Nonlinear Phenomena in Power Electronics Bifurcations, Chaos, Control, and Applications

, by Banerjee, Soumitro; Verghese, George C.

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ISBN: 9780780353831 | 0780353838
Cover: Hardcover
Copyright: 7/16/2001

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Brings the knowledge of 24 experts in this maturing field out from the narrow confines of academic circles, and makes it accessible to graduate students and power electronics professionals alike. * Provides practicing engineers with the knowledge to predict power requirement behavior. * The insights gained from this all-inclusive compilation will ultimately lead to better design methodologies.

Preface

Acknowledgments

xix

List of Contributing Authors

xxi

Introduction

(24)

D. C. Hamill

S. Banerjee

G. C. Verghese

Introduction to Power Electronics

(4)

Power Switching Devices

(2)

Sources of Nonlinearity in Power Electronics

(1)

Power Converters

(1)

Electrical Machines and Drives

(1)

Power Systems

(1)

An Example: The Buck DC/DC Converter

(10)

Conventional Model of the Buck Converter

(1)

Continuous Conduction Mode

(2)

Actual System Behavior

(1)

Nonlinear Map-Based Model of the Buck Converter

(4)

Discontinuous Conduction Mode

(1)

Limitations and Extensions of Average Models

(1)

Study of Nonlinear Dynamics and Chaos in Power Electronics

(5)

Conclusions

(5)

Dynamic Models of Power Converters

(412)

Introduction to Power Electronic Converters and Models

(13)

G. C. Verghese

A. M. Stankovic

Introduction

(1)

Types of Power Electronic Converters

(1)

High-Frequency PWM DC/DC Converters

(1)

Other High-Frequency PWM Converters

(1)

Other Inverters

(1)

Resonant Converters

(1)

Phase-Controlled Converters

(1)

AC/AC Converters

(1)

Averaged and Sampled-Data Models for Analysis, Simulation, and Control of Converter Dynamics

(2)

Switched State-Space Model for the Boost Converter

(2)

Averaged Model for the Boost Converter

(1)

Averaged Model for Current-Mode Control of the Boost Converter

(1)

Sampled-Data Models for the Boost Converter

(2)

Extensions

(1)

Generalized Averaging

(1)

Generalized State-Space Models

(2)

A Closer Look at Sampled-Data Models for Power Converters

(15)

F. Vasca

M. di Bernardo

G. Olivar

Introduction

(1)

Poincare Maps for Smooth and Nonsmooth Dynamical Systems

(2)

Piecewise-Smooth Power Electronic Circuits

(3)

Power Electronic Systems as Hybrid Systems

(3)

Stroboscopic Maps

(2)

Periodic Phase Sequence and Time-Varying Inputs

(1)

A Stroboscopic Nonswitching Map

(1)

Closed-Loop Maps

(1)

Switching Maps

(1)

S-Switching Maps for DC/DC Converters

(1)

A-Switching Maps for DC/DC Converters

(1)

Simplified Maps

(1)

Conclusions

(2)

Basics of Bifurcation and Chaos Theory

Introduction to Nonlinear Dynamics and Chaos

(14)

S. Banerjee

System State, and

(2)

State-Space Models

Autonomous Systems and Nonautonomous Systems

(1)

Vector Fields of Linear, Linearized, and Nonlinear Systems

(2)

Attractors in Nonlinear Systems

(2)

Chaos

(2)

Poincare Map

(1)

Dynamics of Discrete-Time Systems

(2)

Fractal Geometry

(2)

Lyapunov Exponent

(1)

Bifurcation

(1)

Bifurcations of Smooth Maps

(6)

J. H. B. Deane

The Pitchfork Bifurcation

(1)

The Saddle-Node Bifurcation

(1)

The Period-Doubling Bifurcation

(1)

The Neimark Bifurcation

(3)

Bifurcations in Piecewise-Smooth Maps

(16)

S. Banerjee

C. Grebogi

The Normal Form

(2)

Bifurcations in the One-Dimensional Normal Form

(1)

Border Collision Pair Bifurcation

(1)

Border-Crossing Bifurcations

(1)

Bifurcations in the Two-Dimensional Normal Form

(2)

Classification of Border Collision Bifurcations

(1)

Border Collision Pair Bifurcation

(1)

Border-Crossing Bifurcations

(4)

Nonstandard Bifurcations in Discontinuous Maps

(5)

I. Dobson

S. Banerjee

The Method of Schwarzian Derivatives

(7)

C. K. Tse

Background

(1)

Problem Description

(1)

Mechanism of Period Doubling

(1)

Schwarzian Derivative and Period-Doublings ad infinitum

(1)

Application to Power Electronics

(1)

Illustrative Example: The Boost Converter

(2)

Interpretation and Application of the Result

100

(1)

Remarks and Summary

100

(1)

Coexisting Attractors, Basins of Attraction, and Crises

101

(10)

E. Fossas

G. Olivar

Characteristic (Floquet) Multipliers

101

(1)

Invariant Sets and Invariant Manifolds

102

(1)

Homoclinic and Heteroclinic Orbits

103

(1)

Coexisting Attractors

104

(1)

The Role of Invariant Manifolds and Basins of Attraction

105

(1)

Crises

106

(1)

Interior Crises

107

(1)

Boundary Crises

108

(1)

Two-Dimensional Maps

109

(2)

Experimental and Computational Techniques for Investigation of Nonlinear Phenomena

Techniques of Experimental Investigation

111

(12)

C. K. Tse

Introduction

111

(1)

Overview of Simulation Study and Verification

111

(1)

Experimental Investigation

112

(1)

Displaying Time-Domain Waveforms, Attractors, and Spectra

112

(2)

Displaying Poincare Sections

114

(1)

Principle of Poincare Section Measurement

115

(2)

Example: Free-Running Cuk Converter

117

(1)

Poincare Sections for Nonautonomous Circuits

118

(1)

Displaying Bifurcation Diagrams

118

(1)

Basic Operational Requirements

119

(1)

Digital Implementation and Related Issues

120

(1)

Other Methods, Problems, and Practical Issues

121

(1)

Example: Boost Converter Under Current-Mode Control

121

(2)

Techniques of Numerical Investigation

123

(6)

S. Banerjee

D. C. Hamill

Simulation of Power Electronic Circuits

123

(1)

Problems Arising from Varying Topology

124

(1)

Problems Arising from Incompatible Boundary Conditions

125

(1)

Obtaining Bifurcation Diagrams

125

(1)

Plotting Basins of Attraction in Systems with Multiple Attractors

126

(1)

Computing the Maximal Lyapunov Exponent

127

(2)

Computation of Averages Under Chaos

129

(20)

J. L. Rodriguez Marrero

G. C. Verghese

R. Santos Bueno

S. H. Isabelle

Introduction

129

(1)

Chaotic Operation of DC/DC Converters Under Current-Mode Control

130

(1)

Describing Chaotic Behavior Via Densities

130

(4)

Calculation of the Time-Average of the Inductor Current

134

(1)

Analysis of DC/DC Converters

135

(7)

Average Switching Frequency and Average Duty Ratio

142

(1)

Experimental Results

143

(2)

Conclusions

145

(4)

Calculation of Spectral Peaks in a Chaotic DC/DC Converter

149

(16)

J. H. B. Deane

Characterization of Spectral Properties

149

(2)

Motivation and Outline

151

(1)

The Simplified Mapping

152

(2)

Approximation of the Mean State Variables

154

(1)

The Power Density Spectrum of the Inductor Current

155

(3)

The Invariant Density Algorithm

158

(2)

Practical Results

160

(1)

Experimental Results

160

(3)

Discussion

163

(2)

Computer Methods to Analyze Stability and Bifurcation Phenomena

165

(12)

Y. Kuroe

Introduction

165

(1)

Nonlinear Systems and Stability of Periodic Solutions

165

(2)

Computer Methods to Analyze Stability

167

(2)

Computation of the Jacobian Matrix

169

(2)

Analysis Method Based on Transient Simulator

171

(3)

Computer Method to Analyze Bifurcation Phenomena

174

(1)

Classification of Bifurcations

174

(1)

Method to Determine Bifurcation Values

175

(2)

Computation of Operating-Mode Boundaries

177

(15)

Y. Kuroe

T. Kato

G. C. Verghese

Introduction

177

(1)

What Is Operating-Mode Analysis?

178

(1)

Computation of Operating-Mode Boundary by Curve Tracing

179

(1)

Conditions that Define Operating-Mode Boundaries

179

(2)

Numerical Tracing of Boundary Curves

181

(1)

Computation of Steady-State Sensitivities

182

(1)

Numerical Examples

183

(4)

Computation of Operating-Mode Boundaries by a Binary-Box Method

187

(1)

Basic Approach

187

(1)

Binary-Box Method

188

(1)

Application to More Complicated Converters

188

(4)

Nonlinear Phenomena in DC/DC Converters

Border Collision Bifurcations in the Current-Mode-Controlled Boost Converter

192

(7)

S. Banerjee

P. Ranjan

Modeling and Analysis

192

(3)

Analysis of Bifurcations

195

(4)

Bifurcation and Chaos in the Voltage-Controlled Buck Converter with Latch

199

(9)

S. Banerjee

D. Kastha

S. Das

Overview of Circuit Operation

199

(1)

Experimental Results

200

(2)

Coexisting Attractors and Crises

202

(6)

Routes to Chaos in the Voltage-Controlled Buck Converter without Latch

208

(21)

M. di Bernardo

G. Olivar

F. Vasca

Buck Converter Modeling Under Voltage-Mode Control

208

(1)

Differential equations

209

(1)

Discrete-Time Map and Periodic Orbits

210

(2)

Different Types of Periodic Orbits

212

(1)

Analytical Study of Periodic Orbits: Existence and Stability

212

(5)

One-Dimensional Bifurcation Diagrams

217

(1)

The Main Bifurcation Diagram

217

(2)

Secondary Bifurcations

219

(1)

Chaotic Attractors in the Buck Converter

220

(1)

3T-Periodic Orbits and the Three-Piece Chaotic Attractor

220

(1)

Invariant Manifolds and Basins of Attraction

220

(3)

6T-Periodic Orbits

223

(1)

12T-Periodic Orbits

224

(1)

5T-Periodic Orbits and the Jump to Larger Chaos

225

(4)

Saddle-Node and Neimark Bifurcations in PWM DC/DC Converters

229

(11)

C. C. Fang

E. H. Abed

Introduction

229

(1)

General Sampled-Data Model for Closed-Loop PWM Converters

229

(3)

Periodic Solution Before and After Local Bifurcation

232

(1)

Saddle-Node Bifurcation in Buck Converter Under Discrete-Time Control

232

(2)

Neimark Bifurcation in Buck Converter Under Voltage-Mode Control

234

(2)

Neimark Bifurcation in Buck Converter with Input Filter Under Voltage-Mode Control

236

(2)

Neimark Bifurcation in Buck Converter with Input Filter Under Current-Mode Control

238

(2)

Nonlinear Analysis of Operation in Discontinuous-Conduction Mode

240

(8)

C. K. Tse

Review of Operating Modes

240

(1)

Derivation of Discrete-Time Maps

241

(2)

Period-Doubling Bifurcation

243

(2)

Computer Simulations and Experiments

245

(1)

Remarks and Summary

246

(2)

Nonlinear Phenomena in the Cuk Converter

248

(14)

C. K. Tse

Review of the Cuk Converter and its Operation

248

(1)

Discrete-Time Modeling for Fixed Frequency Operation

249

(2)

Free-Running Current-Mode-Controlled Cuk Converter

251

(1)

Autonomous System Modeling

251

(2)

Dimensionless Equations

253

(1)

Stability of Equilibrium Point and Hopf Bifurcation

254

(2)

Local Trajectories from Describing Equation

256

(2)

Computer Simulation Study

258

(4)

Nonlinear Dynamics in Thyristor and Diode Circuits

I. Dobson

Introduction

262

(1)

Ideal Diode and Thyristor Switching Rules

263

(1)

Static VAR System Example

263

(2)

Poincare Map

265

(2)

Jacobian of Poincare Map

267

(7)

Thyristor Current Function and Transversality

268

(1)

Relations Between On and Off Systems

269

(1)

Derivation of Jacobian Formula

270

(1)

Interval Containing a Switch-On

270

(1)

Interval Containing a Switch-Off

271

(1)

Assembling the Jacobian

272

(1)

Discussion of Jacobian Formula

273

(1)

Switching Damping

274

(4)

Simple Example

274

(1)

Switching Damping in the SVC Example

275

(1)

Variational Equation

276

(2)

Switching Time Bifurcations

278

(8)

Switching Time Bifurcations and Instability

278

(2)

Switching Time Bifurcations for Transients

280

(1)

Misfire Onset as a Transcritical Bifurcation

281

(2)

Noninvertibility and Discontinuity of the Poincare Map

283

(1)

Multiple Attractors and Their Basin Boundaries

284

(2)

Diode Circuits

286

(3)

Transversality and Poincare Map Jacobian Formula

286

(1)

Poincare Map Jacobian for the DC/DC Buck-Boost Converter in Discontinuous Mode

286

(2)

Poincare Map Continuity and Switching Time Bifurcations

288

(1)

Firing Angle Control

289

(3)

Nonlinear Phenomena in Other Power Electronic Systems

Modeling a Nonlinear Inductor Circuit

292

(6)

J. H. B. Deane

Introduction

292

(1)

The Circuit

292

(1)

Saturating and Hysteretic Inductor Modeling

293

(1)

Differential Equation for the Circuit

294

(1)

Results

295

(1)

Bifurcation Diagram Comparison

296

(1)

Poincare Section Comparison

297

(1)

Conclusions

298

(1)

Inverters Under Tolerance Band Control

298

(15)

A. Magauer

Introduction

298

(1)

Functioning Principle

299

(1)

System Model and Equation

300

(2)

Poincare Map

302

(1)

Circuit Realization

303

(1)

Mode of Oscillations, Bifurcations, and Crises

303

(1)

Chaotic Mode

304

(1)

Symmetry-Breaking Bifurcation

305

(2)

Merging Crisis

307

(1)

Interior Crisis

308

(1)

Saddle-Node Bifurcation and Square-Wave Mode

308

(3)

Boundary Crisis

311

(1)

Period-Doubling Bifurcations

312

(1)

Conclusions

312

(1)

Nonlinear Noise Effects in Power Converters

313

(15)

P. T. Krein

P. Midya

Introduction

313

(1)

Discussion of Switching Noise

314

(1)

External Noise Effects in Open-Loop Converters

315

(1)

External Noise Action

315

(1)

Background of Analysis

316

(2)

Summary of Assumptions

318

(1)

Probability Density Function of Switch Timing

318

(1)

Implications of the Non-Gaussian Duty Ratio Distribution with Latch

319

(2)

External Noise Effects in Closed-Loop DC/DC Converters

321

(1)

The Nature of Closed-Loop Noise Effects

321

(1)

The Closed-Loop Process and System Model

321

(1)

Time Domain Noise Analysis

322

(1)

Confirmation

323

(1)

Frequency Domain Analysis

324

(3)

Summary

327

(1)

Nonlinear Phenomena in the Current Control of Induction Motors

328

(10)

I. Nagy

Z. Suto

System Model

329

(1)

Voltage Source Converter (VSC)

329

(1)

AC Side

330

(1)

Hysteresis Current Control (HCC)

331

(1)

Poincare Map

332

(1)

Nonlinear Phenomena

333

(1)

Sensitive Dependence on Initial Condition

333

(1)

Period Doubling Bifurcation

333

(2)

Intermittency

335

(1)

Coexisting Attractors

336

(1)

Numerical Values

336

(1)

Conclusions

337

(1)

Analysis of Stability and Bifurcation in Power Electronic Induction Motor Drive Systems

338

(15)

Y. Kuroe

Introduction

338

(1)

Model of Power Electronic Induction Motor Drive Systems

338

(1)

Model of Induction Motor and its Mechanical Load

338

(2)

Model of Inverter and Rectifier

340

(3)

Poincare Map and Periodicity of Steady States

343

(1)

Case I

343

(1)

Case II

344

(1)

Case III

345

(1)

Stability Analysis

345

(3)

Analysis of Bifurcations

348

(5)

Nonlinear Control and Control of Chaos

Conventional Nonlinear Controls in Power Electronics

353

(4)

P. T. Krein

Introduction

353

(1)

Hysteresis Controllers

354

(1)

Nonlinear Modulation

354

(2)

Multipliers in the Loop

356

(1)

Sliding Mode and Switching Surface Control

357

(14)

P. T. Krein

Introduction

357

(1)

Hysteresis Control

358

(2)

Switching Surface Control Analysis

360

(1)

Trajectories and Equilibria

360

(1)

Switching Surface-Based Control Laws

361

(1)

Necessary Conditions for Switching Surface Controls

362

(1)

Sample Outputs and Hysteresis Design Approaches

363

(2)

Global Stability Considerations

365

(1)

Successor Points

365

(1)

Behavior Near a Switching Surface

366

(1)

Choosing a Switching Surface

367

(2)

Higher Dimensions

369

(1)

Summary

369

(2)

Energy-Based Control in Power Electronics

371

(15)

A. M. Stankovic

G. Escobar

R. Ortega

S. R. Sanders

Introduction

371

(1)

Circuit-Theoretic Approaches

372

(1)

Basic Control

372

(2)

Adaptation

374

(1)

Estimation and Output Feedback

375

(1)

Passivity-Based Control

376

(1)

Basic Controller

376

(2)

Adaptation

378

(1)

Hamiltonian Control

379

(1)

Connections with Sliding-Mode Control

380

(1)

Sliding-Mode Controller Revisited

381

(1)

Passivity-Based Sliding-Mode Controller

382

(1)

Combining SMC with Prediction

383

(1)

Conclusions

384

(2)

Ripple Correlation Control

386

(7)

P. T. Krein

Background

386

(1)

Ripple-Based Control

386

(1)

Ripple Correlation

387

(2)

Some Application Examples

389

(1)

Adaptive Dead Time

389

(1)

Solar Power Processing

390

(1)

Motor Power Minimization in Drives

391

(1)

Summary

392

(1)

Control of Chaos

393

(13)

M. di Bernardo

G. Olivar

C. Batlle

Introduction

393

(1)

A Combination of OGY and Pyragas Methods

394

(1)

Application to the Current-Mode-Controlled Boost Converter

395

(3)

Controlling Border-Collision Bifurcations

398

(1)

Local Feedback Strategy

399

(1)

An Example: A Two-Dimensional Map

400

(1)

Time-Delay Control of Chaos

401

(1)

An Example: TDAS for the Current-Mode Boost Converter

402

(2)

Conclusions

404

(2)

Closed-Loop Regulation of Chaotic Operation

406

(12)

J. L. Rodriguez Marrero

R. Santos Bueno

G. C. Verghese

Introduction

406

(1)

Dynamics and Control

406

(2)

Experimental Results

408

(1)

The OGY Method

409

(2)

Review of the OGY Method

411

(1)

Controlling DC/DC Converters

412

(3)

Integral Control

415

(2)

Conclusions

417

(1)

Control of Bifurcation

418

(10)

C. K. Tse

Y.-M. Lai

Background

418

(1)

Controlling Bifurcation in Discontinuous-Mode Converters

419

(1)

Controlling Bifurcation in Current-Mode-Controlled DC/DC Converters

420

(1)

Use of Compensating Ramp for Controlling Bifurcation

420

(4)

Effects on Dynamical Response

424

(1)

Experimental Measurements

425

(1)

Variable Ramp Compensation

426

(2)

Synchronization of Chaos

428

(9)

C. K. Tse

Background

428

(1)

The Drive-Response Concept

429

(1)

Synchronization in Chaotic Free-Running Cuk Converters

430

(1)

Derivation of the Conditional Lyapunov Exponents

431

(1)

Numerical Calculation of the Conditional Lyapunov Exponents

432

(1)

Computer Simulations

433

(1)

Remarks on Practical Synchronization

433

(4)

Index

437

(4)

About the Editors

441

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Nonlinear Phenomena in Power Electronics Bifurcations, Chaos, Control, and Applications

Summary

Author Biography

Table of Contents

Supplemental Materials