Fundamentals of Magnetic Theory

Basic Laws of Magnetic Theory

(4)

Ampere's Law and Magnetomotive Force

(3)

Faraday's Law and EMF

(1)

Lenz's Law and Gauss's Law for Magnetic Circuits

(1)

Magnetic Materials

(12)

Ferromagnetic Materials

(2)

Magnetization Processes

(1)

Hysteresis Loop

(3)

Permeability

(2)

Complex Permeability

(2)

Hysteresis Material Constant

(1)

Magnetic Circuits

(14)

Basic Laws for Magnetic Circuits

(2)

Inductance

(1)

Flux Linkage

(1)

Inductance: Definitions

(1)

Inductance: Additional Considerations

(1)

Self-inductance and Mutual Inductance

(1)

Transformer Models

(1)

Ideal Transformer

(1)

Practical Transformer

(2)

Magnetic and Electrical Analogy

(1)

References

(3)

Fast Design Approach Including Eddy Current Losses

Fast Design Approach

(36)

Non-Saturated Thermal Limited Design

(1)

Step 1) Choose a Core Material and Size

(3)

Step 2) Calculate the Heat Dissipation Capability Ph

(1)

Step 3) Copper Loss/Core Loss Ratio

(1)

Step 4) Calculate the Specific Core Losses Pfe, sp

(1)

Step 5) Find the Peak Induction Bp, g from Graphical Data

(1)

Step 6) Check if the Peak Induction Bp is Higher Than the Saturation Value Bsat

(1)

Symmetrical Waveforms

(1)

Asymmetrical Waveforms

(1)

Step 7) Calculate the Winding Turns Ni

(1)

Step 8) Distribute Allowed Total Copper Losses Ph, cu Among the Windings

(1)

Step 9) Determine Wire Diameter di

(1)

Step 10) Calculate the Actual Copper Losses Pcu

(1)

Ohmic Copper Losses

(1)

Low-Frequency Transverse Field Eddy Current Losses

(6)

Wide Frequency Eddy Current Losses

(9)

Total Copper Losses

(1)

Step 11) Check if the Copper Losses Pcu are Lower Than the Allowed Copper Dissipation Ph, cu

(1)

Step 12) Is Improvement Possible?

(1)

Step 12a) Optimize the Diameter and Winding Arrangement

(1)

Transformers

(1)

Inductors

(1)

Step 13) Check the Copper Filling Factor

(1)

Step 13a) Choose a Larger Core

(1)

Step 14) Check if the Chosen Core Size in Step 1) is not Too High

(1)

Step 14a) Choose a Smaller Core

(1)

Step 15) Calculate the Total Air Gap Length Σlg

(2)

Saturated Thermally Limited Design

(1)

Step 1') Find the Peak-to-Peak Induction Bpp

(1)

Step 2') Choose a Core, Material, and Size

(2)

Step 3') Find the Core Losses Pfe from Graphical Data

(1)

Step 4') Find the Heat Dissipation Capability Ph of the Component

(1)

Step 5') Check the Ratio Pfe/Ph

(1)

Step 6') Estimate the Allowed Copper Dissipation Capability

(1)

Signal Quality Limited Design

(1)

Examples

(18)

Non-Saturated Thermally Limited Design Example

(1)

Design Steps

(5)

Conclusions

(2)

Improvements of the Design

(1)

Measuring and Validation of the Design

(2)

Saturated Thermal Limited Design Example

(2)

Design Procedure

(3)

Equation Approach

(1)

Measurements and Validation of the Eddy Current Losses

(2)

Conclusions

(1)

Core Size Scale Law for Ferrites in Non-Saturated Thermal Limited Design

(2)

Eddy Current Losses for Wide Frequency

(4)

Approximation of kc

(1)

Transformers

(1)

Direct Calculations

(2)

Graphical Transformer Method

(1)

Inductors

(1)

Direct Calculations

(1)

Graphical Inductor Method

(1)

Mathcad Example Files

(5)

References

(2)

Soft Magnetic Materials

Magnetic Core Materials

(14)

Iron-Based Soft Magnetic Materials

(1)

Laminated Cores

(2)

Powdered Iron and Carbonyl Iron Cores

100

(1)

Powdered Iron

100

(1)

Carbonyl Iron

100

(1)

Amorphous Alloys

101

(1)

Production Process and Microstructure Characteristics

101

(1)

Magnetic Properties

102

(1)

Applications

102

(1)

Shapes

102

(1)

Nanocrystalline Magnetic Materials

103

(1)

Production Process and Microstructure Characteristics

104

(1)

Magnetic Properties

104

(1)

Temperature Behavior

105

(1)

Shapes

106

(1)

Applications

106

(1)

Ferrites

107

(1)

Production Process and Microstructure Characteristics

108

(1)

Magnetic Properties

108

(1)

Low Induction Level (Signal Level) Parameters

109

(1)

High Inductions Level (Power Level) Parameters

109

(1)

Shapes

110

(1)

Comparison and Applications of the Core Materials in Power Electronics

111

(1)

Losses in Soft Magnetic Materials

112

(5)

Simplified Approach for Laminated Steel Cores

112

(1)

Hysteresis Losses

112

(1)

Eddy-Current Losses

113

(2)

Eddy Current Losses in Laminated Cores

115

(1)

Low Frequency Approximation of Eddy Current Losses in Laminated Cores

115

(1)

Eddy Current Losses in Laminated Cores at Arbitrary Frequencies

116

(1)

Anomalous (Residual, Excess) Losses

117

(1)

Ferrite Core Losses with Non-Sinusoidal Voltage Waveforms

117

(5)

Identification of the Steinmetz Equation

118

(1)

Natural Steinmetz Extension for Ferrite Core Losses with Non-Sinusoidal Voltage Waveforms

118

(4)

Wide Frequency Model of Magnetic Sheets Including Hysteresis Effects

122

(9)

Constant Loss Angle Impedance

123

(1)

Transmission Line Approach with Constant Loss Angle Material

124

(1)

Wide Frequency Complex Permeability Function

124

(1)

Real, Reactive, and Apparent Power

125

(1)

Dependence on Saturation Level

126

(1)

Wide Frequency Model Curves of Typical Materials

126

(1)

Silicon Steel

126

(1)

Nanocrystalline Material

127

(4)

Wide Frequency Model for Ferrites

131

(1)

Power and Impedance of Magnetic Sheets

131

(8)

References

135

(4)

Coil Winding and Electrical Insultion

Filling Factor

139

(7)

Round Wires

141

(1)

Square Fitting

141

(1)

Hexagonal Fitting

142

(1)

Practical Case

143

(1)

Foil Windings

144

(1)

Wires with Rectangular Cross Section

145

(1)

Litz Wires

146

(1)

Wire Length

146

(2)

Circular Coil Formers

146

(1)

Rectangular Coil Formers

147

(1)

Physical Aspects of Breakdown

148

(4)

Breakdown Voltage in Air

148

(2)

Breakdown Voltage in Solid Insulation Material

150

(2)

Corona Discharge

152

(1)

Insulation Requirements and Standards

152

(4)

Basic, Supplementary, and Reinforced Insulation

152

(1)

Standard Insulation Distances

153

(1)

Clearance

153

(1)

Creepage Distance

154

(1)

Electric Strength Tests

155

(1)

Leakage Currents

155

(1)

Thermal Requirements and Standards

156

(4)

Thermal Evaluation of Insulation Materials and Systems

156

(1)

Requirements and Standards for Inductive (Magnetic) Modules

157

(1)

Standards for Wires

158

(1)

Bare Material Diameter

158

(1)

Enamel Thickness

158

(1)

Resistance Per Meter

159

(1)

Thermal Classes of Magnet Wires

159

(1)

Magnetic Component Manufacturing Sheet

160

(3)

Coupling

160

(1)

Air Gaps

160

(1)

Impregnating

160

(1)

Partially Filled Layer

161

(1)

References

162

(1)

Eddy Currents in Conductors

Introduction

163

(2)

Current Power Electronics Needs

163

(1)

Skin Effect

163

(1)

Proximity Effect

164

(1)

Air Gap Effects

164

(1)

Eddy Current Losses in Conductors

164

(1)

Basic Approximations

165

(3)

Low Frequency Approximation

165

(1)

High Frequency Approximation

166

(1)

Superposition of Losses

167

(1)

Wide Frequency Approximation

168

(1)

Losses in Rectangular Conductors

168

(6)

Exact Solution For a Current Carrying Rectangular Conductor in a Transverse Field

168

(2)

Low Frequency Approximation

170

(1)

Current Carrying Conductor Without Transverse Field

170

(1)

Conductor Without Current in a Transverse Field

171

(1)

High Frequency Approximation

172

(1)

Ideal Case

172

(1)

Spaced Conductors

173

(1)

Classical Approach

173

(1)

Low Filling Factor and High Frequency

174

(1)

Quadrature of the Circle Method for Round Conductors

174

(8)

Equivalent Rectangle Principle

175

(1)

Adapted Equations

175

(1)

Low Frequency Approximation

176

(1)

Accuracy of Dowell Method

177

(1)

Improved Quadrature of the Circle Method

178

(3)

Discussion of Quadrature of the Circle Methods

181

(1)

Conclusions for Classical Dowell Method

181

(1)

Conclusions for IQOC Method

182

(1)

Losses of a Current Carrying Round Conductor in 2-D Approach

182

(4)

Exact Solution

182

(2)

Low and High Frequency Approximation

184

(1)

Wide Frequency Approximation

185

(1)

Losses of a Round Conductor in a Uniform Transverse AC Field

186

(4)

Exact Solution

186

(1)

Low Frequency Approximation

187

(1)

High Frequency Approximation

188

(1)

Wide Frequency Approximation

189

(1)

Discussion

190

(1)

Low Frequency 2-D Approximation Method for Round Conductors

190

(6)

Direct Integration Method for Round Wires

190

(2)

Three-Field Approximation

192

(2)

Solution in a Magnetic Window Using Mirroring

194

(1)

Suppression of the First Infinite Sum

195

(1)

Wide Frequency Method for Calculating Eddy Current Losses in Windings

196

(18)

High Frequency Effect of Other Wires, Using Dipoles

196

(3)

Wide Frequency Method, Tuning with Finite Element Solutions

199

(1)

A Wire in a Transverse Field

199

(3)

A Wire in a Half Layer

202

(1)

Conclusions of the Comparisons

202

(1)

Losses in the General Case of a Transformer Winding

203

(2)

Losses in an Inductor Winding

205

(3)

High Frequency, High Filling Factor Relations

208

(1)

Summary of the Wide Frequency Method

209

(1)

Comparison of Analytically Based Methods

209

(1)

Low Frequency Methods

209

(2)

Wide Frequency Method and Quadrature of Circle Methods

211

(3)

Losses in Foil Windings

214

(7)

Homogenous Field Parallel to the Foil

214

(1)

Induced Losses by Air Gaps

215

(1)

Analytical Modeling

215

(3)

Tip Currents in Foil Conductors

218

(1)

Foil Inductors

219

(1)

Foil Transformers

220

(1)

Conclusions Concerning Tip Currents

220

(1)

Conclusions for Foil Windings

220

(1)

Losses in Planar Windings

221

(1)

Advantages of the Planar Cores

221

(1)

Losses in Planar Magnetic Components

222

(1)

Specifics

222

(1)

Eddy Current 1-D Model for Rectangular Conductors

222

(13)

Basic Derivations

223

(5)

Single Conductor in a Slot

228

(2)

Superimposed Rectangular Conductors in a Slot

230

(2)

Taylor Expansion and Low Frequency Approximation for Superimposed Rectangular Conductors in a Slot

232

(2)

Approximation for Rectangular Conductors with Air

234

(1)

Classical Approach

234

(1)

Low Frequency 2-D Models for Eddy Current Losses in Round Wires

235

(9)

Low Frequency Approach

235

(1)

Defining a 2-D Winding Arrangement

236

(1)

Eddy Current Losses by The Direct Integration Method

237

(2)

The Proposed Three Orthogonal Fields Method

239

(1)

The Field of the Conductor

240

(1)

The Transverse Field

240

(1)

The Hyperbolic Field

241

(1)

Residual Field

241

(1)

Eddy Current Losses by the Three Orthogonal Fields

242

(1)

Validation of the Proposed 3-Field Approximation

243

(1)

Extension of the Obtained Solution

244

(1)

Field Factor For Inductors

244

(9)

2-D Analytical Approximation of the Field Factor kF

244

(3)

Simplified Approach

247

(1)

Parallel and Perpendicular Components of kF

248

(3)

References

251

(2)

Thermal Aspects

Fast Thermal Design Approach (Level 0 Thermal Design)

253

(3)

Specific Dissipation p for Ferrites

255

(1)

Conclusion About Level 0 Thermal Design

256

(1)

Single Thermal Resistance Design Approach (Level 1 Thermal Design)

256

(2)

Classic Heat Transfer Mechanisms

258

(4)

Conduction Heat Transfer

258

(2)

Convection Heat Transfer

260

(1)

Natural and Forced Convection

260

(1)

Convection Heat Transfer Coefficient hc

260

(1)

Radiation Heat Transfer

261

(1)

Thermal Design Utilizing a Resistance Network

262

(5)

Level 2 Thermal Design

262

(1)

Thermal Resistances

263

(3)

Finding Temperature Rise

266

(1)

Contribution to Heat Transfer Theory of Magnetic Components

267

(9)

Practical Experience

269

(1)

Precise Expression of the Natural Convection Coefficient hc

269

(1)

Derivation of Convection Coefficient hc

269

(2)

Dependencies of hc on the Parameter L and on the Position and Shape

271

(2)

Forced Convection

273

(1)

Classical Approach

273

(2)

Adapted Approach

275

(1)

Relationship with Thermal Resistance Networks

276

(1)

Transient Heat Transfer

276

(4)

Thermal Capacitances in Magnetic Components

276

(1)

Transient Heating

277

(2)

Adiabatic Loading Conditions

279

(1)

Summary

280

(1)

Accurate Natural Convection Modeling for Magnetic Components

281

(8)

Experimental Set Up

281

(1)

Thermal Measurements with the Box-Type Model

282

(1)

Thermal Measurements with the EE Transformer Type Model

282

(1)

Thermal Measurements at an Ambient Temperature of 25°C

282

(1)

Thermal Measurements at an Ambient Temperature of 60°C

282

(1)

Derivation of an Accurate Presentation of the Convection Coefficient hc

283

(3)

Comparison of the Experimental Results and Proposed Thermal Modeling

286

(1)

References

287

(2)

Parasitic Capacitances in Magnetic Components

Capacitance Between Windings: Inter Capacitance

289

(3)

Effects of the Inter Capacitance

290

(1)

Calculating Inter Capacitances and the Equivalent Voltage

290

(1)

Measuring Inter Capacitances

291

(1)

Self-Capacitance of a Winding: Intra Capacitance

292

(4)

Effects of Intra Capacitance

292

(1)

Calculating Intra Capacitances of a Winding

293

(1)

Measuring Intra Capacitances of Windings

294

(1)

Single Parasitic Capacitance Model

294

(1)

Model with a Parasitic Capacitance for Each Winding

295

(1)

Capacitance Between the Windings and the Magnetic Material

296

(1)

Practical Approaches for Decreasing the Effects of Parasitic Capacitances

296

(5)

Low Intra-Capacitance Windings

296

(1)

Decreasing the Effects of the Inter Capacitance

297

(2)

Screening

299

(1)

References

299

(2)

Inductor Design

Air Coils and Related Shapes

301

(5)

Air Coils

301

(1)

Solenoids

302

(1)

Toroidal Coils

303

(1)

Coils with Rectangular Cross Sections

304

(1)

General Case

304

(1)

`Four Square' Cylindrical Air Coil

305

(1)

Inductor Shapes

306

(2)

Typical Ferrite Inductor Shapes

308

(1)

Fringing in Wire-Wound Inductors with Magnetic Cores

308

(8)

Center Gapped, Spacer and Side Gapped Inductors

308

(2)

Simplified Approach to the Center Gapped Inductors

310

(2)

Improved Approximation for Fringing Permeances of Gapped Inductors

312

(1)

Fringing Coefficients

312

(1)

Equivalent Surface

313

(1)

Single and Multiple Air Gap Cases

313

(3)

Eddy Currents in Inductor Windings

316

(1)

Referring to Described Methods

316

(1)

Multiple Air Gap Inductors

316

(1)

Avoiding Winding Close to the Air Gap

316

(1)

Foil Wound Inductors

317

(5)

Foil Inductor---Ideal Case

318

(1)

Single and Multiple Air Gap Design in Foil Inductors

319

(1)

Eddy Current Losses in Foil Windings of Gapped Inductors

320

(1)

Planar Inductors

320

(2)

Inductor Types Depending on Application

322

(4)

DC Inductors

322

(1)

HF Inductors

323

(1)

Combined DC-HF Inductor

324

(1)

Classical Solutions

324

(1)

Special, Combined Design: Litz Wire-Full Wire Inductor Winding

324

(1)

Analytical Modeling of the Combined Full-Wire--Litz Wire Inductor

325

(1)

Design Examples of Different Types of Inductors

326

(7)

Boost Converter Inductor Design

326

(2)

Coupled Inductor Design

328

(2)

Flyback Transformer Design

330

(3)

Fringing Coefficients For Gapped-Wire-Wound Inductors

333

(9)

Basic Cases

333

(1)

Basic Case 1

333

(1)

Basic Case 2

334

(2)

Basic Case 3

336

(1)

Basic Case 4

337

(1)

Symmetrical Cases

338

(1)

Case 1s

339

(1)

Case 2s

339

(1)

Case 3s

339

(1)

Case 4s

339

(1)

Application to Gapped Rectangular Cores

340

(1)

Application to Center Gapped Rectangular Cores

340

(1)

Application to Center Gapped Round Cores

341

(1)

Analytical Modeling of Combined Litz-Wire--Full-Wire Inductors

342

(7)

Example of a Combined Litz-Wire-Full-Wire Inductor

344

(1)

Experimental Results

345

(1)

Conclusion

346

(1)

References

346

(3)

Transformer Design

Transformer Design in Power Electronics

349

(1)

Magnetizing Inductance

349

(3)

Basics

349

(2)

Design

351

(1)

Leakage Inductance

352

(5)

Leakage Inductance of Concentric Windings

352

(2)

Leakage Inductance of Windings in Separate Rooms

354

(1)

General Case

354

(1)

Axis-Symmetrical Case

354

(2)

Leakage Inductance in T, L and M Models of Transformers

356

(1)

T Transformer Model

356

(1)

L Transformer Model

356

(1)

M Transformer Model

356

(1)

Using Parallel Wires and Litz Wires

357

(3)

Parallel Wires

357

(1)

Low Frequency Case: d < 1.6δ

357

(1)

High Frequency Case: d > 2.7δ

358

(1)

Parallel Windings Using Symmetry in the Magnetic Path

358

(1)

Using Litz Wire

359

(1)

Example in the Low-Frequency Approximation

359

(1)

Half Turns

360

(1)

Interleaved Windings

360

(1)

Superimposing Frequency Components

361

(2)

Magnetic Materials

361

(1)

Eddy Currents in Conductors

361

(1)

General Solution

362

(1)

Superimposing Modes

363

(4)

References

366

(1)

Optimal Copper/Core Loss Ratio in Magnetic Components

Simplified Approach

367

(3)

Transformer

368

(2)

Inductor

370

(1)

Loss Minimization in the General Case

370

(1)

Loss Minimization Without Eddy Current Losses

371

(2)

Constant Copper Volume

371

(1)

Constant Wire Cross Section

372

(1)

Equal Core and Copper Surface Temperatures

372

(1)

Loss Minimization Including Low-Frequency Eddy Current Losses

373

(4)

Constant Copper Wire Cross Section

373

(2)

Constant Copper Wire Volume

375

(1)

Variable Wire Cross Section and Number of Turns

375

(2)

More General Problems with Eddy Currents

377

(1)

Summary

377

(1)

Examples

378

(3)

References

379

(2)

Measurements

Introduction

381

(1)

Temperature Measurements

382

(4)

Thermocouple Measurement

382

(1)

PT100 Thermistor Temperature Measurement

383

(1)

NTC Thermistor Temperature Measurement

384

(1)

Glass Fiber Optic Temperature Measurement

384

(1)

Infrared Surface Temperature Measurement

385

(1)

Thermal Paint and Strips

385

(1)

Winding Resistance Measurement Method

385

(1)

Power Losses Measurements

386

(8)

Circuit Wattmeter Measurement

386

(1)

Oscilloscope Measurements

387

(1)

Example of the Accuracy Problem in Oscilloscope Measurement

387

(1)

Impedance Analyzers and RLC Meters

387

(1)

Impedance Analyzers

387

(1)

RLC Meters

388

(1)

Q-factor Test of LC Networks

388

(1)

Power Loss Estimation by Thermal Resistance

389

(1)

Calorimetric Power Loss Measurement

389

(1)

Inertia Calorimeter

390

(1)

Flow Calorimeter

391

(1)

Principle of Operation

391

(1)

Accuracy of Flow Calorimeters

391

(1)

Practical Flow Calorimeter

392

(1)

Conclusions

393

(1)

Measurement of Inductances

394

(4)

Measurement of the Inductance of an Inductor

394

(1)

No Load Test of Transformers

394

(1)

Short Circuit Test

394

(1)

Measurement of the Inductances in Transformers

395

(2)

Measurement of Low Inductances

397

(1)

Core Loss Measurements

398

(6)

Classical Four-Wire Method

398

(2)

Two-Wire Method

400

(1)

Osciloscope Based Measurement

400

(1)

Wide Band Current Probe

401

(1)

Corresponding Voltage Probe

402

(1)

Flux Measurement Probe

403

(1)

Practical Ferrite Power Loss Measurement Set Up

403

(1)

Measurement of Parasitic Capacitances

404

(3)

Measurement of Capacitance Between Windings

404

(1)

Measurement of the Equivalent Parallel Capacitance of a Winding

405

(2)

Combined Measuring Instruments

407

(2)

References

407

(2)

Appendix A RMS Values of Waveforms

A.1 Definitions

409

(1)

Physical Meaning of the RMS Value

409

(1)

RMS Value in the Frequency Domain

409

(1)

RMS Value in the Time Domain

410

(1)

A.2 RMS Values of Some Basic Waveforms

410

(2)

A.2.1 Discontinuous Waveforms

410

(1)

A.2.2 Repeating Line Waveforms

411

(1)

A.2.3 Waveforms Consisting of Different Repeating Line Parts

411

(1)

A.3 RMS Values of Common Waveforms

412

(5)

A.3.1 Sawtooth Wave, Fig. A.4

412

(1)

A.3.2 Clipped Sawtooth, Fig. A.5

412

(1)

A.3.3 Triangular Waveform, No DC Component, Fig. A.6

413

(1)

A.3.4 Triangular Waveform with DC Component, Fig. A.7

413

(1)

A.3.5 Clipped Triangular Waveform, Fig. A.8

413

(1)

A.3.6 Square Wave, Fig. A.9

414

(1)

A.3.7 Rectangular Pulse Wave, Fig. A.10

414

(1)

A.3.8 Sine Wave, Fig. A.11

414

(1)

A.3.9 Clipped Sinusoid, Full Wave, Fig. A.12

415

(1)

A.3.10 Clipped Sinusoid, Half Wave, Fig. A.13

415

(1)

A.3.11 Trapezoidal Pulse Wave, Fig. A.14

415

(2)

Appendix B Magnetic Core Data

B.1 ETD Core Data (Economic Transformer Design Core)

417

(3)

B.2 EE Core Data

420

(1)

B.3 Planar EE Core Data

421

(2)

B.4 ER Core Data

423

(1)

B.5 UU Core Data

424

(1)

B.6 Ring Core Data (Toroid Core)

424

(2)

B.7 P Core Data (Pot Core)

426

(2)

B.8 PQ Core Data

428

(1)

B.9 RM Core Data

429

(3)

B.10 Other Information

432

(1)

Appendix C Copper Wires Data

C.1 Round Wire Data

433

(2)

C.2 American Wire Gauge Data

435

(2)

C.3 Litz Wire Data

437

(4)

Appendix D Mathematical Functions

References

439

(2)

Index

441

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Inductors and Transformers for Power Electronics

9781574446791

1574446797

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