High-Frequency Magnetic Components

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

1.4 Magnetic Circuits 6

1.5 Magnetic Laws 9

1.6 Eddy Currents 29

1.7 Core Saturation 32

1.8 Inductance 40

1.9 Air Gap in Magnetic Core 51

1.10 Fringing Flux 54

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

1.15 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

2.9 Hysteresis 96

2.10 Low-Frequency Core Permeability 98

2.11 Core Geometries 99

2.12 Ferromagnetic Core Materials 103

2.13 Superconductors 108

2.14 Hysteresis Loss 109

2.15 Eddy-Current Core Loss 113

2.16 Steinmetz Empirical Equation for Total Core Loss 129

2.17 Core Losses for Nonsinusoidal Inductor Current 135

2.18 Complex Permeability of Magnetic Materials 136

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

3.3 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

3.7 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

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

3.12 Summary 218

3.13 Appendix 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

4.6 Antiproximity and Skin Effects in Two Parallel Plates 244

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

4.12 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

5.16 Winding AC Resistance with Uniform Foil Thickness 300

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

5.18 Winding AC Resistance of Rectangular Conductor 309

5.19 Winding Resistance of Square Wire 318

5.20 Winding Resistance of Round Wire 326

5.21 Inductance 335

5.22 Solution for Round Conductor Winding in Cylindrical Coordinates 338

5.23 Litz Wire 338

5.24 Winding Power Loss for Inductor Current with Harmonics 351

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

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

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

5.32 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

6.6 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

7.26 Saturable Reactors 454

7.27 Transformer Winding Power Losses with Harmonics 455

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

8.17 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

9.7 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

9.13 Self-Capacitance of Single-Layer Inductors 553

9.14 T-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

10.6 Temperature Rise of Inductors 581

10.7 Mean Turn Length of Inductors 585

10.8 Area Product Method 586

10.9 Design of AC Inductors 590

10.10 Inductor Design for Buck Converter in CCM 603

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

10.12 Core Geometry Coefficient Kg Method 654

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

11.3 Optimum Flux Density 673

11.4 Area Product Ap for Sinusoidal Voltages 674

11.5 Transformer Design for Flyback Converter in CCM 675

11.6 Transformer Design for Flyback Converter in DCM 689

11.7 Geometrical Coefficient Kg Method 702

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