Transformers and Inductors for Power Electronics Theory, Design and Applications

Based on the fundamentals of electromagnetics, this clear and concise text explains basic and applied principles of transformer and inductor design for power electronic applications. It details both the theory and practice of inductors and transformers employed to filter currents, store electromagnetic energy, provide physical isolation between circuits, and perform stepping up and down of DC and AC voltages.

The authors present a broad range of applications from modern power conversion systems. They provide rigorous design guidelines based on a robust methodology for inductor and transformer design.  They offer real design examples, informed by proven and working field examples.

Key features include: 

• emphasis on high frequency design, including optimisation of the winding layout and treatment of non-sinusoidal waveforms
• a chapter on planar magnetic with analytical models and descriptions of the processing technologies
• analysis of the role of variable inductors, and their applications for power factor correction and solar power
• unique coverage on the measurements of inductance and transformer capacitance, as well as tests for core losses at high frequency
• worked examples in MATLAB, end-of-chapter problems, and an accompanying website containing solutions, a full set of instructors? presentations, and copies of all the figures.

Covering the basics of the magnetic components of power electronic converters, this book is a comprehensive reference for students and professional engineers dealing with specialised inductor and transformer design. It is especially useful for senior undergraduate and graduate students in electrical engineering and electrical energy systems, and engineers working with power supplies and energy conversion systems who want to update their knowledge on a field that has progressed considerably in recent years.

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

2.3.1 Why Use a Core? 35

2.3.2 Distributed Gap 38

2.4 Self and Mutual Inductance of Circular Coils 39

2.4.1 Circular Filaments 39

2.4.2 Circular Coils 40

2.5 Fringing Effects around the Air Gap 48

2.6 Problems 51

References 53

Further Reading 54

Chapter 3 Inductor Design 55

3.1 The Design Equations 55

3.1.1 Inductance 55

3.1.2 Maximum Flux Density 55

3.1.3 Winding Loss 56

3.1.4 Optimum Effective Permeability 57

3.1.5 Core Loss 58

3.1.6 The Thermal Equation 58

3.1.7 Current Density in the Windings 59

3.1.8 Dimensional Analysis 61

3.2 The Design Methodology 61

3.3 Design Examples 64

3.3.1 Example 3.1: Buck Converter with a Gapped Core 64

3.3.2 Example 3.2: Forward Converter with a Toroidal Core 69

3.4 Multiple Windings 74

3.4.1 Example 3.3: Flyback Converter 75

3.5 Problems 84

References 89

Further Reading 89

SECTION II TRANSFORMERS 93

Chapter 4 Transformers 95

4.1 Ideal Transformer 96

4.1.1 No Load Conditions 97

4.1.2 Load Conditions 98

4.1.3 Dot Convention 99

4.1.4 Reflected Impedance 100

4.1.5 Summary 101

4.2 Practical Transformer 102

4.2.1 Magnetizing Current and Core Loss 102

4.2.2 Winding Resistance 105

4.2.3 Magnetic Leakage 105

4.2.4 Equivalent Circuit 107

4.3 General Transformer Equations 109

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

7.5 Problems 214

References 216

Further Reading 216

SECTION III ADVANCED TOPICS 219

Chapter 8 Measurements 221

8.1 Measurement of Inductance 221

8.1.1 Step Voltage Method 222

8.1.2 Incremental Impedance Method 223

8.2 Measurement of the B-H Loop 225

8.3 Measurement of Losses in a Transformer 227

8.3.1 Short-Circuit Test (Winding/Copper Loss) 228

8.3.2 Open-Circuit Test (Core/ Iron Loss) 229

8.3.3 Core Loss at High Frequencies 232

8.3.4 Leakage Impedance at High Frequencies 235

8.4 Capacitance in Transformer Windings 237

8.4.1 Transformer Effective Capacitance 238

8.4.2 Admittance in the Transformer Model 239

8.5 Problems 244

References 245

Further Reading 245

Chapter 9 Planar Magnetics 247

9.1 Inductance Modelling 248

9.1.1 Spiral Coil in Air 249

9.1.2 Spiral Coil on a Ferromagnetic Substrate 253

9.1.3 Spiral Coil in a Sandwich Structure 261

9.2 Fabrication of Spiral Inductors 265

9.2.1 PCB Magnetics 265

9.2.2 Thick Film Devices 267

9.2.3 LTCC Magnetics 270

9.2.4 Thin Film Devices 271

9.2.5 Summary 274

9.3 Problems 275

References 298

Further Reading 299

Chapter 10 Variable Inductance 301

10.1 Saturated Core Inductor 303

10.2 Swinging Inductor 309

10.3 Sloped Air Gap Inductor 312

10.4 Applications 315

10.4.1 Power Factor Correction 315

10.4.2 Harmonic Control with Variable Inductance 317

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

William Gerard Hurley, Department of Electrical and Electronic Engineering, National University of Ireland, Galway

Professor Hurley is the Founder/Director of the Power Electronics Research Centre at NUI, Galway. He has over 35 years of experience in the field of Power Electronics, specifically dealing with Magnetics, and is a Fellow of the Institution of Engineers of Ireland. He is currently an Associate Editor of the Journal of Advances in Power Electronics. He has published more than 100 papers in peer reviewed journals and conference proceedings with over 700 citations.

Werner H. Wölfle, Convertec Ltd., Ireland

Dr. Wölfe is currently a Managing Director of Convertec Ltd., a company that develops high-reliability power converters for industrial applications. He has been involved with designing magnetic components for power electronics for over 30 years and is also an Adjunct Professor of Electrical Engineering at the National University of Ireland, Galway.