1.5.2 Example: Manipulation of the Buelt-Boost Convettet Modei

itito Canoaieai Foitn 250

7.5.3 CatioQital Ciftruit Parameter Values for Some Common Ctmverters 252

7.6 IModeiing the Pulse-Width Modulator 253

7.7 Summary of Key Ptjints 256 References 257 Problems 258

8 Converter Transfer Funttions 265

8.1 Review of Bode Plots 267

8.1.1 Single Ptjle ResptJtise 269

8.1.2 Single Zero Re.sponse 275

8.1.3 Right Half-Plane Zero 276

8.1.4 Frequency Inversion 277

8.1.5 Combinations 278 8.1 .Й Quadratic Pole Response: Resonance 282

8.1.7 The Ltiw-Q Approximation 287

8.1.8 Approximate Roots of an Arbittttty-Degtee Polynomial 289

8.2 Analysis tif Converter Transfer Functions 293

8.2.1 Exatnple: Transfer FiiQctinns ofthe Buclf-Boost Converter 294

8.2.2 Transfer Functions of Some Basic CCM Converters 300

8.2.3 Physical Origins of the RHP Zero iti Converters 300

8.3 Graphical Construction of Impedances and Transfer Functions ЗИ2

8.3.1 Seties Impedances: Addition of Asytnptotes 303

8.3.2 Series Resonant Circuit Example 305

8.3.3 Parallel Impedances: Inverse .Addition of Asymptotes 308

8.3.4 Parallel Resonaat Circuit Example 309

8.3.5 Voltage Divider Transfer Functions: Division of Asymptotes 311

8.4 Graphical CtJtistruction of Converter Transfer Functions 313

8.5 Measiiretnent of AC Tran.sfer Funttions and Ltnpedaiites 317

8.6 Summary of Key Points 321 References 322 Problems 322

9 Controller Design 331

9.1 Introduction 331

9.2 Effect nf Negative Feedback on the Network Transfer Fuatrfions 334

9.2.1 Feedback Reduces the Transfer Functions

from Disturbances to the Output 335

9.2.2 Feedbaclt Causes the Transfer Functitm ftcmi the Reference Input to the Output to be Insensitive to Variations in the Gains tu the

Forward Path of the Loop 337

9.3 Coii.struction of the Important Quantities 1/(1 + T) and 77(1 + Г)

and the Ciosed-ljoop Transfer FutitrtioQS 337

9.4 Stability 340

Ctiiiteiits xi

9.4.1 The Phase Margin Test 341 9.42 The Relationship Between PhiLse Margin

and Closed-Loop Damping Factor 342

9.4.3 Transient Respon.se vs. Damping Factor 346

9.5 Regulator Design 347

9.5.1 Le;id (PD) Compensator 348

9.5.2 Lag (Pf) Compensator 351

9.5.3 Combined (PID) Compensator 353

9.5.4 Design Example 354

9.6 Measurement of Ltwp Ctiins 362

9.6.1 Voltage Injection 364

9.6.2 Current Injection 367 9.fi.3 Measurement of Unstable Systems

9.7 Summary of Key Points 369 Refeiences 369 Picjblems 369

10 Input Filter D gn 377

10.1 Introduction 377

101.1 Conducteti EMI 377

101.2 The Input Filter Design Problem 379

10.2 Effect of an Input Filter on Converter Transfer Functions 381

10.2.1 Discussion 382

10.2.2 Impedance Inequalities 384 103 Buck Converter Example 385

10.3.1 Effect of Undamped Itiput Fi Iter 385

10.3.2 Damping the Input Filter 391 10.4 Design of a Damped Input Filter 392

10.4.1 Л-С, Parallel Damping 395

10.4.2 Parallel Damping 396

10.4.3 R-Lj, Series Damping 398

10.4.4 Cascading Filter Sections 39S

10.4.5 Example: Two Stage Input Filter 400 105 Summary of Key Points 403 References 405 Problems 406

11 AC and DC Eqtiivalent Circuit Modeling of the DiscAntJnuouu Conduction Mode 409

11.1 DCM Averaged Switch Model 410

11.2 Small-Signal AC Modeling of the DCM Switch Network 420

11.2.1 Exatnple: Control-to-Output Frequency Response

of a DCM B(X)st Converter 428

11.2.2 Example: Control-to-output Frequency Responses

of a ССМЛ5СМ SEPIC 429

из High-Frequency Dynamics, of Converters ill DCM 431

11.4 Suinmary of Key Points 434

References 434

Problems 435

12 Current Programmed Control 439

12.1 Oscillation for D > 0.5 441

12.2 A Simple First-Order Model 449

12.2.1 Simple Modei via Algebraic Approach: Buck-Boost Example 450

12.2.2 Averaged Switch Modeling 454

12.3 A More Accurate Model 459

12.3.1 Current-Programmed Controller Model 459

12.3.2 Solution ofthe CPM Transfer Functions 462

12.3.3 Discussion 465

12.3.4 Current-Programmed Transfer Functions ofthe CCM Buck Converter 466

12.3.5 Results for Basic Converters 469

12.3.6 Quantitative Effects of Current-Programmed Control

on the Converter Transfer Functions 471

12.4 Disciintiiiuous Coiidnction Mode 473

12.5 Summary of Key Points 480 References 481 Problems 482

III Magnetics 489

13 Bask Magnetics Theory 491

13.1 Review of Basic Magnetics 491

13.1.1 Biisit Reiation.ships 491

13.1.2 Magnetic Circuits 498

13.2 Transformer Modeling 501

13.2.1 The Ideal TransfDimer 502

13.2.2 The Magnetizing Inductance 502

13.2.3 Leakage Inductances 504

13.3 Loaf, Mechanisms in Magnetic Devices 506 133.1 CiireLt}ss 506 13.3.2 Low-Frequency Copper Loss 508

13.4 Eddy Currents in Winding Conductor.s 508

13.4.1 Introduction to the Skin and Priiximity Effects 508

13.4.2 Leakage Flux in Windings 512

13.4.3 Fod Windings and Layers 514

13.4.4 Power Loss in a Layer 515

13.4.5 Example: Power Laws in a Transformer Winding 518

13.4.6 Interleaving the Windings 520

53.4.7 PWM Waveform Harnnonics 522