Transformers and Inductors for Power Electronics - W. G. Hurley - ebook

Transformers and Inductors for Power Electronics ebook

W. G. Hurley

349,99 zł


Based on the fundamentals of electromagnetics, this clear andconcise text explains basic and applied principles of transformerand inductor design for power electronic applications. It detailsboth the theory and practice of inductors and transformers employedto filter currents, store electromagnetic energy, provide physicalisolation between circuits, and perform stepping up and down of DCand AC voltages. The authors present a broad range of applications from modernpower conversion systems. They provide rigorous design guidelinesbased on a robust methodology for inductor and transformerdesign. They offer real design examples, informed by provenand working field examples. Key features include: * emphasis on high frequency design, including optimisation ofthe winding layout and treatment of non-sinusoidal waveforms * a chapter on planar magnetic with analytical models anddescriptions of the processing technologies * analysis of the role of variable inductors, and theirapplications for power factor correction and solar power * unique coverage on the measurements of inductance andtransformer capacitance, as well as tests for core losses at highfrequency * worked examples in MATLAB, end-of-chapter problems, and anaccompanying website containing solutions, a full set ofinstructors' presentations, and copies of all thefigures. Covering the basics of the magnetic components of powerelectronic converters, this book is a comprehensive reference forstudents and professional engineers dealing with specialisedinductor and transformer design. It is especially useful for seniorundergraduate and graduate students in electrical engineering andelectrical energy systems, and engineers working with powersupplies and energy conversion systems who want to update theirknowledge on a field that has progressed considerably in recentyears.

Ebooka przeczytasz w aplikacjach Legimi na:

czytnikach certyfikowanych
przez Legimi

Liczba stron: 361



Title Page



About the Authors





Chapter 1: Introduction

1.1 Historical Context

1.2 The Laws of Electromagnetism

1.3 Ferromagnetic Materials

1.4 Losses in Magnetic Components

1.5 Magnetic Permeability

1.6 Magnetic Materials for Power Electronics

1.7 Problems


Further Reading

Section I: Inductors

Chapter 2: Inductance

2.1 Magnetic Circuits

2.2 Self and Mutual Inductance

2.3 Energy Stored in the Magnetic Field of an Inductor

2.4 Self and Mutual Inductance of Circular Coils

2.5 Fringing Effects around the Air Gap

2.6 Problems


Further Reading

Chapter 3: Inductor Design

3.1 The Design Equations

3.2 The Design Methodology

3.3 Design Examples

3.4 Multiple Windings

3.5 Problems


Further Reading

Section II: Transformers

Chapter 4: Transformers

4.1 Ideal Transformer

4.2 Practical Transformer

4.3 General Transformer Equations

4.4 Power Factor

4.5 Problems


Further Reading

Chapter 5: Transformer Design

5.1 The Design Equations

5.2 The Design Methodology

5.3 Design Examples

5.4 Transformer Insulation

5.5 Problems

Further Reading

Chapter 6: High Frequency Effects in the Windings

6.1 Skin Effect Factor

6.2 Proximity Effect Factor

6.3 Proximity Effect Factor for an Arbitrary Waveform

6.4 Reducing Proximity Effects by Interleaving the Windings

6.5 Leakage Inductance in Transformer Windings

6.6 Problems


Further Reading

Chapter 7: High Frequency Effects in the Core

7.1 Eddy Current Loss in Toroidal Cores

7.2 Core Loss

7.3 Complex Permeability

7.4 Laminations

7.5 Problems


Further Reading

Section III: Advanced Topics

Chapter 8: Measurements

8.1 Measurement of Inductance

8.2 Measurement of the B-H Loop

8.3 Measurement of Losses in a Transformer

8.4 Capacitance in Transformer Windings

8.5 Problems


Further Reading

Chapter 9: Planar Magnetics

9.1 Inductance Modelling

9.2 Fabrication of Spiral Inductors

9.3 Problems


Further Reading

Chapter 10: Variable Inductance

10.1 Saturated Core Inductor

10.2 Swinging Inductor

10.3 Sloped Air Gap Inductor

10.4 Applications

10.5 Problems


Further Reading

Appendix A


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

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.

MATLAB® 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 MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

Library of Congress Cataloging-in-Publication Data

Hurley, William G.

Transformers and inductors for power electronics: theory, design andapplications / W.G. Hurley, W.H. Wölfle.

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. Wölfle, Werner H. II. Title.

TK2551.H87 2013



ISBN 978-1-119-95057-8

To Our Families

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 Wölfle 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.


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 Ó'Mathúna, 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, Université Paul Sabatier, Toulouse and Lionel Breuil, University of Nantes. Dr Pádraig Ó'Catháin 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.


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 Wölfle 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 Wölfle 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 Wölfle, 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 Wölfle 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 Wölfle 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. KassakianProfessor of Electrical EngineeringThe Massachusetts Institute of Technology


1. McLyman, Colonel W.T. (1978) Transformer and Inductor Design Handbook. Marcel Dekker, Inc., New York.


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 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 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 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 WölfleNational University of Ireland, Galway, IrelandMarch 2013


The following is a list of symbols used in this book, and their meanings.

AAverage or geometric mean radiusAcCross-sectional area of magnetic coreAgCross-sectional area of the gapALInductance per turnAmEffective cross-sectional area of magnetic circuitApProduct of window winding area × cross-sectional areaAtSurface area of wound transformerAwBare wire conduction areaaTransformer turns ratioa1, a2Inside and outside radii of a coilBmaxMaximum flux densityBoOptimum flux densityBsatSaturation flux densitybWinding dimension: see Figure 6.4CeffEffective capacitance of a transformerDDuty cycledThickness of foil or layerd1, d2Height of filaments or coil centres above ferromagnetic substrateΦMagnetomotive force, mmffFrequency in hertzG, gMaximum and minimum air gap lengthsGMDGeometric mean distance between coilsg(x)Air-gap length at xhWinding dimension: see Figure 2.14hcCoefficient of heat transfer by convectionh1, h2Coil heights in axial directionÎPeak value of the current waveformIdcAverage value of currentInRMS value of the nth harmonic of currentIn(x), Kn(x)Modified Bessel functions of the first and second kind, respectivelyI'rmsRMS value of the derivative of the current waveformIrmsRMS value of the current waveformJoCurrent densityJ(r)Current density at radius rJ0(x), J1(x)Bessel functions of the first kindKcMaterial parameterK(f), E(f)Complete elliptic integrals of the first and second kind, respectivelyKiCurrent waveform factorKt48.2 × 103KvVoltage waveform factorkCoupling coefficientka, kc, kwDimensionless constants (see Equations 3.25, 3.26 and 3.27)kfCore stacking factor Am/AckiDefined in Figure 7.28kpPower factorkpnRatio of the AC resistance to DC resistance at nth harmonic frequencyksSkin-effect factorkuWindow utilization factorLSelf-inductanceLeffEffective inductanceLlLeakage inductanceLmMagnetizing inductanceLsAdditional coil inductance due to ferromagnetic substratelcMagnetic path length of coreMMutual inductanceMLTMean length of a turnm√(jωμ0σ)NNumber of turns in coilnHarmonic numberPcuCopper or winding lossPfeIron or core lossPoOutput powerPvPower loss per unit volumepNumber of layersRAverage or geometric mean radiusReluctanceRacAC resistance of a winding with sinusoidal excitationRdcDC resistance of a windingReffEffective AC resistance of a winding, with arbitrary current waveformRδDC resistance of a winding of thickness δ0RθThermal resistancer1, r2Inside and outside radii of a coilroRadius of bare wiresSubstrate separation in sandwich structureTPeriod of a waveformTaAmbient temperatureTmaxMaximum operating temperaturetSubstrate thicknesstrRise time (0–100%)VrmsRMS value of the voltage waveformVAVoltampere rating of windingVcVolume of coreVoDC output voltageVsDC input voltageVwVolume of windingAverage value of voltage over time τWaWindow winding area of coreWcElectrical conduction areaWmStored energy in a magnetic fieldwWinding dimension: see Figure 6.4ZImpedanceZiInternal impedance of a conductorzAxial separationα, βMaterial constantsα20Temperature co-efficient of resistivity at 20°CΔRatio d/δ0ΔBFlux density rippleΔTTemperature riseΔVOutput voltage rippleδSkin depthδ0Skin depth at fundamental frequencyδnSkin depth at the nth harmonic frequencyϕFluxϕ(k)Defined in Equation 9.49Defined in Equation 9.58γRatio of iron loss to copper lossΛDefined in Equation 9.36λFlux linkageμStatic or absolute permeabilityμ0Magnetic permeability of free space 4π × 10–7 H/mμeffEffective relative permeabilityμiInitial permeabilityμincIncremental permeabilityμoptOptimum value of effective relative permeabilityμrRelative permeabilityμrsComplex relative permeabilityηPorosity factorρ20Electrical resistivity at 20 °CρwElectrical resistivityσElectrical conductivityτTime for flux to go from zero to its maximum valueΨ(5p2–1)/15ωAngular frequency (rad/s)



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.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!