Preface xvii

About the Author xix

List of Symbols xxi

**1 Fundamentals of Magnetic Devices 1**

1.1 Introduction 1

1.2 Fields 2

1.3 Magnetic Relationships 2

1.3.1 Magnetomotive Force 2

1.3.2 Magnetic Field Intensity 3

1.3.3 Magnetic Flux 3

1.3.4 Magnetic Flux Density 4

1.3.5 Magnetic Flux Linkage 5

1.4 Magnetic Circuits 6

1.4.1 Reluctance 6

1.4.2 Magnetic KVL 8

1.4.3 Magnetic Flux Continuity 8

1.5 Magnetic Laws 9

1.5.1 Amp`ere’s Law 9

1.5.2 Faraday’s Law 13

1.5.3 Lenz’s Law 15

1.5.4 Volt–Second Balance 16

1.5.5 Ohm’s Law 16

1.5.6 Biot–Savart’s Law 18

1.5.7 Maxwell’s Equations 19

1.5.8 Maxwell’s Equations for Good Conductors 24

1.5.9 Poynting’s Vector 24

1.5.10 Joule’s Law 26

1.6 Eddy Currents 29

1.7 Core Saturation 32

1.7.1 Core Saturation for Sinusoidal Inductor Voltage 34

1.7.2 Core Saturation for Square-Wave Inductor Voltage 36

1.7.3 Core Saturation for Rectangular Wave Inductor Voltage 38

1.8 Inductance 40

1.8.1 Definitions of Inductance 40

1.8.2 Inductance of Solenoid 45

1.8.3 Inductance of Inductor with Toroidal Core 47

1.8.4 Inductance of Inductor with Torus Core 49

1.8.5 Inductance of Inductor with Pot Core 49

1.8.6 Inductance Factor 49

1.9 Air Gap in Magnetic Core 51

1.9.1 Inductance 51

1.9.2 Magnetic Field in Air Gap 53

1.10 Fringing Flux 54

1.10.1 Fringing Flux Factor 54

1.10.2 Effect of Fringing Flux on Inductance for Round Air Gap 57

1.10.3 Effect of Fringing Flux on Inductance for Rectangular Air Gap 58

1.10.4 Method of Effective Air Gap Cross-Sectional Area 60

1.10.5 Method of Effective Length of Air Gap 60

1.10.6 Patridge’s Fringing Factor 60

1.10.7 Distribution of Fringing Magnetic Field 62

1.11 Inductance of Strip Transmission Line 62

1.12 Inductance of Coaxial Cable 62

1.13 Inductance of Two-Wire Transmission Line 63

1.14 Magnetic Energy and Magnetic Energy Density 64

1.14.1 Magnetic Energy Density 64

1.14.2 Magnetic Energy Stored in Inductors with Ungapped Core 64

1.14.3 Magnetic Energy Stored in Inductors with Gapped Core 65

1.15 Self-Resonant Frequency 69

1.16 Quality Factor of Inductors 69

1.17 Classification of Power Losses in Magnetic Components 69

1.18 Noninductive Coils 71

1.19 Summary 71

1.20 References 74

1.21 Review Questions 76

1.22 Problems 78

**2 Magnetic Cores 81**

2.1 Introduction 81

2.2 Properties of Magnetic Materials 81

2.3 Magnetic Dipoles 83

2.4 Magnetic Domains 89

2.5 Curie Temperature 90

2.6 Magnetic Susceptibility and Permeability 91

2.7 Linear, Isotropic, and Homogeneous Magnetic Materials 93

2.8 Magnetic Materials 93

2.8.1 Ferromagnetic Materials 93

2.8.2 Antiferromagnetic Materials 94

2.8.3 Ferrimagnetic Materials 95

2.8.4 Diamagnetic Materials 95

2.8.5 Paramagnetic Materials 96

2.9 Hysteresis 96

2.10 Low-Frequency Core Permeability 98

2.11 Core Geometries 99

2.11.1 Toroidal Cores 99

2.11.2 CC and UU Cores 100

2.11.3 Pot Cores 100

2.11.4 PQ and RM Cores 101

2.11.5 EE and EDT Cores 102

2.11.6 Planar Cores 103

2.12 Ferromagnetic Core Materials 103

2.12.1 Iron Cores 104

2.12.2 Ferrosilicon Cores 104

2.12.3 Amorphous Alloy Cores 104

2.12.4 Nickel–Iron and Cobalt–Iron Cores 105

2.12.5 Ferrite Cores 105

2.12.6 Powder Cores 106

2.12.7 Nanocrystalline Cores 107

2.13 Superconductors 108

2.14 Hysteresis Loss 109

2.15 Eddy-Current Core Loss 113

2.15.1 General Expression for Eddy-Current Core Loss 113

2.15.2 Eddy-Current Core Loss for Sinusoidal Inductor Voltage 115

2.15.3 Eddy-Current Power Loss in Round Core for Sinusoidal Flux Density 117

2.15.4 Total Core Power Loss for Sinusoidal Inductor Voltage 118

2.15.5 Eddy-Current Core Loss for Square-Wave Inductor Voltage 122

2.15.6 Eddy-Current Core Loss for Rectangular Inductor Voltage 124

2.15.7 Eddy-Current Power Loss in Laminated Cores 128

2.15.8 Excess Core Loss 129

2.16 Steinmetz Empirical Equation for Total Core Loss 129

2.16.1 Losses of Ungapped Cores 129

2.16.2 Losses of Gapped Cores 134

2.17 Core Losses for Nonsinusoidal Inductor Current 135

2.18 Complex Permeability of Magnetic Materials 136

2.18.1 Series Complex Permeability 137

2.18.2 Loss Angle and Quality Factor 141

2.18.3 Complex Reluctance 144

2.18.4 Complex Inductance 145

2.18.5 Complex Impedance of Inductor 145

2.18.6 Approximation of Series Complex Permeability 146

2.18.7 Parallel Complex Permeability 148

2.18.8 Relationships Between Series and Parallel Complex Permeabilities 150

2.19 Cooling of Magnetic Cores 151

2.20 Summary 152

2.21 References 157

2.22 Review Questions 160

2.23 Problems 161

**3 Skin Effect 163**

3.1 Introduction 163

3.2 Resistivity of Conductors 164

3.2.1 Temperature Dependance of Resistivity 164

3.3 Skin Depth 166

3.4 AC-to-DC Winding Resistance Ratio 173

3.5 Skin Effect in Long Single Round Conductor 173

3.6 Current Density in Single Round Conductor 175

3.6.1 Bessel Differential Equation 175

3.6.2 Kelvin Functions 176

3.6.3 Approximations of Bessel’s Equation Solution 177

3.6.4 Current Density J (r)/J (0) 177

3.6.5 Current Density J (r)/J (ro ) 178

3.6.6 Current Density J (r)/JDC 180

3.6.7 Approximation of Current Density in Round Conductor 183

3.6.8 Impedance of Round Conductor 184

3.6.9 Approximation of Resistance and Inductance of Round Conductor 189

3.6.10 Simplified Derivation of Round Wire Resistance 191

3.7 Magnetic Field Intensity for Round Wire 193

3.8 Other Methods of Determining the Round Wire Inductance 195

3.9 Power Loss Density in Round Conductor 200

3.10 Skin Effect in Single Rectangular Plate 204

3.10.1 Magnetic Field Intensity in Single Rectangular Plate 204

3.10.2 Current Density in Single Rectangular Plate 206

3.10.3 Power Loss in Single Rectangular Plate 208

3.10.4 Impedance of Single Rectangular Plate 211

3.11 Skin Effect in Rectangular Foil Conductor Placed Over Ideal Core 215

3.12 Summary 218

3.13 Appendix 220

3.13.1 Derivation of Bessel Equation for Long Round Wire 220

3.14 References 222

3.15 Review Questions 223

3.16 Problems 224

**4 Proximity Effect 226**

4.1 Introduction 226

4.2 Orthogonality of Skin and Proximity Effects 227

4.3 Proximity Effect in Two Parallel Round Conductors 227

4.4 Proximity Effect in Coaxial Cable 228

4.5 Proximity and Skin Effects in Two Parallel Plates 230

4.5.1 Magnetic Field in Two Parallel Plates 230

4.5.2 Current Density in Two Parallel Plates 231

4.5.3 Power Loss in Two Parallel Plates 235

4.5.4 Impedance of Each Plate 243

4.6 Antiproximity and Skin Effects in Two Parallel Plates 244

4.6.1 Magnetic Field in Two Parallel Plates 244

4.6.2 Current Density in Two Parallel Plates 247

4.6.3 Power Loss in Two Parallel Plates 248

4.7 Proximity Effect in Open-Circuit Conductor 249

4.8 Proximity Effect in Multiple-Layer Inductor 250

4.9 Self-Proximity Effect in Rectangular Conductors 256

4.10 Summary 259

4.11 Appendix 260

4.11.1 Derivation of Proximity Power Loss 260

4.12 References 261

4.13 Review Questions 263

4.14 Problems 263

**5 Winding Resistance at High Frequencies 265**

5.1 Introduction 265

5.2 Eddy Currents 265

5.3 Magnetic Field Intensity in Multilayer Foil Inductors 266

5.4 Current Density in Multilayer Foil Inductors 274

5.5 Winding Power Loss Density in Individual Foil Layers 278

5.6 Complex Winding Power in nth Layer 281

5.7 Winding Resistance of Individual Foil Layers 282

5.8 Orthogonality of Skin and Proximity for Individual Foil Layers 284

5.9 Optimum Thickness of Individual Foil Layers 286

5.10 Winding Inductance of Individual Layers 291

5.11 Power Loss in All Layers 292

5.12 Impedance of Foil Winding 293

5.13 Resistance of Foil Winding 294

5.14 Dowell’s Equation 294

5.15 Approximation of Dowell’s Equation 298

5.15.1 Approximation of Dowell’s Equation for Low and Medium Frequencies 298

5.15.2 Approximation of Dowell’s Equation for High Frequencies 299

5.16 Winding AC Resistance with Uniform Foil Thickness 300

5.16.1 Optimum Uniform Foil Thickness of Inductor Winding for Sinusoidal Inductor Current 301

5.16.2 Boundary Between Low and Medium Frequencies for Foil Windings 306

5.17 Transformation of Foil Conductor to Rectangular, Square, and Round Conductors 308

5.18 Winding AC Resistance of Rectangular Conductor 309

5.18.1 Optimum Thickness of Rectangular Conductor for Sinusoidal Inductor Current 315

5.18.2 Boundary Between Low and Medium Frequencies for Rectangular Wire Winding 318

5.19 Winding Resistance of Square Wire 318

5.19.1 Winding AC Resistance of Square Conductor 320

5.19.2 Optimization of Square Wire Winding at Fixed Pitch 321

5.19.3 Optimization of Square Wire Winding at Fixed Porosity Factor 322

5.19.4 Critical Thickness of Square Winding Resistance 324

5.19.5 Boundary Between Low and Medium Frequencies for Square Wire Winding 325

5.20 Winding Resistance of Round Wire 326

5.20.1 AC Resistance of Round Wire Winding 329

5.20.2 Optimum Diameter of Round Wire at Fixed Pitch 331

5.20.3 Optimum Diameter of Round Wire at Fixed Porosity Factor 332

5.20.4 Critical Round Wire Diameter 334

5.20.5 Boundary Between Low and Medium Frequencies for Round Wire Winding 3355.21 Inductance 335

5.22 Solution for Round Conductor Winding in Cylindrical Coordinates 338

5.23 Litz Wire 338

5.23.1 Litz-Wire Construction 338

5.23.2 Model of Litz-Wire and Multistrand Wire Windings 339

5.23.3 Litz-Wire Winding Resistance 341

5.23.4 Optimum Strand Diameter at Fixed Porosity Factor 345

5.23.5 Approximated Optimum Strand Diameter 346

5.23.6 Optimum Strand Diameter at Variable Porosity Factor 348

5.23.7 Boundary Between Low and Medium Frequencies for Litz-Wire Windings 349

5.23.8 Approximation of Litz-Wire Winding Resistance for Low and Medium Frequencies 349

5.24 Winding Power Loss for Inductor Current with Harmonics 351

5.24.1 Copper Power Loss in PWM DC–DC Converters for Continuous Conduction Mode 353

5.24.2 Copper Power Loss in PWM DC–DC Converters for DCM 360

5.25 Winding Power Loss of Foil Inductors Conducting DC and Harmonic Currents 364

5.25.1 Optimum Foil Thickness of Inductors Conducting DC and Harmonic Currents 365

5.26 Winding Power Loss of Round Wire Inductors Conducting DC and Harmonic Currents 366

5.26.1 Optimum Diameter of Inductors Conducting DC and Harmonic Currents 367

5.27 Effective Winding Resistance for Nonsinusoidal Inductor Current 367

5.28 Thermal Effects on Winding Resistance 370

5.29 Thermal Model of Inductors 373

5.30 Summary 374

5.31 Appendix 375

5.31.1 Derivation of Dowell’s Equation Approximation 375

5.32 References 377

5.33 Review Questions 381

5.34 Problems 381

**6 Laminated Cores 383**

6.1 Introduction 383

6.2 Low-Frequency Eddy-Current Laminated Core Loss 384

6.3 Comparison of Solid and Laminated Cores 389

6.4 Alternative Solution for Low-Frequency Eddy-Current Core Loss 389

6.4.1 Sinusoidal Inductor Voltage 391

6.4.2 Square-Wave Inductor Voltage 393

6.4.3 Rectangular Inductor Voltage 393

6.5 General Solution for Eddy-Current Laminated Core Loss 393

6.5.1 Magnetic Field Distribution at High Frequencies 393

6.5.2 Power Loss Density Distribution at High Frequencies 397

6.5.3 Lamination Impedance at High Frequencies 400

6.6 Summary 408

6.7 References 409

6.8 Review Questions 410

6.9 Problems 411

**7 Transformers 412**

7.1 Introduction 412

7.2 Transformer Construction 413

7.3 Ideal Transformer 413

7.4 Voltage Polarities and Current Directions in Transformers 416

7.5 Nonideal Transformers 417

7.6 Neumann’s Formula for Mutual Inductance 422

7.7 Mutual Inductance 424

7.8 Magnetizing Inductance 425

7.9 Coupling Coefficient 427

7.10 Leakage Inductance 429

7.11 Dot Convention 432

7.12 Series-Aiding and Series-Opposing Connections 435

7.13 Equivalent T Network 435

7.14 Energy Stored in Coupled Inductors 436

7.15 High-Frequency Transformer Model 437

7.16 Stray Capacitances 438

7.17 Transformer Efficiency 438

7.18 Transformers with Gapped Cores 438

7.19 Multiple-Winding Transformers 439

7.20 Autotransformers 439

7.21 Measurements of Transformer Inductances 440

7.22 Noninterleaved Windings 442

7.23 Interleaved Windings 444

7.24 Wireless Energy Transfer 446

7.25 AC Current Transformers 446

7.25.1 Principle of Operation 446

7.25.2 Model of Current Transformer 447

7.25.3 Low-Frequency Response 448

7.25.4 High-Frequency Response 449

7.25.5 Maximum Power Transfer by Current Transformer 452

7.26 Saturable Reactors 454

7.27 Transformer Winding Power Losses with Harmonics 455

7.27.1 Winding Power Losses with Harmonics for CCM 455

7.27.2 Winding Power Losses with Harmonics for DCM 460

7.28 Thermal Model of Transformers 464

7.29 Summary 465

7.30 References 467

7.31 Review Questions 470

7.32 Problems 471

**8 Integrated Inductors 472**

8.1 Introduction 472

8.2 Skin Effect 472

8.3 Resistance of Rectangular Trace with Skin Effect 474

8.4 Inductance of Straight Rectangular Trace 477

8.5 Inductance of Rectangular Trace with Skin Effect 478

8.6 Construction of Integrated Inductors 480

8.7 Meander Inductors 481

8.8 Inductance of Straight Round Conductor 485

8.9 Inductance of Circular Round Wire Loop 486

8.10 Inductance of Two-Parallel Wire Loop 486

8.11 Inductance of Rectangle of Round Wire 486

8.12 Inductance of Polygon Round Wire Loop 486

8.13 Bondwire Inductors 487

8.14 Single-Turn Planar Inductor 488

8.15 Inductance of Planar Square Loop 490

8.16 Planar Spiral Inductors 490

8.16.1 Geometries of Planar Spiral Inductors 490

8.16.2 Inductance of Square Planar Inductors 493

8.16.3 Inductance of Hexagonal Spiral Inductors 502

8.16.4 Inductance of Octagonal Spiral Inductors 503

8.16.5 Inductance of Circular Spiral Inductors 504

8.17 Multimetal Spiral Inductors 505

8.18 Planar Transformers 506

8.19 MEMS Inductors 507

8.20 Inductance of Coaxial Cable 509

8.21 Inductance of Two-Wire Transmission Line 509

8.22 Eddy Currents in Integrated Inductors 509

8.23 Model of RF-Integrated Inductors 510

8.24 PCB Inductors 512

8.25 Summary 514

8.26 References 515

8.27 Review Questions 518

8.28 Problems 519

**9 Self-Capacitance 520**

9.1 Introduction 520

9.2 High-Frequency Inductor Model 520

9.3 Self-Capacitance Components 530

9.4 Capacitance of Parallel-Plate Capacitor 531

9.5 Self-Capacitance of Foil Winding Inductors 532

9.6 Capacitance of Two Parallel Round Conductors 533

9.6.1 Potential of Infinite Single Straight Round Conductor with Charge 533

9.6.2 Potential Between Two Infinite Parallel Straight Round Conductors with Nonuniform Charge Density 533

9.6.3 Capacitance of Two Parallel Wires with Nonuniform Charge Density 536

9.7 Capacitance of Round Conductor and Parallel Conducting Plane 539

9.8 Capacitance of Straight Parallel Wire Pair Over Ground 540

9.9 Capacitance Between Two Parallel Straight Round Conductors with Uniform Charge Density 540

9.10 Capacitance of Cylindrical Capacitor 542

9.11 Self-Capacitance of Single-Layer Inductors 542

9.12 Self-Capacitance of Multilayer Inductors 545

9.12.1 Exact Equation for Self-Capacitance of Multilayer Inductors 545

9.12.2 Approximate Equation for Turn-to-Turn Self-Capacitance of Multilayer Inductors 550

9.13 Self-Capacitance of Single-Layer Inductors 553

9.13.1 Exact Equation for Self-Capacitance of Single-Layer Inductors 553

9.13.2 Approximate Equation for Turn-to-Turn Self-Capacitance of Single-Layer Inductors 555

9.14 -to-Y Transformation of Capacitors 557

9.15 Overall Self-Capacitance of Single-Layer Inductor with Core 557

9.16 Measurement of Self-Capacitance 559

9.17 Inductor Impedance 560

9.18 Summary 564

9.19 References 565

9.20 Review Questions 566

9.21 Problems 566

**10 Design of Inductors 568**

10.1 Introduction 568

10.2 Magnet Wire 569

10.3 Wire Insulation 572

10.4 Restrictions on Inductors 572

10.5 Window Utilization Factor 574

10.5.1 Wire Insulation Factor 574

10.5.2 Air and Wire Insulation Factor 575

10.5.3 Air Factor 576

10.5.4 Bobbin Factor 577

10.5.5 Edge Factor 578

10.5.6 Number of Turns 578

10.5.7 Window Utilization Factor 579

10.5.8 Window Utilization Factor for Foil Winding 581

10.6 Temperature Rise of Inductors 581

10.6.1 Expression for Temperature Rise of Inductors 581

10.6.2 Surface Area of Inductors with Toroidal Core 582

10.6.3 Surface Area of Inductors with Pot Core 583

10.6.4 Surface Area of Inductors with PQ Core 584

10.6.5 Surface Area of Inductors with EE Core 585

10.7 Mean Turn Length of Inductors 585

10.7.1 Mean Turn Length of Inductors with Toroidal Cores 585

10.7.2 Mean Turn Length of Inductors with PC and PQ Cores 586

10.7.3 Mean Turn Length of Inductors with EE Cores 586

10.8 Area Product Method 586

10.8.1 General Expression for Area Product 586

10.8.2 Area Product for Sinusoidal Inductor Voltage 587

10.9 Design of AC Inductors 590

10.9.1 Optimum Magnetic Flux Density 590

10.9.2 Examples of AC Inductor Designs 591

10.10 Inductor Design for Buck Converter in CCM 603

10.10.1 Derivation of Area Product Ap for Square-Wave Inductor Voltage 603

10.10.2 Inductor Design for Buck Converter in CCM Using Area Product Ap Method 603

10.11 Inductor Design for Buck Converter in DCM Using Ap Method 619

10.12 Core Geometry Coefficient Kg Method 654

10.12.1 General Expression for Core Geometry Coefficient Kg 654

10.12.2 AC Inductor with Sinusoidal Current and Voltage 655

10.12.3 Inductor for PWM Converter in CCM 656

10.12.4 Inductor for PWM Converter in DCM 656

10.13 Inductor Design for Buck Converter in CCM Using Kg Method 658

10.14 Inductor Design for Buck Converter in DCM Using Kg Method 660

10.15 Summary 663

10.16 References 664

10.17 Review Questions 666

10.18 Problems 666

**11 Design of Transformers 668**

11.1 Introduction 668

11.2 Area Product Method 668

11.2.1 Derivations of Core Area Product Ap 668

11.2.2 Core Window Area Allocation for Transformer Windings 670

11.3 Optimum Flux Density 673

11.4 Area Product Ap for Sinusoidal Voltages 674

11.5 Transformer Design for Flyback Converter in CCM 675

11.5.1 Practical Design Considerations of Transformers 675

11.5.2 Area Product Ap for Transformer Square Wave Voltages 675

11.5.3 Area Product Ap Method 676

11.6 Transformer Design for Flyback Converter in DCM 689

11.7 Geometrical Coefficient Kg Method 702

11.7.1 Derivation of Geometrical Coefficient Kg 702

11.7.2 Kg for Transformer with Sinusoidal Currents and Voltages 704

11.7.3 Transformer for PWM Converters in CCM 704

11.7.4 Transformer for PWM Converters in DCM 705

11.8 Transformer Design for Flyback Converter in CCM Using Kg Method 705

11.9 Transformer Design for Flyback Converter in DCM Using Kg Method 709

11.10 Summary 714

11.11 References 714

11.12 Review Questions 715

11.13 Problems 715

Appendix A Physical Constants 717

Appendix B Maxwell’s Equations 718

Answers to Problems 719

Index 725