag(L12))/(2*pi*frequency)+L12_air; L_total(i)=L1+L2+2*L12; i=i+1; end x=0.01e-3:0.01e-3:10e-3; y=L_total./L1; semilogx(x,y,’-’) axis([0.01e-3,10e-3,0,100]) %File to deﬁne the function air_mutual function y=air_mutual(r1,r2,a1,a2,h1,h2,d1,d2) %This function is used to calculate the mutual inductance in air core %condition. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %z is the axis separation. z=0 for self-inductance calculation; z=|d2d1| %for mutual inductance calculation. global uo; uo=4*pi*1e-7; g=@(k)aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end Planar Magnetics 291 function y=aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k) z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; y=S1.*S2.*Q.*exp(-z.*k); end %File to deﬁne the function impedance_sandwich function y=impedance_sandwich(r1,r2,a1,a2,h1,h2,d1,d2,t1,t2,s,ur, sigma,freq) %This function is used to calculate the additional mutual impedance to the %presence of the substrate in sandwich structures. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %t1 and t2 are thickness of bottom and upper substrates %s substrate separation %ur relative permeability of the magnetic substrate %sigma relative permeability of the magnetic substrate %freq operation frequency global uo; uo=4*pi*1e-7; omega=2*pi*freq; g=@(k)integrand(r1,r2,a1,a2,h1,h2,d1,d2,t1,t2,s,ur,sigma,freq,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); 292 Transformers and Inductors for Power Electronics if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=1j.*omega.*uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/ r1)); end function y=integrand(r1,r2,a1,a2,h1,h2,d1,d2,t1,t2,s,ur,sigma, freq,k) global uo; z=d2-d1; d1_s=s-d1; d2_s=s-d2; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; eta=sqrt((1j*2*pi.*freq*sigma*ur*4*pi*1e-7)+(k.^2)); phi=(ur.*k-eta)./(ur.*k+eta); lambda1=phi.*(1-exp(-2*t1.*eta))./(1-(phi.^2).*exp(-2*t1.*eta)); lambda2=phi.*(1-exp(-2*t2.*eta))./(1-(phi.^2).*exp(-2*t2.*eta)); denominator=1-lambda1.*lambda2.*exp(-2*s.*k); F=(lambda1.*exp(-(d1+d2).*k)+lambda2.*exp(-(d1_s+d2_s).*k))./ denominator; G=2.*lambda1.*lambda2.*exp(-2*s.*k).*cosh((d2-d1).*k)./ denominator; y=S1.*S2.*Q.*(F+G); end Planar Magnetics 293 MATLAB Program for Example 9.4(b) %This MATLAB program is used to calculate the self impedance and plot the %results in Example 9.4(b) %The parameters are shown in Example 9.2 sandwiched between two ferrite %substrates rin=[1.15e-3 2e-3]; rout=[1.75e-3 2.6e-3]; height=[15e-6 15e-6]; d=[7.5e-6 7.5e-6]; L1_air=air_mutual(rin(1),rout(1),rin(1),rout(1),height(1),height (1),d(1),d(1)); L2_air=air_mutual(rin(2),rout(2),rin(2),rout(2),height(2),height (2),d(2),d(2)); L12_air=air_mutual(rin(1),rout(1),rin(2),rout(2),height(1),height (2),d(1),d(2)); L1=L1_air+L2_air+2*L12_air; ur=1000; sigma=10; t=0.5e-3; s=15e-6; Rdc=0.093; L_s=L1.*ones(1,991); L_single=L1.*ones(1,991); Rac_s=Rdc.*ones(1,991); Rac_single=Rdc.*ones(1,991); i=1; for frequency=1e6:0.1e6:100e6; L1_s=impedance_sandwich(rin(1),rout(1),rin(1),rout(1),height(1), height(1),d(1),d(1),t,t,s,ur,sigma,frequency); R1_s=real(L1_s); L1_s=(imag(L1_s))/(2*pi*frequency)+L1_air; L2_s=impedance_sandwich(rin(2),rout(2),rin(2),rout(2),height(2), height(2),d(2),d(2),t,t,s,ur,sigma,frequency); R2_s=real(L2_s); L2_s=(imag(L2_s))/(2*pi*frequency)+L2_air; L12_s=impedance_sandwich(rin(1),rout(1),rin(2),rout(2),height(1), height(2),d(1),d(2),t,t,s,ur,sigma,frequency); R12_s=real(L12_s); L12_s=(imag(L12_s))/(2*pi*frequency)+L12_air; 294 Transformers and Inductors for Power Electronics L_s(i)=L1_s+L2_s+2*L12_s; Rac_s(i)=R1_s+R2_s+2*R12_s; L1_single=impedance_substrate(rin(1),rout(1),rin(1),rout(1),height (1),height(1),d(1),d(1),t,ur,sigma,frequency); R1_single=real(L1_single); L1_single=(imag(L1_single))/(2*pi*frequency)+L1_air; L2_single=impedance_substrate(rin(1),rout(1),rin(1),rout(1),height (1),height(1),d(1),d(1),t,ur,sigma,frequency); R2_single=real(L2_single); L2_single=(imag(L2_single))/(2*pi*frequency)+L2_air; L12_single=impedance_substrate(rin(1),rout(1),rin(1),rout(1), height(1),height(1),d(1),d(1),t,ur,sigma,frequency); R12_single=real(L12_single); L12_single=(imag(L12_single))/(2*pi*frequency)+L12_air; L_single(i)=L1_single+L2_single+2*L12_single; Rac_single(i)=R1_single+R2_single+2*R12_single; i=i+1; end x=1e6:0.1e6:100e6; y1=L_s./L1; y2=L_single./L1; z1=Rac_s./Rdc+1; z2=Rac_single./Rdc+1; semilogx(x,y1,’-’,x,y2,’-’) axis([1e6,100e6,0,100]) ﬁgure; loglog(x,z1,’-’,x,z2,’-’) axis([1e6,100e6,1,2e3]) %File to deﬁne the function air_mutual function y=air_mutual(r1,r2,a1,a2,h1,h2,d1,d2) %This function is used to calculate the mutual inductance in air core %condition. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %z is the axis separation. z=0 for self-inductance calculation; z=|d2-d1| %for mutual inductance calculation. Planar Magnetics 295 global uo; uo=4*pi*1e-7; g=@(k)aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/r1)); end function y=aircoremul(r1,r2,a1,a2,h1,h2,d1,d2,k) z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; y=S1.*S2.*Q.*exp(-z.*k); end %File to deﬁne the function impedance_sandwich function y=impedance_sandwich(r1,r2,a1,a2,h1,h2,d1,d2,t1,t2,s,ur, sigma,freq) %This function is used to calculate the additional mutual impedance to the %presence of the substrate in sandwich structures. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. 296 Transformers and Inductors for Power Electronics %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %t1 and t2 are thickness of bottom and upper substrates %s substrate separation %ur relative permeability of the magnetic substrate %sigma relative permeability of the magnetic substrate %freq operation frequency global uo; uo=4*pi*1e-7; omega=2*pi*freq; g=@(k)integrand(r1,r2,a1,a2,h1,h2,d1,d2,t1,t2,s,ur,sigma,freq,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=1j.*omega.*uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/ r1)); end function y=integrand(r1,r2,a1,a2,h1,h2,d1,d2,t1,t2,s,ur,sigma,freq, k) global uo; z=d2-d1; d1_s=s-d1; d2_s=s-d2; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; Planar Magnetics 297 S2(1)=0; end; eta=sqrt((1j*2*pi.*freq*sigma*ur*4*pi*1e-7)+(k.^2)); phi=(ur.*k-eta)./(ur.*k+eta); lambda1=phi.*(1-exp(-2*t1.*eta))./(1-(phi.^2).*exp(-2*t1.*eta)); lambda2=phi.*(1-exp(-2*t2.*eta))./(1-(phi.^2).*exp(-2*t2.*eta)); denominator=1-lambda1.*lambda2.*exp(-2*s.*k); F=(lambda1.*exp(-(d1+d2).*k)+lambda2.*exp(-(d1_s+d2_s).*k))./ denominator; G=2.*lambda1.*lambda2.*exp(-2*s.*k).*cosh((d2-d1).*k)./denominator; y=S1.*S2.*Q.*(F+G); end %File to deﬁne the function impedance_substrate function y=impedance_substrate(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,sigma, freq) %This function is used to calculate the additional mutual impedance to the %presence of the substrate in sandwich structures. %r1,r2,h1,d1 are the inside radius, outside radius, height and upright %position of the cross-section 1. %a1,a2,h2,d2 are the inside radius, outside radius, height and upright %position of the cross-section 2. %t is the thickness of the substrate %ur relative permeability of the magnetic substrate %sigma relative permeability of the magnetic substrate %freq operation frequency global uo; uo=4*pi*1e-7; omega=2*pi*freq; g=@(k)integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,sigma,freq,k); for upper=1000:1000:1000000 [integalresult,err] = quadgk(g,0,upper); if err<0.01*integalresult integalresult_real=integalresult; else break; end end y=1j.*omega.*uo.*pi.*integalresult_real./(h1*log(a2/a1)*h2*log(r2/ 298 Transformers and Inductors for Power Electronics r1)); end function y=integrand(r1,r2,a1,a2,h1,h2,d1,d2,t,ur,sigma,freq,k) global uo; z=d2-d1; if z==0 Q=2.*(h1.*k+exp(-h1.*k)-1)./(k.^2); else Q=2.*(cosh(0.5.*(h1+h2).*k)-cosh(0.5.*(h1-h2).*k))./(k.^2); end S1=(besselj(0,r2.*k)-besselj(0,r1.*k))./k; S2=(besselj(0,a2.*k)-besselj(0,a1.*k))./k; if k(1)==0 Q(1)=0; S1(1)=0; S2(1)=0; end; eta=sqrt((1j*2*pi.*freq*sigma*ur*4*pi*1e-7)+(k.^2)); phi=(ur.*k-eta)./(ur.*k+eta); lambda=phi.*(1-exp(-2*t.*eta))./(1-(phi.^2).*exp(-2*t.*eta)); y=S1.*S2.*Q.*lambda.*exp(-(d1+d2).*k); end References 1. 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(2011) Microfabricated V-groove power inductors using multilayer Co-Zr-O thin ﬁlms for very-high-frequency DC-DC converters. Proceedings of the IEEE Energy Conversion Congress and Exposition, ECCE, pp. 1845–1852. 5. Dwight, H.B. (1919) Some new formulas for reactance coils. Transactions of the American Institute of Electrical Engineers, XXXVIII (2) 1675–1696. 6. Ferreira, J.A. (1994) Improved analytical modeling of conductive losses in magnetic components. IEEE Transactions on Power Electronics 9 (1), 127–131. 7. Gray, A. (1893) Absolute Measurements in Electricity and Magnetism, MacMillan, London. 8. Grover, F.W. (2004) Inductance Calculations: Working Formulas and Tables, Dover Publications Inc., New York. 9. Hui, S.Y.R. and Ho, W.W.C. (2005) A new generation of universal contactless battery charging platform for portable consumer electronic equipment. IEEE Transactions on Power Electronics 20 (3), 620–627. 10. Hui, S.Y.R. and Liu, X. (2010) Semiconductor transformers. 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IEEE Transactions on Industrial Electronics 59 (7), 2800–2810. 300 Transformers and Inductors for Power Electronics 26. Ouyang, Z., Zhang, Z., Andersen, M.A.E., and Thomsen, O.C. (2012) Four quadrants integrated transformers for dual-input isolated DC-DC converters. IEEE Transactions on Power Electronics 27 (6), 2697–2702. 27. Ouyang, Z., Zhang, Z., Thomsen, O.C., and Andersen, M.A.E. (2011) Planar-integrated magnetics (PIM) module in hybrid bidirectional DC-DC converter for fuel cell application. IEEE Transactions on Power Electronics 26 (11), 3254–3264. 28. Quinn, C., Rinne, K., O’Donnell, T. et al. (2001) A review of planar magnetic techniques and technologies. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 1175–1183. 29. Rodriguez, R., Dishman, J., Dickens, F., and Whelan, E. (1980) Modeling of two-dimensional spiral inductors. IEEE Transactions on Components, Hybrids, and Manufacturing Technology, 3 (4), 535–541. 30. Roshen, W.A. (1990) Effect of ﬁnite thickness of magnetic substrate on planar inductors. IEEE Transactions on Magnetics 26 (1), 270–275. 31. Roshen, W.A. (1990) Analysis of planar sandwich inductors by current images. IEEE Transactions on Magnetics 26 (5), 2880–2887. 32. Roshen, W.A. and Turcotte, D.E. (1988) Planar inductors on magnetic substrates. IEEE Transactions on Magnetics 24 (6), 3213–3216. 33. Sato, T., Hasegawa, M., Mizoguchi, T., and Sahashi, M. (1991) Study of high power planar inductor. IEEE Transactions on Magnetics 27 (6), 5277–5279. 34. Su, Y.P., Xun, L., and Hui, S.Y. (2008) Extended theory on the inductance calculation of planar spiral windings including the effect of double-layer electromagnetic shield. IEEE Transactions on Power Electronics 23 (4), 2052–2061. 35. Su, Y.P., Xun, L., and Hui, S.Y.R. (2009) Mutual inductance calculation of movable planar coils on parallel surfaces. IEEE Transactions on Power Electronics 24 (4), 1115–1123. 36. Sullivan, C.R. (2009) Integrating magnetics for on-chip power: Challenges and opportunities. Proceedings of the Custom Integrated Circuits Conference, CICC, pp. 291–298. 37. Sullivan, C.R. and Sanders, S.R. (1993) Microfabrication of transformers and inductors for high frequency power conversion. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 33–41. 38. Sullivan, C.R. and Sanders, S.R. (1996) Design of microfabricated transformers and inductors for highfrequency power conversion. IEEE Transactions on Power Electronics 11 (2), 228–238. 39. Tang, S.C., Hui, S.Y., and Chung, H.S.H. (1999) Coreless printed circuit board (PCB) transformers with multiple secondary windings for complementary gate drive circuits. IEEE Transactions on Power Electronics 14 (3), 431–437. 40. Tang, S.C., Hui, S.Y., and Chung, H.S.H. (2000) Coreless planar printed-circuit-board (PCB) transformers-a fundamental concept for signal and energy transfer. 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Proceedings of the IEEE Power Electronics and Drive Systems, PEDS, pp. 1143–1148. 10 Variable Inductance1 An inductor with an air gap of ﬁxed length, and operated so that the ﬂux density in the core is below the saturation value, has a ﬁxed value of inductance determined by the length of the air gap (the analysis is provided in Section 2.1). When the current is increased to the point where the ﬂux density enters the saturation region, the increased reluctance of the core (due to the reduced value of relative permeability) reduces the overall inductance. When the core material enters saturation, the operation of the inductor becomes non-linear and, while the resulting characteristics have advantages in certain applications, new challenges arise in terms of the circuit analysis. In this chapter, we explore the main physical manifestations of variable inductance, we examine their applications and analyze the approach to circuit simulation. The ﬂux level in the core is determined by the magnetic ﬁeld intensity, as dictated by Ampere’s law, and therefore by the current level in the coil. In saturation, the inductance decreases with increasing current as a consequence of progressively increasing saturation. In the case of a ﬁxed air gap, the saturation is dictated by the B-H characteristic of the magnetic core material, which means that we have no control of the inductance/current (L / i) characteristic once saturation is reached. A small measure of control may be introduced by using a stepped air gap; this is called a swinging inductor. A logical extension of the stepped air gap is to introduce an inﬁnite number of steps in the form of a sloped air gap (SAG) so that the onset of saturation takes place progressively from the narrow end of the gap to the wide end and, as we shall see later, this gives us a measure of control over the L/ i characteristic to suit the intended application. In a powder iron core, the gap is distributed and the manufacturer supplies a curve of magnetic permeability versus magnetic ﬁeld intensity, from which an L / i characteristic may be constructed using the analysis of Chapter 2. There are many applications where the required inductance is a function of the load level, and an L/i characteristic where the inductance falls off with increasing current is desirable. Such an L / i characteristic has an 1 Parts of this chapter are reproduced with permission from [1] Wolﬂe, W.H. and Hurley, W.G. (2003) Quasi-active power factor correction with a variable inductive ﬁlter: theory, design and practice. IEEE Transactions on Power Electronics 18 (1), 248–255; [2] Zhang, L., Hurley, W.G., and W€olﬂe, W.H. (2011) A new approach to achieve maximum power point tracking for PV system with a variable inductor. IEEE Transactions on Power Electronics 26 (4), 1031–1037. Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 302 Transformers and Inductors for Power Electronics advantage in that the largest inductance for the smallest value of current means that the stored energy (in the form of 1/2Li2) is lower at the higher current level. This, in turn, means that the overall size of the inductor is reduced compared to the size of an inductor based on the largest L and the largest i. In this chapter, we will introduce each type of variable inductor along with an application: the swinging inductor for voltage regulation, the SAG inductor for power factor correction and the powder iron core for maximum power point tracking in a solar PV system. We will show that a considerable reduction in the size of the inductor may be achieved while lower energy is stored in a saturated inductor at high current. The swinging inductor has a stepped air gap. The core may be considered to have two parallel reluctance paths, with each path having two reluctances in series, the core and the gap. As the current increases, the path containing the smaller gap reaches saturation ﬁrst and the increased reluctance reduces the overall inductance. This device has been employed for voltage regulation in rectiﬁer circuits, as it achieves continuous current even at small values of current. The sloped air gap (SAG) inductor [1] operates on the same principle as the swinging inductor. The air gap increases from its minimum value to its maximum value in a graded shape rather than in a discrete step. This means that the variation in inductance with current is more gradual, and we shall see later that the gradual characteristic of the SAG inductor is more suitable for power factor correction than the more abrupt transition afforded by a swinging inductor. The powder iron core is used in high frequency applications and the inductance is normally found from the manufacturer’s data sheet. In solar photovoltaic (PV) systems, impedance matching between the output of the solar panels and the load for maximum power transfer is normally achieved by varying the duty cycle of a buck converter [2]. The minimum inductance required for continuous conduction in maximum power point tracking (MPPT) is a function of the solar insolation and, therefore, a variable inductance is perfectly suited to this application. The voltage across an inductor is related to its ﬂux linkage and this, in turn is related to the current. The dependence of the inductance on its current must be taken into account. Recall from Chapter 2: l ¼ LðiÞi ð10:1Þ Invoking Faraday’s law: V ¼ dl ¼ LðiÞ di þ i dLðiÞ ¼ dt dt dt LðiÞ þ i dLðiÞ di di dt ¼ Leff di dt Leff ¼ LðiÞ þ i dLðiÞ di ð10:2Þ ð10:3Þ Leff in Equation 10.2 is readily found from the L / i characteristic of the inductor. The Leff versus current characteristic is more insightful, since most simulation models automatically assume this relationship. Most simulation packages assume that the inductance value is ﬁxed for variable inductance. We will need to take the variation with current into account. The ElectroMagnetic Transient Program (EMTP) [3] is particularly suited for this purpose, and it will be introduced later in the chapter. Variable Inductance 303 10.1 Saturated Core Inductor The inductance of a gapped core is normally dominated by the air gap. The overall inductance consists of the core and the air gap, with the dimensions shown in Figure 10.1. Including the reluctance of the core, the overall inductance is: L ¼ Rc N2 þ Rg ¼ N2 Rg 1 Rc Rg þ 1 ð10:4Þ The reluctance of the core is: Rc ¼ lc mrm0Ac ð10:5Þ and the reluctance of the gap is: Rg ¼ g m0Ag ð10:6Þ The overall inductance may be expressed in terms of the gap. Fringing may be neglected if the gap length is much smaller than the dimensions of the core cross-section: L ¼ N2 Rg 1 1 þ meff mr where the effective relative permeability is meff ¼ lc, g as described in Chapter 2. ð10:7Þ φ i + _e N turns g lc Ag Figure 10.1 Fixed air gap inductor. 304 Transformers and Inductors for Power Electronics When meff ( mr the inductance is dominated by the gap. Increasing the current leads to increased magnetic ﬁeld intensity and a drop in mr. In saturation, mr decreases towards meff and the value of inductance falls. The manufacturer normally supplies the variation of relative permeability with magnetic ﬁeld intensity in graphical form. An empirical relationship may be established in the following form, which is amenable to analysis in predicting inductance as a function of current: mr ¼ H Hm þ H0 ð10:8Þ Hm and H0 are constants. Example 10.1 Establish the constants H0 and Hm for M530-50A material whose permeability versus magnetic ﬁeld intensity is shown in Figure 10.2. Take two data points at: H ¼ 400; mr ¼ 2600 A=m H ¼ 5000; mr ¼ 255 A=m Inserting both values into Equation 10.8: 2; 600 ¼ Hm 400 þ H0 255 ¼ 5; Hm 000 þ H0 Solving yields H0 ¼ 100 A/m and Hm ¼ 1.3 Â 106 A/m. 8000 7000 6000 5000 Relative Permeability 4000 3000 2000 1000 0 102 103 104 H (A/m) Figure 10.2 Relative permeability data for M530-50A laminated steel. Variable Inductance 305 8000 7000 6000 5000 Relative 4000 Permeability 3000 2000 1000 Data Empirical 0 102 103 104 H (A/m) Figure 10.3 Empirical prediction of relative permeability. The empirical curve in Example 10.1 is compared with the actual data in Figure 10.3. At very low values of H (and current), mr is smaller than the value given by the empirical formula. However, this is not important, since the calculations of interest will be in the range of H above 200 A/m, where the agreement is very good. Typically, meff is of the order of 100, so that the region of the curve above 2000 A/m would have an impact on the inductance, as indicated by Equation 10.7. Invoking Ampere’s law for a gapped inductor with current i: Ni ¼ Hclc þ B m0 g ð10:9Þ Noting that B ¼ mrm0Hc and using the empirical relationship in Equation 10.8 yields a quadratic equation for mr: H0m2r þ Ni g þ H0meff À Hm mr À Hmmeff ¼ 0 ð10:10Þ Equation 10.10 is a quadratic equation of the form ax2 þ bx þ c ¼ 0 with: a ¼ H0 b¼ Ni g þ H0meff À Hm c ¼ ÀHmmeff ð10:11Þ ð10:12Þ ð10:13Þ mr is now pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ mr ¼ Àb Æ b2 À 4ac 2a ð10:14Þ 306 Transformers and Inductors for Power Electronics Solving for mr in Equation 10.10 and substituting into Equation 10.7 yields the desired value of inductance. The effective inductance described in Equation 10.3 may be calculated as follows: Leff ¼ LðiÞ þ i d LðiÞ di ¼ LðiÞ þ i dL d mr d mr di ð10:15Þ The individual terms on the right hand side of Equation 10.15 may be deduced from Equation 10.7: meff dL d mr ¼ L 1 þmm2rmerff ð10:16Þ and from Equation 10.10: " # dmr ¼ À 1 þ pﬃﬃﬃbﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N di 2a 2a b2 À 4ac g ð10:17Þ Example 10.2 An inductor constructed with an EI assembly EI42 (corresponding to IEC YEI 1-14) with M530-50A material, as described in Example 10.1, was constructed with g ¼ 0.5 mm, core length lc ¼ 8.4 cm, core cross-sectional area Ag ¼ 2.072 cm2 and with N ¼ 365 turns. Calculate the inductance L and the effec- tive inductance Leff as a function of current for the range 0.25 to 4 A. For low values of current before the onset of saturation, the inductance is a function of the gap and is: Lmax ¼ N2 Rg ¼ m0N2Ag g ¼ ð4p Â 10À7Þð365Þ2ð2:072 0:5 Â 10À3 Â 10À4Þ Â 103 ¼ 69:4 mH L is found from Equation 10.7: And for i ¼ 1.5 A: meff ¼ lc g ¼ 84 0:5 ¼ 168 a ¼ H0 ¼ 100 A=m b ¼ Ni g þ H0meff À Hm ¼ ð365Þð1:5Þ 0:5 Â 10À3 þ ð100Þð168Þ À 1:3 Â 106 ¼ À18:82 Â 104 A=m c mr ¼ ¼ ÀÀbHþmmpe2ffﬃbaﬃﬃ¼2ﬃﬃﬃÀﬃﬃÀﬃﬃﬃ4ﬃðﬃa1ﬃﬃcﬃ:ﬃ3¼Âð11086:8Þð21Â681Þ0¼4ÞÀþ2q:18ﬃðﬃ4ÀﬃﬃﬃﬃÂ1ﬃﬃ8ﬃﬃ1ﬃ:ﬃ8ﬃ0ﬃﬃ28ﬃðﬃﬃÂA2ﬃﬃﬃÞﬃ=ﬃð1ﬃm1ﬃ0ﬃﬃ0ﬃ4ﬃ0ﬃÞﬃ2ﬃÞﬃﬃÀﬃﬃﬃﬃﬃðﬃﬃ4ﬃﬃÞﬃﬃðﬃﬃ1ﬃﬃﬃ0ﬃﬃ0ﬃﬃÞﬃﬃðﬃﬃÀﬃﬃﬃﬃ2ﬃﬃ:ﬃ1ﬃﬃﬃ8ﬃﬃ4ﬃﬃﬃﬃÂﬃﬃﬃﬃ1ﬃﬃﬃ0ﬃﬃ8ﬃﬃÞﬃﬃ ¼ 2693 L ¼ N2 Rg 1 1 þ meff mr ¼ ð69:4 Â 10À3Þ 1 1 þ 168 2693 ¼ 65:3 mH Variable Inductance 307 80 70 60 50 Inductance (mH) 40 30 20 10 Inductance Effective Inductance 0 0.5 1 1.5 2 2.5 3 3.5 4 Current (A) Figure 10.4 Inductance and effective inductance of an inductor with a constant air gap. Leff is found from Equation 10.15: dL d mr ¼ L 1 meff =m2r þ meff =mr ¼ ð0:0653Þ 168=ð2693Þ2 1 þ 168=2693 ¼ 1:424 Â 10À6 H " # dmr ¼ À 1 þ pﬃﬃﬃbﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N di 2a 2a b2 À 4ac g 2 3 ¼ 64À 1 ð2Þð100Þ þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀﬃﬃﬃﬃ1ﬃﬃﬃ8ﬃﬃ:ﬃ8ﬃﬃﬃ2ﬃﬃﬃÂﬃﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃﬃ4ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ75 ð2Þð100Þ ð18:82 Â 104Þ2 À ð4Þð100ÞðÀ2:184 Â 108Þ Â 0:5 365 Â 10À3 ¼ À5:61 Â 103 AÀ1 Leff ¼ LðiÞ þ i dL dmr ¼ 0:0653 þ ð1:5Þð1:424 Â 10À6ÞðÀ5:61 Â 103Þ ¼ 53:3 mH dur di This process is repeated for the range of current and the plots of L and Leff are shown in Figure 10.4. The low current asymptotic value of inductance (Lmax in Figure 10.4) is determined solely by the air gap, as indicated by Equation 10.7 with mr ) meff, The inductance is reduced to 80% of Lmax where mr ¼ 4meff. Substituting this result into Equation 10.10 with Hm ) H0 yields the roll-off current: In Example 10.2: I80 ¼ 5Hmg 4N : ð10:18Þ I80 ¼ 5Hmg 4N ¼ ð5Þð1:3 Â 106Þð0:5 ð4Þð365Þ Â 10À3Þ ¼ 2:2 A: 308 Transformers and Inductors for Power Electronics At low values of current, Leff and L coincide as expected. Once saturation sets in, the shape of the characteristic is largely determined by the mr characteristic of the core material. The characteristic inductance versus current curve maybe be further modiﬁed or controlled by introducing a stepped air gap, whereby one or more gaps are operating under saturation. This type of inductor is called a swinging inductor. Example 10.3 Consider a Micrometals toroidal iron powder core with 52/m75 material. The initial permeability versus magnetic ﬁeld intensity curve is given in Figure 10.5. An inductor was constructed using a toroid with core length lc ¼ 4.23 cm, core cross-sectional area Ac ¼ 0.179 cm2 and with N ¼ 72 turns. Calculate the inductance L and the effective inductance Leff as a function of current for the range 1 to 4 A. For H ¼ 30 Oe, mr is read from Figure 10.5: i ¼ Hlc ¼ ð2387Þð4:23 Â 10À2Þ ¼ 1:403 A LðiÞ ¼ N mrðiÞ Â m0 Â 72 N2 Â Ac lc ¼ ð56:1Þð4p Â 10À7Þð72Þ204::12739ÂÂ1100ÀÀ24 ¼ 154:7 mH Leff ¼ LðiÞ þ i DL Di ¼ ð154:7 Â 10À6Þ þ ð1:403Þ ð154:7 À 176:7Þ Â 10À6 ð1:403 À 0:935Þ ¼ 88:7 mH The full set of calculations are summarized in Table 10.1, and L and Leff are plotted in Figure 10.6. Figure 10.5 Initial permeability curve for iron powder material. Reproduced with permission from Micrometals, Inc. Copyright 2007 Micrometals, Inc. Variable Inductance Table 10.1 Calculations for inductance and effective inductance H (Oe) H (A/m) mr i (A) 1 80 75 0.047 1.4 111 75 0.065 2 159 74.9 0.094 3 239 74 0.140 5 398 73.5 0.234 7 557 72.5 0.327 10 796 71.1 0.468 14 1114 68.4 0.655 20 1592 64.1 0.935 30 2387 56.1 1.403 50 3979 44 2.338 70 5571 35 3.273 100 7958 27 4.675 L (mH) 206.8 206.8 206.5 204.0 202.6 199.9 196.0 188.6 176.7 154.7 121.3 96.5 74.4 309 Leff (mH) 206.8 206.8 205.6 196.6 199.2 190.2 183.1 162.5 137.2 88.7 37.9 9.6 0.9 250 200 150 Inductance (μH) 100 50 Inductance Effective Inductance 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Current (A) Figure 10.6 Inductance and effective inductance of an inductor with an iron powder core. 10.2 Swinging Inductor The swinging inductor has a two-stepped air gap, as shown in Figure 10.7. The equivalent magnetic circuit may be obtained by dividing the core into two sections corresponding to the steps in the gap, as shown in Figure 10.8. Each branch represents one step and the overall inductance is found from the equivalent reluctance: L ¼ N2 R g1 1 1 þ meff1 mr1 þ N2 R g2 1 1 þ meff2 mr2 ð10:19Þ 310 Transformers and Inductors for Power Electronics i φ + _e N turns A g1 lc2 Ag 2 Figure 10.7 Swinging inductor. g1 g2 + Ni _ c1 c2 g1 g2 Figure 10.8 Equivalent magnetic circuit for a swinging inductor. where Rg1 is the reluctance of the gap for step 1, and likewise for step 2. L may be considered the series combination of two gapped inductances, with the core divided into two sec- tions. The analysis of the saturated core inductor in Section 10.0 above may be applied to each step. Example 10.4 An inductor constructed with an EI assembly EI42 (corresponding to IEC YEI 1-14) with M530-50A material described in Example 10.1 was constructed with g1 ¼ 0.39 mm, Ag1 ¼ 1:036 cm2, g2 ¼ 0.69 mm, Ag2 ¼ 1:036 cm2, core length lc ¼ 8.4 cm and with N ¼ 365 turns. Calculate the inductance L and the effective inductance Leff as a function of current for the range 0.25 to 4 A. For low values of current before the onset of saturation, the inductance is a function of the gaps. We can treat the swinging inductor as two inductors in series. For g1 ¼ 0.39 mm: Lmax1 ¼ N2 R g1 ¼ m0N2Ac g1 ¼ ð4p Â 10À7Þð365Þ2ð1:036 0:39 Â 10À3 Â 10À4Þ Â 103 ¼ 44:47 mH meff1 ¼ l c1 g1 ¼ 84 0:39 ¼ 215 Variable Inductance 311 And for i ¼ 2.0 A: a ¼ H0 ¼ 100 A=m b1 ¼ Ni g1 þ H0meff À Hm ¼ ð365Þð2:0Þ 0:39 Â 10À3 þ ð100Þð215Þ À ð1:3 Â 106Þ ¼ 5:93 Â 105 A=m c1 mr1 ¼ ¼ ÀHmmeff1 ¼ Àð1:3 Â 106Þð215Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Àb þ b2 À 4ac ¼ Àð5:93 Â 105Þ 2a À2:80 Â 108 A=m qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ ð5:93 Â 105Þ2 À ð4Þð100ÞðÀ2:80 Â 108Þ ð2Þð100Þ ¼ 439 L1 ¼ Lmax1 1 1 þ meff1 mr1 ¼ ð44:47Þ 1 1 þ 215 439 ¼ 29:8 mH dL d mr ¼ L "1 meff =m2r þ meff =mr ¼ ð0:0298Þ 215=ð439Þ2 1 þ 215=439 # ¼ 2:23 Â 10À5 H dmr ¼ À 1 þ pﬃﬃﬃbﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N di 2a 2 2a b2 À 4ac g 3 ¼ 64À 1 ð2Þð100Þ þ ð2Þð100Þqﬃðﬃ5ﬃﬃﬃ:ﬃ9ﬃﬃ3ﬃﬃﬃﬃÂﬃﬃﬃﬃﬃ1ﬃ5ﬃ0ﬃﬃ:ﬃ59ﬃÞﬃ3ﬃ2ﬃﬃﬃÂÀﬃﬃﬃﬃ1ﬃðﬃﬃ04ﬃﬃ5Þﬃﬃðﬃﬃ1ﬃﬃ0ﬃﬃﬃ0ﬃﬃÞﬃﬃðﬃﬃÀﬃﬃﬃﬃ2ﬃﬃ:ﬃ8ﬃﬃﬃ0ﬃﬃﬃÂﬃﬃﬃﬃﬃ1ﬃﬃ0ﬃﬃﬃ8ﬃﬃÞﬃ750:393Â6510À3 ¼ À604=A Leff1 ¼ LðiÞ þ i dL dmr dur di ¼ 0:0298 þ ð2:0Þð2:23 Â 10À5ÞðÀ604Þ ¼ 2:9 mH For: g2 ¼ 0.69 mm: And for i ¼ 2.0 A: Lmax2 ¼ 25:14 mH meff2 ¼ 122 a¼ b2 ¼ c2 ¼ mr2 ¼ L2 ¼ dL d mr ¼ dmr ¼ di Leff2 ¼ H0 ¼ 100 A=m À2:299 Â 105 A=m À1:583 Â 108 A=m 2853 24:1 mH 3:458 Â 10À7 H À4:429 Â 103 =A 21 mH The low current asymptotic value of the inductance in the swinging inductor is the sum of these two values: L ¼ L1 þ L2 ¼ 29:8 þ 24:1 ¼ 53:9 mH Leff ¼ 2:9 þ 21 ¼ 23:9 mH This process is repeated for the range of current, and the plot of L and Leff is shown in Figure 10.9. 312 Transformers and Inductors for Power Electronics 80 70 60 50 Inductance (mH) 40 30 20 10 Inductance Effective Inductance 0 0.5 1 1.5 2 2.5 3 3.5 4 Current (A) Figure 10.9 Inductance and effective inductance of a swinging inductor. The low current asymptotic value of inductance (Lmax in Figure 10.7) is given by Equation 10.19 with mr1 ) meff1 and mr2 ) meff2 . It is straightforward to establish the roll-off current in this case, by adopting the approach in Section 10.0. I80 is given by Equation 10.18 with [5/4] replaced by [5/(4 À g1/g2)] and g ¼ g1, yielding: I80 ¼ 5Hmg1 4 À g1 N g2 ð10:20Þ In Example 10.4: I80 ¼ ð5Þð1:3 Â 106Þð0:39 Â 10À3Þ 4 À 0:39 0:69 ð365Þ ¼ 2:0 A Before the onset of saturation in the larger gap, the overall inductance is determined by the relative permeability of the material (mr1) and the larger air gap (g2). The designer has additional control over the L/i characteristic by judicious selection of g1 and g2, so it seems logical to introduce on inﬁnite number of gaps or a sloped air gap. 10.3 Sloped Air Gap Inductor The inductor with a sloped air gap is illustrated in Figure 10.10, along with its dimensions. Neglecting fringing, the ﬂux linkage dl in the element dx Â D (assuming mr ) meff) is: dl ¼ N2i d RgðxÞ ¼ m0N2iD gðxÞ dx ð10:21Þ Variable Inductance 313 i + e_ N turns lc D φ l(0) g g(x) x dx d 0 Figure 10.10 Sloped air gap (SAG) inductor. where g(x) is the length of the air gap at x, given by: gðxÞ ¼ G À ðG À gÞx d ð10:22Þ G and g are the maximum and minimum dimensions of the sloped gap. The ﬂux linkage between x ¼ 0 and x is: lðxÞ ¼ ðx 0 m0N2iD gðxÞ dx ð10:23Þ The total ﬂux linkage is found, by performing the integration Equation 10.17 with x ¼ d, the total inductance is l(d)/i: Ld ¼ m0 N 2 Ag GÀg G ln g ð10:24Þ This is the total inductance before any part of the core becomes saturated. Saturation effects may be treated in the same manner described in Section 10.1. Consider again the element dx and apply Ampere’s law (Equation 10.9), with g replaced by g(x). Equation 10.10 applies with meff ¼ lc/g(x). H0mrðxÞ2 þ Ni gðxÞ þ H0meff À Hm mrðxÞ À Hmmeff ¼ 0 ð10:25Þ 314 Transformers and Inductors for Power Electronics The inductance of the element is: dL ¼ m0N2D gðxÞ 1 þ meff mrðxÞ dx ð10:26Þ The total inductance is obtained by integrating Equation 10.21 between x ¼ 0 and x ¼ d. This cannot be achieved analytically, since mr must be obtained from Equation 10.25. The core must be discretized into m elements (normally m ¼ 10 elements will sufﬁce). The total inductance is then: L ¼ m0N2D X mÀ1 i¼0 Dx gðxiÞ 1 þ meffi mri ð10:27Þ For m equal elements: Dx ¼ d m xi ¼ iDx þ DðxÞ 2 meffi ¼ lc gðxiÞ ð10:28Þ ð10:29Þ ð10:30Þ mri is obtained from Equation 10.25. Example 10.5 An inductor constructed with an EI assembly EI42 (corresponding to IEC YEI 1-14), with M530-50A material as described in Example 10.1, was constructed with G ¼ 1.0 mm, g ¼ 0.2 mm, lc ¼ 8.4 cm, Ag ¼ 2.027 cm2, d¼1.4394 cm and N ¼ 365 turns. Calculate the inductance L and the effective inductance Leff as a function of current for the range 0.25 to 4 A. For low values of current before the onset of saturation, the inductance is a function of the gap and is given by Equation 10.24: Ld ¼ m0 Á N2 Á Ag GÀg G ln g ¼ ð4p Â 10À7Þð365Þ2ð2:072 Â ð1:0 À 0:2Þ Â 10À3 10À4Þ 1:0 ln 0:2 Â 103 ¼ 69:8 mH We may divide the core into ten discrete elements, so that Dx ¼ 0.1 d. The MATLAB program to perform these calculations is included at the end of this chapter. The plot of L and Leff is shown in Figure 10.11. The shape of the graph, in this case, is controlled by the sloped gap, which means that we can achieve the desired characteristic by controlling the geometry of the gap. Variable Inductance 315 80 70 60 50 Inductance (mH) 40 30 20 10 Inductance Effective Inductance 0 0.5 1 1.5 2 2.5 3 3.5 4 Current (A) Figure 10.11 Inductance and effective inductance of a SAG inductor. 10.4 Applications 10.4.1 Power Factor Correction A typical AC/DC converter with an output buffer capacitor and a passive inductor for power factor correction is shown in Figure 10.12. P represents the input power to a second DC-DC converter stage. The output voltage ripple and the hold-up time of the circuit are determined by the time constant of the capacitor and load resistance. When the output voltage ripple is less than 10% of the peak DC output voltage, approximate analysis [1] shows that: C ¼ PoutT 2V Á DV ð10:31Þ where T is the mains period, V is the peak value of the input voltage and DV is the output voltage ripple. The value of the capacitance is proportional to the output power. For DV/ V ¼ 10%, the hold-up time for a drop of 20% in the output voltage is approximately equal to T or 20 ms in the case of a 50 Hz supply. iin + vin _ iL L iP iC + C vout P _ Figure 10.12 Rectiﬁer circuit with passive power factor correction. 316 Transformers and Inductors for Power Electronics Table 10.2 Harmonic limits for mains current in class D equipment (EN61000-3-2) Harmonic order Maximum permissible harmonic current (rms) At 75 W(A) 75 W < P < 600 W(mA/W) At 600 W(A) 3 0.255 3.4 2.04 5 0.142 1.9 1.14 7 0.075 1.0 0.60 9 0.037 0.5 0.30 Equation 10.31 shows that the ﬁlter inductor L plays no role in the output voltage ripple and hold-up time (for discontinuous conduction). However, it does determine the rectiﬁer line current harmonics and power factor. Limits for harmonic levels in the mains current are speciﬁed in the international standard EN61000-3-2 and are listed in Table 10.2 for class D equipment, which covers rectiﬁer circuits used in TVs and computer equipment. The standard does not require the power supply to meet the limits over the entire power range, but every customer will operate a standard power supply at a different power level. The nominal power of the end users equipment can be anywhere from 25% to 100% of the maximum rating of the power supply. It is reasonable, then, to assume that the power supply is designed to meet the limits between 75 W and the maximum power rating. The circuit in Figure 10.12 was simulated using the ElectroMagnetic Transient Program [3]. The input was 220 V at 50 Hz and the capacitance C was chosen for 10% output ripple. L was selected so that the limits in Table 10.2 were not exceeded. The simulations where carried out for the output load range from 75 W to 600 W. The values of L and C and the peak value of the input current are listed in Table 10.3. The capacitance values, predicted by the approximate formula in Equation 10.31, are higher than the exact values in Table 10.3. In practice, L is largely independent of C. The required value of L is a function of the current ﬂowing through it. The variation of ﬁlter inductance with load is illustrated in Figure 10.13 for the data in Table 10.3. A conventional inductor, which would ensure compliance to the harmonic limits for a power range of 100 to 600 W, would have a value of 51 mH. At 600 W, the 51 mH inductor would carry a peak current of 6 A and the stored energy in the inductor would be 918 mJ. An inductor with an inductance versus current characteristics as shown in Figure 10.13 would have its maximum stored energy at 600 W (i.e. 9 mH at 8.3 A), which represents a stored energy of 310 mJ. The size of an inductor is directly proportional to its energy storage capacity, so that the variable inductor would occupy less than 35% of the volume of a conventional inductor. Table 10.3 Minimum ﬁlter inductance as a function of load power Load (W) 75 100 200 300 400 C (mF) 54 71 143 215 285 L (mH) 68 51 26 17 13 Ipeak (A) 1 1.4 2.8 10.2 5.6 500 359 11 6.9 600 430 9 8.3 Variable Inductance 317 80 70 60 50 Minimal Imductance 40 (mH) 30 20 10 0 100 200 300 400 500 600 Power (W) 01 2 3 45 6 7 8 Current Peak (A) Figure 10.13 Variation of ﬁlter inductance with load power and input peak current. 10.4.2 Harmonic Control with Variable Inductance The circuit in Figure 10.14 has a variable inductance driven by a sinusoidal source of voltage. The variable inductance characteristic is shown in Figure 10.15 and may be described as: L ¼ Lmax 1 À 1 jij 2 Imax ð10:32Þ This inductance described by Equation 10.32 and illustrated in Figure 10.15 is an idealized form of variable inductance. The inductance falls to zero at i ¼ 2Imax. The effective inductance is obtained from Equation 10.3: Leff ¼ Lmax 1 À jij I max : ð10:33Þ In this case, the effective inductance falls to zero at i ¼ Imax, as shown in Figure 10.15. + Vsin ωt _ + i (t ) vL = di Leff dt _ Figure 10.14 Circuit with variable inductance. 318 Transformers and Inductors for Power Electronics Inductance Lmax L Leff 0 I max 2Imax Current Figure 10.15 Variable inductance as a function of current. The solution for the current i(t) in Figure 10.14 is straightforward and is given by: h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi iðtÞ ¼ Àsgnðcos ðtÞÞImax 1 À 1 À kjcos ðvtÞj ð10:34Þ where k ¼ 2I0 Imax and I0 ¼ V vLmax is the peak current for constant inductance Lmax. i(t)/ Imax is illustrated in Figure 10.16 for several values of k. For small values of k, the current is almost sinusoidal and therefore would have very low harmonics. The Fourier series for i(t) is given by [1]: X 1 iðtÞ ¼ a2nþ1cosð2n þ 1Þvt n¼0 ð10:35Þ 1.0 0.8 0.6 0.4 0.2 Normalized current 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 k=1 k = 0.8 k = 0.6 k = 0.4 k = 0.2 Time Figure 10.16 Current in the variable inductor. Variable Inductance 319 where: a2nþ1 ¼ " 4 ðÀ1Þn p 2n þ 1 À 1 þ 3F2 À 1 ; 1 ; 1; n 44 þ 3;1 22 À # n; k2 À 0 2@ 2n 1 2 þ 1 1Ak2nþ1 2 2 F 1n þ 1 4 ; n þ 3 4 ; 2n þ 2; k2 ð10:36Þ ÀÁ 2F1 and 3F2 are hypergeometric functions of type (2, 1) and (3, 2) respectively, and n m is a binomial coefﬁcient. This may be simpliﬁed for k > 0.5 to yield: " # iðtÞ % À p 4 3 k2 cos vt þ X 1 n¼1 k2nþ1 ð2n þ 1Þ2 cos ð2n þ 1Þvt ð10:37Þ which clearly show the dependency of the harmonics on k. As the peak current in the circuit approaches Imax (k ! 1), the harmonic content increases as illustrated in Figure 10.17 (values shown are normalized to the fundamental in each case). Adopting the conventional deﬁnition of total harmonic distortion (THD): THD ¼ v u u u tnﬃPﬃ16¼ﬃﬃﬃ1ﬃﬃIﬃﬃ2nﬃﬃ I 2 1 ð10:38Þ where In is the amplitude of the nth harmonic. 0.20 0.18 0.16 0.14 Normalized 0.12 Harmonics 0.10 Current 0.08 0.06 0.04 0.02 0.00 2 3 4 5 6 7 Harmonics Figure 10.17 Harmonics of the current. k=1 k = 0.8 k = 0.6 k = 0.4 k = 0.2 8 9 320 Transformers and Inductors for Power Electronics Total Harmonics Distortion 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Current Factor k Figure 10.18 Total harmonic distortion. The severity of the harmonÂic problem Àis illupstﬃrﬃﬃaﬃﬃtﬃeﬃﬃdﬃﬃﬃÁiÃn Figure 10.18, particularly as the peak current in the circuit, Ipeak ¼ Imax 1 À 1 À k , approaches Imax, that is k ! 1, where Imax is interpreted as the value of current where Leff becomes zero. In the practical inductors of Examples 10.2, 10.4 and 10.5, Imax could be taken as 2.2 A, 3.0 A and 3.5 A for the saturated core inductor (Figure 10.4), the swinging inductor (Figure 10.9) and the SAG inductor (Figure 10.11) respectively. This analysis shows that the effectiveness of the variable inductance in reducing unwanted harmonics is determined by the relationship between the peak current in the circuit and the roll-off of the inductance as current increases. Example 10.6 The rectiﬁer in Figure 10.12 has the speciﬁcations given in Table 10.4. Determine the harmonics when the SAG inductor of Example 10.5 is used at the input. For simulation purposes, the circuit is represented in Figure 10.19; the input represents the rectiﬁer output. PSPICE and other simulation packages are based on constant inductance. An alternative approach is to use discretized equations in the time domain. The method of simulation is based on the ElectroMagnetic Transient Program (EMTP) by Dommel [3], which can evaluate the variable inductance at each value of current. Table 10.4 Speciﬁcations Input voltage (rms) Output capacitance Inductor resistance Frequency Output power 220 V 330 mF 2.5 V 50 Hz 200 W Variable Inductance 321 1 i R2 + |Vsin ωt | _ L 3 iP + iC C vP _ Figure 10.19 Simulated rectiﬁer circuit. The general solution for the three nodes in Figure 10.19, recognizing that there are no input currents at nodes 2 and 3, is: 2 32 32 3 2 3 i1ðtnÞ Y11 Y 12 Y 13 v1ðtnÞ I 1 ðtnÀ1 Þ 664 0 775 ¼ 664 Y21 Y22 Y23775 Á 664 v2ðtnÞ775 þ 664 I2ðtnÀ1Þ775 0 Y 31 Y 32 Y 33 v3ðtnÞ I 3 ðtnÀ1 Þ The terms in the admittance matrix are: Y 11 ¼ 1 R Y 21 ¼ À 1 R Y31 ¼ 0 Y 12 ¼ À1 R Y 22 ¼ 1 R þ Dt 2Leff Y 32 ¼ Àt 2Leff Y13 ¼ 0 Y 23 ¼ À Dt 2Leff Y 33 ¼ Dt 2Leff þ 2C Dt tn ¼ nDt Leff must be calculated at i1(tnÀ1) according to Equation 10.15. The current sources Ii(tnÀ1) are known from past history: I1ðtnÀ1Þ ¼ 0 I2ðtnÀ1Þ ¼ I 2 ðtnÀ2 Þ þ Dt Leff ½v2ðtnÀ1Þ À v3ðtnÀ1Þ n ¼ 2; 3 . . . I3(tn–1) includes contributions from Leff and C and, since we are dealing with a constant power output rather than a load resistor, this effect may be included: I3ðtnÀ1Þ ¼ I30ðtnÀ1Þ À I 2 ðtnÀ1 Þ þ P0 v3ðtnÀ1Þ n ¼ 2; 3 . . . I30ðtnÀ1Þ ¼ ÀI30ðtnÀ2Þ À 4C Dt v3ðtnÀ1Þ n ¼ 2; 3 . . . 322 Transformers and Inductors for Power Electronics 8 7 6 5 Current (A) 4 3 2 1 SAG 0.2/1 mm Voltage Fixed gap 0.5 mm 0 1 2 3 4 5 6 7 8 9 10 Time (ms) Figure 10.20 Simulation results for 200 W power supply. The initial conditions are: v2ð0Þ ¼ v3ð0Þ ¼ iið0Þ ¼ 0; I2ð0Þ ¼ 0; I30ð0Þ ¼ 0I3ð0Þ ¼ 0: If i1 results in a negative value, it is reset to zero and both I2 and I3 are adjusted accordingly, since in reality the diodes block negative current at the output of the rectiﬁer. The results are shown in Figure 10.20 for one half cycle. The peak current in the circuit (3.0 A) is less than Imax (3.5 A in Figure 10.11). The harmonic spectrum of the current waveform is obtained by the Fast Fourier Transform and is shown in Figure 10.21. The harmonic currents are normalized to the fundamental current in each case. The limits from Table 10.2 are also shown, and the harmonics are within the allowable limits 0.8 0.6 Normalized Harmonic 0.4 Current 0.2 Limit Simulated 0 2 3 4 5 6 7 8 9 10 11 12 13 14 1516 Harmonics Figure 10.21 Harmonics with the variable inductor. Variable Inductance 323 10.4.3 Maximum Power Point Tracking Maximum power point tracking (MPPT) [2] is implemented in solar photovoltaic (PV) systems to achieve maximum power output as the ambient conditions, such as incident solar radiation and temperature, change. MPPT is normally achieved by either the perturb and observe method (P&O) or by the incremental conductance method (ICM). In the ICM approach, the output resistance of the PV panel is equal to the load resistance, as expected from the celebrated maximum power transfer theorem; this may be shown by linearizing the I-V output characteristic of a PV panel about the operating point, as illustrated in Figure 10.22. Thus, the equivalent resistance r at the maximum power point should meet the following equation: Àr ¼ À DV DI ¼ RLR ¼ VP IP ð10:39Þ Where RLR is the regulated resistance in order to achieve MPPT, and VP and IP are the PV voltage and current at maximum power. The actual load resistance RL is matched to RLR by a buck converter, as shown in Figure 10.23. The regulated resistance is related to the load resistance by: RLR ¼ 1 D2 RL ð10:40Þ where D is the duty cycle of the buck converter. The value of D is between 0 and 1, therefore RLR has a value between RL and inﬁnity. Consider two levels of illumination intensity at points (1) and (2) in Figure 10.22. The current at the MPP decreases going from (1) to point (2), which changes the value of the PV resistance at the MPP. In order to achieve MPPT, the regulated resistance RLR should be adjusted by changing the duty cycle D in Equation 10.40. The minimum inductance in a buck converter in continuous conduction mode (CMM) is given by: Lmin ¼ RLð1 À 2f s DÞ ð10:41Þ where fs is the switching frequency. ip Maximum Power Point 1 I sc RLR I p1 1 1 Ip2 2 r Vp1 Vp2 0 Voc Voc' Figure 10.22 MPPT based on impedance matching. 324 Transformers and Inductors for Power Electronics iP LC iL + M1 vP D1 C1 RL _ PV Panel Buck Converter for MPPT Figure 10.23 Maximum power transfer in a PV module. Combine Equations 10.39, 10.40 and 10.41 to yield: Lmin ¼ D2ð1 À DÞVP 2f sIP ð10:42Þ The PV voltage is relatively constant over the full range of solar intensity [2], so Equation 10.42 shows that the minimum inductance is a function of duty cycle D and the output current of the PV panel Ip under constant switching frequency. The minimum inductance to achieve CCM falls off with increasing PV current as the solar intensity increases. Conversely, the higher value of inductance required at light loads may be achieved without increasing the volume of the inductor. The role of the variable inductor in the stable operation of the buck converter is explained by reference to Figure 10.24. Continuous conduction can only be achieved with inductance L Traditional Inductor 1 L1 c Inductance Variable Inductor b L2 0 I o1 a Traditional Inductor 2 Current Continuous Boundary Io2 Io Current Figure 10.24 Comparisons of CCM conditions in an MPPT DC/DC converter with a variable inductance. Variable Inductance Table 10.5 Parameters under different load conditions Insolation (W/m2) VP (V) IP (A) 800 41.3 4.1 600 41.4 3.1 400 41.6 2.0 200 41.6 1.0 325 Maximum output power (W) 169 128 83.2 41.6 values above the dashed line in Figure 10.24 (the shaded area is off limits). The lower limit of load current (corresponding to low solar insolation) is given by IO1 as long as the inductance is greater than L1. At higher currents (and higher insolation levels), say, IO2, a smaller inductor L2 would sufﬁce, with the added advantage of a reduced volume occupied by the inductor. Conversely, setting the inductance at L2 would limit the lower load range to values of current (and solar insolation) greater than IO2. Example 10.7 The voltage and current at the maximum power point in a 200 W solar panel are listed in Table 10.5, along with the maximum power for various level of solar insolation. Calculate the internal resistance of the panel for each level of solar power, the duty cycle of the buck converter for impedance matching with a load resistance of 8 V operating at 20 kHz. Calculate the minimum inductance for continuous conduction. The internal resistance is given by Equation 10.39: r ¼ VP IP ¼ 41:3 4:1 ¼ 10:07 V The remaining values are 13.35, 20.80, and 41.60 V. The duty cycle is given by Equation 10.40: rﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D¼ RL ¼ r 8:0 10:07 ¼ 0:89 The remaining values are 0.77, 0.62, and 0.44. The minimum value of inductance to ensure continuous conduction is given by Equation 10.42: Lmin ¼ D2ð1 À DÞVP 2f sIP ¼ ð0:89Þ2ð1 À 0:89Þð41:3Þ ð2Þð20 Â 103Þð4:1Þ Â 106 ¼ 21:9 mH The remaining values are 45.5, 76.0, and 112.8 mH. Example 10.8 The inductance characteristic of the inductor in Example 10.3 satisﬁes the criteria for the inductancecurrent pairs in Example 10.7. 326 Transformers and Inductors for Power Electronics Table 10.6 Speciﬁcations Input voltage Output capacitance Inductor resistance Load resistance Frequency 41.6 V 80 mF 1V 8V 20 kHz The buck converter in Figure 10.23 has the speciﬁcations given in Table 10.6. Simulate the circuit and plot the inductor current for each of the following values of inductance: 1. constant inductance of 21.9 mH at solar insolation of 800 W/m2 2. constant inductance of 21.9 mH at solar insolation of 200 W/m2 3. variable inductance of Figure 10.6 at 200 W/m2 The circuit of Figure 10.23 has been simulated for the purposes of evaluating the response for the variable inductor. The method of simulation is based on the electromagnetic transient program (EMPT). The simulated circuit is shown in Figure 10.25. The series coil resistance R, which has a damping effect on the inductor current, is included and, for convenience, the solar panels and MOSFET voltage are represented by a rectangular PWM waveform with a variable duty cycle. The appropriate value of the duty cycle was calculated in Example 10.7. The general solution for the three nodes in Figure 10.25, recognizing that there are no input currents at node 2 and 3, is: 2 i1ðtnÞ 3 2 3 Y11 Y 12 Y 13 2 v1ðtnÞ 3 2 I1ðtnÀ1Þ 3 6664 i2ðtnÞ 7775 ¼ 6664 Y21 Y22 Y23 7775 Á 6664 v2ðtnÞ 7775 þ 6664 I2ðtnÀ1Þ 7775 i3ðtnÞ Y 31 Y 32 Y 33 v3ðtnÞ I 3 ðtnÀ1 Þ 1 i R2 + _ L 3 iL + iC C v RL _ Figure 10.25 Circuit for simulation. Variable Inductance 327 The terms in the admittance matrix are: Y 11 ¼ 1 R Y 12 ¼ À1 R Y13 ¼ 0 Y 21 ¼ À1 R Y 22 ¼ 1þ R Dt 2Leff Y 23 ¼ À Dt 2Leff Y31 ¼ 0 Y 32 ¼ À Dt 2Leff Y 33 ¼ Dt 2Leff þ 2C Dt tn ¼ nDt Leff must be calculated at i1(tnÀ1) according to Equation 10.15. The current sources I1(tnÀ1) are known from past history: I1ðtnÀ1Þ ¼ 0 I 2 ðtnÀ1 Þ ¼ I2ðtnÀ2Þ þ Dt Leff ½v2ðtnÀ1Þ À v3ðtnÀ1Þ n ¼ 2; 3 . . . I 3 ðtnÀ1 Þ ¼ I 30 ðtnÀ1 Þ À I 2 ðtnÀ1 Þ þ v3 ðtnÀ1 Þ RL n ¼ 2; 3 . . . I 30 ðtnÀ1 Þ ¼ ÀI30ðtnÀ2Þ À 4C Dt v3ðtnÀ1Þ n ¼ 2; 3 . . . We deﬁne the initial conditions of voltages and currents equal to 0. If i1 results in a negative value, it is reset to zero and I2 and I3 are adjusted. This is because, in reality, the diode D is in series with the PV panel, as shown in Figure 10.23, and will block the current ﬂowing into the PV panel. 1. Figure 10.26 shows the inductor current for 21.9 mH, at 800 W/m2 (this corresponds to point a in Figure 10.24 and the current is continuous). 10 5 Inductor Current (A) 0 9.75 9.8 9.85 9.9 9.95 10 Time (ms) Figure 10.26 Inductor current for 21.9 mH at 800 W/m2. 328 Transformers and Inductors for Power Electronics 10 5 Current (A) 0 –5 9.75 9.8 9.85 9.9 9.95 10 Time (ms) Figure 10.27 Inductor current for 21.9 mH at 200 W/m2. 2. Figure 10.27 shows the inductor current for 21.9 mH, at 200 W/m2 (point b in Figure 10.24) and, as expected, the converter is operating in discontinuous conduction mode. 3. Figure 10.28 shows that the variable inductor whose characteristic is shown in Figure 10.6 restores continuous conduction. The current waveform has sharp peaks and wide valleys that arise because of the lower value of inductance at higher current. The inductor current range is 2.27 A to 4.61 A, corresponding to a solar insolation level from 200 W/m2 to 800 W/m2. In this range, the minimum inductor falls from 111 mH to 21.9 mH. A conventional inductor would have 111 mH at 4.61 A, corresponding to a stored energy of 1.2 mJ. With a variable inductor, the stored energy at 111 mH and 2.27 A is 0.29 mJ and, at 21.9 mH with 4.61 A, the stored energy is 0.25 mJ. The size of an inductor is directly proportional to its stored energy, so that the variable inductor would occupy 25% of the volume of a conventional ﬁxed value inductor. 10 5 Current (A) 0 –5 9.75 9.8 9.85 9.9 9.95 10 Time (ms) Figure 10.28 Inductor current for variable inductance at 200 W/m2. Variable Inductance 329 10.4.4 Voltage Regulation In the AC/DC converter with an output buffer capacitor in Figure 10.12, we showed that the output voltage ripple is determined by the time constant of the capacitor and load resistance. We also showed, in Equation 10.31, that the ﬁlter inductor L plays no role in the output voltage ripple for discontinuous conduction of the input current; in this case, the average output voltage is approximately equal to the peak value of the input voltage (adjusted for the voltage ripple). All of the analysis of Section 10.4.1 is based on the assumption of discontinuous conduction. However, it is possible to introduce continuous conduction with a sufﬁciently large ﬁlter inductor. The advantage of this mode of operation is that the average output voltage is now independent of the load current and is simply given by the average of the rectiﬁed voltage waveform. This is possible because the input voltage is always connected to the output; in discontinuous conduction, when the load is disconnected from the input, the output voltage is determined by the characteristics of the load. The high peak charging current typical of a circuit with a buffer capacitor is avoided, the harmonics associated with the peak current are greatly reduced and this leads to an improvement in the input power factor. This inductor is sized according to the expected rated load current. However, at light loads, this inductance would be too small to maintain continuous conduction; the operation would revert to the discontinuous mode and the output voltage would rise towards the peak value of the input voltage. In the past, this situation was avoided by shunting the buffer capacitor with a bleeding resistor, with a consequential loss of efﬁciency. Another approach is to have the inductance value ‘swing’ to a higher value at low current, which led to the advent of the swinging inductor. The maximum inductance at low current ensured continuous conduction while improving voltage regulation over a wider current range. Traditionally, since these are 50 Hz or 60 Hz applications, laminated cores are used. This result would apply in the case of a purely resistive load and for an inductive load as long as continuous conduction is maintained. The Fourier series for the output voltage of a full wave rectiﬁer is: v ¼ 2V p À 4V p X 1 n¼1 cos ð2nvtÞ 1 À 4n2 ð10:43Þ where V is the peak value of the input voltage waveform to the full wave rectiﬁer. The average output voltage across the output ﬁlter inductor is zero, so therefore the average output voltage of a full wave rectiﬁer is: V dc ¼ 2V p ð10:44Þ – thus ensuring good output voltage regulation. The average current through the load resistance R is simply: Idc ¼ 2V pR ð10:45Þ 330 Transformers and Inductors for Power Electronics However, harmonic currents will arise from the harmonic voltages in Equation 10.43, and the sum of these harmonics must not exceed the DC current if continuous conduction is to be maintained. The amplitude of the harmonics of voltage decrease in the order 1/3 (n ¼ 1), 1/15 (n ¼ 2) and so on. The lowest frequency of the current is 2f, due to rectiﬁer action, and therefore it is a reasonable assumption to ensure that this current harmonic should be less than the DC component to ensure continuous conduction. The impedance of the output circuit consisting of the inductor, capacitor and load resist- ance in Figure 10.12 is: rﬃ1ﬃﬃﬃþﬃﬃﬃﬃﬃxﬃﬃ2ﬃﬃhﬃﬃﬃ1ﬃﬃﬃÀﬃﬃﬃﬃﬃyﬃﬃﬃﬃþﬃﬃﬃﬃyﬃﬃ2ﬃﬃiﬃﬃ Z¼R x 1 þ y2 ð10:46Þ where: x ¼ vL ¼ XL RR y ¼ vRC ¼ R XC ð10:47Þ ð10:48Þ At frequency 2f, this becomes: rﬃ1ﬃﬃﬃþﬃﬃﬃﬃﬃ4ﬃﬃxﬃﬃﬃ2ﬃﬃhﬃﬃ1ﬃﬃﬃÀﬃﬃﬃﬃﬃyﬃﬃﬃﬃþﬃﬃﬃﬃﬃ4ﬃﬃyﬃﬃ2ﬃﬃiﬃﬃ Z2 ¼ R x 1 þ 4y2 The amplitude of this harmonic of current is found from Equation 10.43: ð10:49Þ I2 ¼ 4V 3pZ2 ð10:50Þ Taking Idc in Equation 10.45 and I2 with Z2 given by Equation 10.49 yields: 5 À 16y2 À 72xy þ 36x2ð1 þ 4y2Þ > 0 ð10:51Þ A good rule of thumb [4] for the design of a choke to give continuous conduction with good regulation, based on this inequality, is: XL ! 0:4 R ð10:52Þ and: R ! 4:0 XC ð10:53Þ Variable Inductance 331 These conditions ensure that the AC component of the current waveform is less than the DC component, which thereby maintains continuous conduction. Example 10.9 Design the ﬁlter inductor for a 25 W load at 25 V rms input at 50 Hz that will maintain conduction down to 10% of rated load. The DC output voltage is: pﬃﬃ Vdc ¼ ð2Þ 2ð25Þ p ¼ 22:5 V The DC output resistance is: R ¼ ð22:5Þ2 ¼ 20:25 V 25 The inductance required by Equation 10.55 is: L ¼ 0:4R 2pf ¼ ð0:4Þð20:25Þ ð2pÞð50Þ Â 103 ¼ 25:8 mH The DC current is: Idc ¼ Vdc R ¼ 22:5 20:25 ¼ 1:11 A At 10% of rated load, the current is 0.111 A and the load resistance is: R ¼ ð22:5Þ2 2:5 ¼ 202:5 V The inductance required by Equation 10.55 is: L ¼ 0:4R 2pf ¼ ð0:4Þð202:5Þ ð2pÞð50Þ Â 103 ¼ 258 mH 10.5 Problems 10.1 Repeat Example 10.2 for a ﬁxed air gap with g ¼ 1.1 mm and comment on the results. 10.2 Repeat Example 10.3 for Micrometals À45 material with the characteristic data in Figure 10.5. 10.3 Repeat Example 10.4 for a swinging inductor with g1 ¼ 0.6 mm and g2 ¼ 0.12 mm. 10.4 Repeat Example 10.5 for a sloped air gap inductor with G ¼ 1.2 mm and g ¼ 0.6 mm. 10.5 Calculate the value of capacitance in Example 10.9 to ensure continuous conduction. 10.6 Design a swinging inductor to meet the requirements of Example 10.9, using the char- acteristics of the inductor in Figure 10.9, by changing the number of turns. 332 Transformers and Inductors for Power Electronics MATLAB Program for Example 10.2 % example 10.2 : Inductance and effective inductance of an inductor with a constant air-gap %constants muo = 4*pi*10^-7; lc = 84e-3; g = 0.5e-3; mueff = lc/g; N = 365; Ag = 2.072e-4; Ho = 100; Hm = 1.3e6; Lmax = muo*N^2*Ag/g; u=0; %loop for i = [0.25:0.1:4]; a = Ho; b = N*i/g+Ho*mueff-Hm; c = -Hm*mueff; mur = (-b+sqrt(b^2-4*a*c))/(2*a); L = Lmax*1/(1+mueff/mur); dL = L*(mueff/mur^2)/(1+mueff/mur); dmur = ((-1/(2*a))+b/(2*a*sqrt(b^2-4*a*c)))*N/g; Leff = L+i*dL*dmur; u = u+1; A(u) = i; B(u) = L; C(u) = Leff; end % plot plot(A,B,’k’,A,C,’b’,’LineWidth’,2) title(’Inductance and effective inductance of an inductor with a constant air-gap’) xlabel(’i’) ylabel(’L’) axis([0 4 0 80e-3]) grid off hold on Variable Inductance 333 MATLAB Program for Example 10.4 % example 10.4 : Inductance and effective inductance of a swinging inductor %constants muo = 4*pi*10^-7; lc = 84e-3; g1 = 0.39e-3; g2 = 0.69e-3; mueff1 = lc/g1; mueff2 = lc/g2; N = 365; Ag1 = 1.036e-4; Ag2 = 1.036e-4; Ho = 100; Hm = 1.3e6; Lmax1 = muo*N^2*Ag1/g1; Lmax2 = muo*N^2*Ag2/g2; u=0; %loop for i = [0.25:0.1:4]; a = Ho; b1 = N*i/g1+Ho*mueff1-Hm; b2 = N*i/g2+Ho*mueff2-Hm; c1 = -Hm*mueff1; c2 = -Hm*mueff2; mur1 = (-b1+sqrt(b1^2-4*a*c1))/(2*a); mur2 = (-b2+sqrt(b2^2-4*a*c2))/(2*a); L1 = Lmax1*1/(1+mueff1/mur1); L2 = Lmax2*1/(1+mueff2/mur2); dL1 = L1*(mueff1/mur1^2)/(1+mueff1/mur1); dL2 = L2*(mueff2/mur2^2)/(1+mueff2/mur2); dmur1 = ((-1/(2*a))+b1/(2*a*sqrt(b1^2-4*a*c1)))*N/g1; dmur2 = ((-1/(2*a))+b2/(2*a*sqrt(b2^2-4*a*c2)))*N/g2; Leff1 = L1+i*dL1*dmur1; Leff2 = L2+i*dL2*dmur2; L = L1+L2; Leff = Leff1+Leff2; u = u+1; A(u) = i; B(u) = L; C(u) = Leff; end 334 Transformers and Inductors for Power Electronics % plot plot(A,B,’k’,A,C,’b’,’LineWidth’,2) title(’Inductance and effective inductance of a swinging inductor’) xlabel(’i’) ylabel(’L’) axis([0 4 0 80e-3]) grid off hold on MATLAB Program for Example 10.5 % example 10.5 : Inductance and effective inductance of a SAG inductor clear all close all %constants muo = 4*pi*10^-7; lc = 84e-3; G = 1e-3; g = 0.2e-3; N = 365; Ag = 2.072e-4; D = 1.4394e-2; d = 1.4394e-2; Ho = 100; Hm = 1.3e6; Ld = (muo*N^2*Ag)/(G-g)*log(G/g); m = 10; deltax = d/m; %loop for different current u=0; for i = [0.25:0.1:4]; L = 0; Leff = 0; %loop of gap division for j = [0:1:m-1]; xj = j*deltax+deltax/2; gxj = G-((G-g)*xj)/d; mueffj = lc/gxj; a = Ho; b = N*i/gxj+Ho*mueffj-Hm; c = -Hm*mueffj; murj = (-b+sqrt(b^2-4*a*c))/(2*a); T = muo*N^2*D*deltax/(gxj*(1+mueffj/murj)); Variable Inductance 335 L = L+T; %————————————————————————————————————————————————% dLj = T*(mueffj/murj^2)/(1+mueffj/murj); dmurj = ((-1/(2*a))+b/(2*a*sqrt(b^2-4*a*c)))*N/gxj; Leffj = T+i*dLj*dmurj; Leff = Leff+Leffj; %————————————————————————————————————————————————% end u = u+1; A(u) = i; B(u) = L; C(u) = Leff; end % plot plot(A,B,’k’,A,C,’b’,’LineWidth’,2) title(’Inductance and effective inductance of a SAG inductor’) xlabel(’i’) ylabel(’L’) axis([0 4 0 80e-3]) grid off hold on References 1. Wolﬂe, W.H. and Hurley, W.G. (2003) Quasi-active power factor correction with a variable inductive ﬁlter: theory, design and practice. IEEE Transactions on Power Electronics 18 (1), 248–255. 2. Zhang, L., Hurley, W.G., and W€olﬂe, W.H. (2011) A new approach to achieve maximum power point tracking for PV system with a variable inductor. IEEE Transactions on Power Electronics 26 (4), 1031–1037. 3. Dommel, H.W. (1969) Digital computer solution of electromagnetic transients in single-and multiphase networks. IEEE Transactions on Power Apparatus and Systems PAS-88 (4), 388–399. 4. Dunham, C.R. (1934) Some considerations in the design of hot-cathode mercury-vapour rectiﬁer circuits. Proceedings of the Institution of Electrical Engineers, Wireless Section 9 (27), 275–285. Further Reading 1. Benavides, N.D. and Chapman, P.L. (2007) Boost converter with a reconﬁgurable inductor. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 1695–1700. 2. Dishman, J.M., Kressler, D.R., and Rodriguez, R. (1981) Characterization, modeling and design of swinging inductors. Proceedings of the Power Conversion Conference, Powercon 8, pp. B3.1-B3.13. 3. Jain, S. and Agarwal, V. (2007) A single-stage grid connected inverter topology for solar PV systems with maximum power point tracking. IEEE Transactions on Power Electronics 22 (5), 1928–1940. 4. Kelley, A.W., Nance, J.L., and Moore, M.D. (1991) Interactive analysis and design program for phase-controlled rectiﬁers. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 271–277. 5. Liserre, M., Teodorescu, R., and Blaabjerg, F. (2006) Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values. IEEE Transactions on Power Electronics 21 (1), 263–272. 336 Transformers and Inductors for Power Electronics 6. Medini, D. and Ben-Yaakov, S. (1994) A current-controlled variable-inductor for high frequency resonant power circuits. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, APEC, pp. 219–225. 7. Patel, H. and Agarwal, V. (2008) Maximum power point tracking scheme for PV systems operating under partially shaded conditions. IEEE Transactions on Industrial Electronics 55 (4), 1689–1698. 8. Redl, R. (1994) Power-factor correction in single phase switching mode power supplies - An overview. International Journal of Electronics 77 (5), 555–582. 9. Redl, R., Balogh, L., and Sokal, N.O. (1994) A new family of single-stage isolated power-factor correctors with fast regulation of the output voltage. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 1137–1144. 10. Wolﬂe, W., Hurley, W.G., and Arnold, S. (2000) Power factor correction for AC-DC converters with cost effective inductive ﬁltering. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 332–337. 11. Wolﬂe, W., Hurley, W.G., and Lambert, S. (2001) Quasi-active power factor correction: the role of variable inductance. Proceedings of the IEEE Power Electronics Specialists Conference, PESC, pp. 2078–2083. Appendix A Table A.1 Wire data AWG IEC Bare Number Diameter (mm) AWG Bare Diameter (mm) 10 2.588 2.5 11 2.308 2.24 12 2.05 2.0 13 1.83 1.8 14 1.63 1.6 15 1.45 1.4 16 1.29 1.25 17 1.15 1.12 18 1.02 1.00 19 0.912 0.9 20 0.813 0.8 21 0.724 0.71 22 0.643 Resistance @ 20 C (mV/m) 3.270 3.480 4.111 4.340 5.211 5.440 6.539 6.720 8.243 8.500 10.42 11.10 13.16 13.90 16.56 17.40 21.05 21.80 26.33 26.90 33.13 34.00 41.78 43.20 52.97 Weight (g/m) 46.76 43.64 37.19 38.14 29.34 27.93 23.38 22.62 18.55 17.87 14.68 13.69 11.62 10.91 9.234 8.758 7.264 6.982 5.807 5.656 4.615 4.469 3.660 3.520 2.887 Overalla Diameter (mm) 2.721 2.631 2.435 2.366 2.171 2.120 1.947 1.916 1.742 1.711 1.557 1.506 1.392 1.351 1.248 1.217 1.114 1.093 1.002 0.9900 0.8985 0.8850 0.8012 0.7900 0.7197 Current @ 5 A/mm2 (A) Turns per cm2 26.30 12 24.54 12 20.92 14 19.70 14 16.50 20 15.71 20 13.15 23 12.72 23 10.43 30 10.38 30 8.256 39 7.700 42 6.535 52 6.140 52 5.193 68 4.930 68 4.086 80 3.930 85 3.266 99 3.180 105 2.596 126 2.510 126 2.058 161 1.980 168 1.624 195 (continued ) Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 338 Appendix A Table A.1 (Continued) AWG IEC Bare Number Diameter (mm) AWG Bare Diameter (mm) 0.63 23 0.574 0.56 24 0.511 0.5 25 0.455 0.45 26 0.404 0.4 27 0.361 0.355 28 0.32 0.315 29 0.287 0.28 30 0.254 0.25 31 0.226 0.224 32 0.203 0.2 33 0.18 0.18 34 0.16 0.16 35 0.142 0.14 36 0.127 0.125 37 0.114 0.112 38 0.102 0.1 39 0.0889 0.08 40 0.0787 41 0.0711 42 0.0635 43 0.0559 aGrade 2 or medium insulation. Resistance @ 20 C (mV/m) Weight (g/m) Overalla Diameter (mm) 54.80 66.47 69.40 83.87 87.10 105.8 108.0 134.2 136.0 168.0 173.0 213.9 219.0 265.9 278.0 339.4 348.0 428.8 434.0 531.4 554.0 675.9 672.0 855.5 850.0 1086.0 1110.0 1358.0 1393.0 1685.0 1735.0 2105.0 2176.0 2771.0 3401.0 3536.0 4328 5443 7016 2.771 2.300 2.190 1.823 1.746 1.445 1.414 1.114 1.117 0.9099 0.8800 0.7150 0.6930 0.5751 0.5470 0.4505 0.4360 0.3566 0.3500 0.2877 0.2790 0.2262 0.2270 0.1787 0.1790 0.1408 0.1370 0.1126 0.1100 0.0907 0.0880 0.0726 0.0700 0.0552 0.0450 0.0433 0.0353 0.0282 0.0218 0.7060 0.6468 0.6320 0.5806 0.5690 0.5213 0.5160 0.4663 0.4620 0.4204 0.4140 0.3764 0.3710 0.3494 0.3340 0.3054 0.3010 0.2742 0.2720 0.2484 0.2450 0.2220 0.2220 0.1990 0.1990 0.1783 0.1760 0.1613 0.1590 0.1455 0.1430 0.1313 0.1290 0.1160 0.1050 0.1036 0.091 0.0855 0.0739 Current @ 5 A/mm2 (A) 1.559 1.294 1.232 1.025 0.982 0.813 0.795 0.6409 0.6280 0.5118 0.4950 0.4021 0.3900 0.3235 0.3080 0.2534 0.2450 0.2006 0.1970 0.1618 0.1570 0.1272 0.1270 0.1005 0.1010 0.0792 0.0770 0.0633 0.0610 0.0510 0.0490 0.0409 0.0390 0.0310 0.0250 0.0243 0.0199 0.0158 0.0123 Turns per cm2 216 247 270 314 340 389 407 492 504 621 635 780 780 896 941 1184 1235 1456 1512 1817 1840 2314 2314 2822 2822 3552 3640 4331 4645 5372 5520 6569 6853 8465 10300 10660 13734 15544 20982 Appendix A Table A.2 List of manufacturers ACME CERAMINC MAGNETICS DMEGC EILOR EPCOS FAIR-RITE FDK FERRITE INT FERRONICS FERROXCUBE HIMAG HITACHI ISKRA ISU JFE(KAWTATETSU) KASCHKE KRVSTINEL MAGNETICS MAGNETICS METALS MICROMETALS MK MAGNETICS NEOSID NICERA ORB ELECTRICAL STEELS PAYTON SAILCREST SAMWHA STEWARD TAKRON (TOHO) TDG TDK THOMSON TOKIN TOMITA TRANSTEK MAGNETICS 339 http://www.acme-ferrite.com.tw http://www.cmi-ferrite.com http://www.chinadmegc.com http://www.magmet.com http://www.epcos.com http://www.fair-rite.com http://www.fdk.com http://www.tscinternational.com http://www.ferronics.com http://www.ferroxcube.com http://www.himag.co.uk http://www.hitachimetals.com http://www.iskra-ferrites.com http://www.isu.co.kr http://www.jfe-frt.com http://www.kaschke.de http://www.mmgna.com http://www.mag-inc.com http://www.magmet.com http://www.micrometals.com http://www.mkmagnetics.com http://www.neosid.com.au http://www.nicera.co.jp http://www.orb.gb.com http://www.paytongroup.com http://www.sailcrestmagnetics.com http://www.samwha.com http://www.steward.com http://www.toho-zinc.co.jp http://www.tdgcore.com http://www.tdk.com http://www.avx.com http://www.nec-tokin.com http://www.tomita-electric.com http://www.transtekmagnetics.com Index AC ﬂux, 12 AC resistance, 11, 105, 163, 181 Air-gap, 18, 56, 96, 232, 301 fringing, 48 length, 28, 48, 82, 301 sloped, 301, 312 stepped, 302, 308 Alloy, 16, 38, 96 amorphous, 17,19 cobalt-iron, 17 laminated iron, 17 nickel-iron, 17 nickel-zinc, 18, 265 manganese-zinc, 17, 265 silicon-iron, 18 Ambient temperature, 64, 128 Ampere’s law, 2, 12, 25, 55, 97, 162 American Wire Gauge (AWG), 337 B-H loop, 8, 17 measurement, 12, 225 Bessel function, 40, 161, 199, 249 Bobbin, 59, 146, 247, 265 Buck converter, see Converter, buck Buck-boost converter, see Converter, buck-boost Capacitance, 102, 183, 221, 248, 315 effective, 238, 242 measurement, 221, 239 series, 238 shunt, 238 Ceramic, 17, 265, 270, 274 Choke, 18, 84, 330 Coil circular, 39, 47, 250 planar, 45, 250, 258 Complex permeability, see Permeability, complex Conduction area, 11, 59, 112, 165 Conductivity, electrical, 11, 161, 203, 248 Continuous conduction mode (CCM), 324 Convection, 58, 248 forced, 59 natural, 59 Converter buck, 64, 84, 302, 323 buck-boost, 75 ﬂyback, 74, 237 forward, 69, 116, 124, 134 full-bridge, 148 half-bridge, 149 push-pull, 124, 140, 181, 207 resonant, 84, 151, 186 Copper loss, 10, 57, 114, 124, 181, 228 eddy current, 49, 159 ohmic, 113 proximity effect, 84, 105, 123, 159 skin effect, 181 Copper wires data, 337 Core cross-sectional area, 97, 113, 200, 306 Core length, 29, 48, 96, 200, 306 Coreless transformer, 248, 270 Core loss, 10, 58, 99, 123, 159, 221 eddy current, 6, 96, 197, 249, 272 hysteresis, 8, 36, 104, 123, 204, 227 laminations, 14, 197, 212 measurement, 114, 229 Transformers and Inductors for Power Electronics: Theory, Design and Applications, First Edition. W. G. Hurley and W. H. W€olﬂe. Ó 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 342 Index Core material, 18, 28, 64, 123, 197, 301 Core shape, 266 Core volume, 38 Core window winding area product, 61, 125 Coupling coefﬁcient, 107 Cross-sectional area, 12, 61 core, 25, 48, 96, 200, 237, 306 gap, 28 conductor, 68, 132 Curie temperature, 9, 15, 20, 123 Current density, 3, 41, 59, 113, 124, 160 Current transformer, 17, 25 Current waveform factor, 56, 78 DC resistance, 11, 68, 159, 229, 252, 270 Diamagnetic, see Magnetic materials, diamagnetic Dielectric, 146, 239, 254, 267, 272 Dipole, 8, 10, 14 Discontinuous conduction mode (DCM), 328 Discrete PCB, 265 Dissipation, 27,56, 109, 123, 159, 168 Domains, magnetic, 8 Dot convention, 99, 107 Dowell, 159, 186 Duty cycle, 71, 136, 181, 302 Eddy-current loss, 10, 18 conductor, round, 160 core, 104, 114, 197 foil, 164 lamination, 13, 96 planar winding, 249 Effective capacitance, 238 Effective inductance, 224, 306 Effective permeability, see Permeability, effective Effective resistance, 10, 173 Efﬁciency, 124, 232, 247, 329 ElectroMagnetic Transient Program (EMTP), 302, 320 Electroplating, 266 Elliptic integrals, 40, 249 Faraday’s law, 5, 30, 69, 97, 136, 213 Ferrites, 4, 17, 39, 96, 197, 271 Ferromagnetic, see Magnetic materials, ferromagnetic Filaments, 39, 45, 248 Flux density, 2, 26, 99, 160, 203, 301 maximum, 55, 82, 97, 202, 226 optimum, 115, 125, 144 remanent, 8 residual, 8, 12, 225 saturation, 8, 56, 115, 123, 225, 265 Flux linkage, 5, 30, 46, 222, 302 Flyback converter, see Converter, ﬂyback Foil, 11, 145, 165, 182, 266 Forward converter, see Converter, forward Fourier series, 159, 171, 318, 329 Free space magnetic permeability, 3 Fringing ﬂux, 49 Full-bridge converter, see Converter, full-bridge Gain-phase analyser, 226, 232 Gapped core, 54 Gauss’s law, 6 General Steinmetz equation (GSE), 14, 69, 114 Geometric Mean Distance (GMD), 42, 47 Half-bridge converter, see Converter, half-bridge Hard magnetic materials, see Magnetic materials, hard Harmonics, 103, 117, 316, 318, 329 Heat conduction, 58 Heat convection, 58, 248 transfer coefﬁcient, 59 High frequency effects, 16, 139, 159, 197 High frequency loss, 63, 109, 129 Hysteresis, 8, 11, 36, 103 loop, 8, 12, 15, 226 loss, 10, 48, 104, 123, 204, 249 Ideal transformer, see Transformer, ideal Impedance, 25, 41, 197, 212 incremental, 223 internal, 162 leakage, 235 load, 100 mutual, 237, 249 reﬂected, 100 self, 200, 261 thermal, 84 Improved general Steinmetz equation (iGSE), 58, 204 Incremental permeability, see Permeability, incremental Inductance, 1, 11, 32, 64, 197, 248 leakage, 105, 184, 236 magnetizing, 96, 103 mutual, 30, 39, 106, 248 Index 343 self, 32, 42, 106, 197, 211, 252 variable, 301, 316, 320 Inductance measurement, 221 incremental impedance method, 223 step voltage method, 222 Inductor design examples buck, 64 ﬂyback, 75 forward, 69 resonant, 84 Initial permeability, see Permeability, initial Insulation, 18, 60, 80, 109, 123, 146 basic, 147 double, 146 reinforced, 146 single, 147 standards, 148 Integrated magnetic, 270 Integrated PCB, 265 Interleaving, 183 Internal impedance, see Impedance, internal Iron alloys, see Alloy Iron loss, see Core loss Isolation, 17, 25, 71, 95, 123, 146 Lamination, 13, 17, 96, 197, 212 Layer thickness, 159, 169, 179 Leakage ﬂux, 37, 102, 105, 184–186 Leakage inductance, see Inductance, leakage Lenz’s law, 5, 10, 30, 97, 197 Litz wire, 11 Loss angle, 209 Low temperature co-ﬁred ceramic (LTCC), 270 Magnetic circuit law, 4, 27, 102, 123 Magnetic dipole, 14 Magnetic energy, 185 Magnetic ﬁeld intensity, 2, 26, 55, 98, 161, 226 Magnetic ﬂux density, 2, 7, 14, 26, 203 Magnetic materials, 1, 9, 16, 36, 247 diamagnetic, 10, 15 ferromagnetic, 7, 12, 27, 204, 249 hard, 9, 13, 16 paramagnetic, 15 soft, 9, 12, 17 Magnetic moment, 8 Magnetic permeability, 7, 14, 99, 301 Magnetic substrate, 25, 253, 268 Magnetic susceptibility, 14 Magnetization, 8, 14, 109 curve, 15, 36, 103, 226 Magnetizing current, 97, 102, 227 Magnetizing inductance, see Inductance, magnetizing Magnetomotive force (mmf), 26, 35, 96, 102, 164, 183 Manganese, 17, 265 Material constants, 14, 126 Maximum ﬂux density, see Flux density, maximum Maximum power point tracking (MPPT), 302, 323 Maxwell’s equations, 1, 39, 160, 198, 254 Mean length per turn (MLT), 56, 113, 127, 185, 186 Measurements B-H loop, 12, 225 capacitance, 221, 239 core loss, 114, 229 inductance, 221 open circuit test, 227 short circuit test, 227 Metallic glass, 19 Microelectromechanical systems (MEMS), 271 Microelectronics, 247, 270 Molybdenum permalloy (MPP), 18, 71 Multiple windings, 74 Mutual inductance, see Inductance, mutual Nanocrystalline materials, 17 Nanotechnology, 247 Non-sinusoidal, 14, 109, 123, 204 Normal magnetisation curve, 15, 103, 226 Open circuit test, see Measurement, open circuit test Operating temperature, 63, 124, 147 Optimum ﬂux density, 115, 125, 144 layer thickness, 159, 175 permeability, 57 Paramagnetic, see Magnetic materials, paramagnetic Parasitic capacitance, 36 Parasitics, 248, 265 Peak working voltage, 146 Permalloy, 20, 266, 272 Permanent magnet, 9, 16 344 Index Permeability, see also Magnetic permeability absolute, 15 complex, 16, 209 effective, 18, 29, 55, 236 incremental, 15 initial, 15, 211, 308 optimum, 57 relative, 3, 26, 48, 96, 160, 202 static, 15 Permeance, 27 Planar coil, 45, 250, 261 Planar inductor, 248 Planar transformer, 266 Porosity, 165, 169 Poynting vector, 167 Powder iron, 17, 38, 71, 237, 301 Power factor correction, 151, 302, 315 Power supply, 51, 222, 316 Power supply on chip (PwrSoC), 248, 265, 271 Primary winding, 75, 95, 139, 163, 228, 242 Printed circuit board (PCB), 265, 274 discrete, 265 integrated, 266 Proximity effect, 11, 64, 113, 129, 163, 183 Pulse-width modulation (PWM), 326 Push-pull converter, see Converter, push-pull Reactance, 104, 163, 186, 228, 235 Rectiﬁer, 17, 117, 129, 148, 302, 329 Reﬂected impedance, see Impedance, reﬂected Regulation, see Voltage regulation Relative permeability, see Permeability, relative Relative permittivity, 243 Reluctance, 27, 36, 48, 96, 102, 301 Remanent magnetism, 9 Resistivity, 17, 56, 63, 113, 197, 210 Resonant frequency, 221, 238 Rise time, 181 Root-mean-square (RMS), 10, 56, 97, 111, 146, 172 Safe extra low voltage (SELV), 147 Saturation ﬂux density, see Flux density, saturation Screen, 242 Screen printing, 267, 270 Secondary winding, 75, 96, 118, 141, 164, 181 Self inductance, 32, 42, 48, 106, 197, 252 Short-circuit test, see Measurements, short circuit test Silicon, 17, 265, 271 Skin depth, 10, 105, 160, 181, 186, 199, 258 Skin effect, 11, 113, 139, 160, 181 Silicon steel, 4, 13, 18, 197, 214 Sinusoidal excitation, 14, 58, 96, 123, 159, 204 Sloped air-gap (SAG), 301, 313, 320 Soft magnetic material, see Magnetic materials, soft Solar, 302, 323 Solenoid, 21, 30, 267 Stacking factor, 109 Steinmetz equation general, 14, 69, 114 improved, 58, 204 Stored energy, 25, 35, 57, 124, 301, 328 Substrate, 249, 253 thickness, 259 Surface area, 59, 159, 248, 265 Susceptibility, 14 Swinging inductor, 301, 308, 320, 329 Temperature coefﬁcient of resistivity, 64 Temperature rise, 55, 61, 109, 124, 159, 265 Thermal resistance, 56, 59 Thermal impedance, 84 Thick ﬁlm, 247, 265, 270 Thickness, see Optimum layer thickness, Substrate thickness Thin ﬁlm, 18, 247, 265, 271 Transformer ideal, 96, 107 practical, 102 Transformer efﬁciency, 109 Transformer tests, see Measurements, open circuit test, short circuit test Turns ratio, 71, 98, 182, 229 VA, 112, 125 Variable inductance, see Inductance, variable Voltage regulation, 109, 302, 329 Voltage ripple, 315, 329 Voltage waveform factor, 110, 128 Winding area, 75 Winding loss, see Copper loss Window utilisation factor, 59, 64, 75, 113, 146 Wire insulation, 147, 243 Working voltage, 146 Zinc, 17, 265

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