Transformers and Inductors for Power Electronics: Theory, Design and Applications

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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
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
viii Contents
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
Contents ix
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
x Contents
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