Designing control loops for linear and switching power supplies : a tutorial guide

A control system is a complex electronics architecture involving setpoints and targets. One simple example is the cruise control system of an automobile. Rather than delving into extensive theory, this book focuses on what power electronics engineers really need to know for compensating or stabilizing a given system.

Author Notes

Christophe Basso is a product engineering director at ON Semiconductor in Toulouse, France. He received his B.S.E.E. in electronics from Montpellier University and his M.S.E.E. in power electronics from the National Polytechnic Institute of Toulouse. A senior member of the IEEE, Mr. Basso is a recognized expert, patent holder, and author in the field.

Foreword	p. xiii
Preface	p. xv
Acknowledgments	p. xvii
Chapter 1 Basics of Loop Control	p. 1
1.1 Open-Loop Systems	p. 1
1.1.1 Perturbations	p. 3
1.2 The Necessity of Control-Closed-Loop Systems	p. 4
1.3 Notions of Time Constants	p. 6
1.3.1 Working with Time Constants	p. 7
1.3.2 The Proportional Term	p. 9
1.3.3 The Derivative Term	p. 10
1.3.4 The Integral Term	p. 11
1.3.5 Combining the Factors	p. 12
1.4 Performance of a Feedback Control System	p. 12
1.4.1 Transient or Steady State?	p. 13
1.4.2 The Step	p. 15
1.4.3 The Sinusoidal Sweep	p. 16
1.4.4 The Bode Plot	p. 17
1.5 Transfer Functions	p. 19
1.5.1 The Laplace Transform	p. 20
1.5.2 Excitation and Response Signals	p. 22
1.5.3 A Quick Example	p. 23
1.5.4 Combining Transfer Functions with Bode Plots	p. 25
1.6 Conclusion	p. 27
Selected Bibliography	p. 27
Chapter 2 Transfer Functions	p. 29
2.1 Expressing Transfer Functions	p. 29
2.1.1 Writing Transfer Functions the Right Way	p. 31
2.1.2 The 0-db Crossover Pole	p. 32
2.2 Solving for the Roots	p. 32
2.2.1 Poles and Zeros Found by Inspection	p. 35
2.2.2 Poles, Zeros, and Time Constants	p. 36
2.3 Transient Response and Roots	p. 39
2.3.1 When the Roots Are Moving	p. 43
2.4 S-Plane and Transient Response	p. 49
2.4.1 Roots Trajectories in the Complex Plane	p. 54
2.5 Zeros in the Right Half Plane	p. 56
2.5.1 A Two-Step Conversion Process	p. 56
2.5.2 The Inductor Current Slew-Rate Is the Limit	p. 58
2.5.3 An Average Model to Visualize RHP Zero Effects	p. 60
2.5.4 The Right Half Plane Zero in the Boost Converter	p. 62
2.6 Conclusion	p. 66
References	p. 66
Appendix 2A Determining a Bridge Input Impedance	p. 67
Reference	p. 69
Appendix 2B Plotting Evans Loci with Mathcad	p. 70
Appendix 2C Heaviside Expansion Formulas	p. 71
Reference	p. 74
Appendix 2D Plotting a Right Half Plane Zero with SPICE	p. 74
Chapter 3 Stability Criteria of a Control System	p. 77
3.1 Building An Oscillator	p. 77
3.1.1 Theory at Work	p. 79
3.2 Stability Criteria	p. 82
3.2.1 Gain Margin and Conditional Stability	p. 84
3.2.2 Minimum Versus Nonminimum-Phase Functions	p. 87
3.2.3 Nyquist Plots	p. 89
3.2.4 Extracting the Basic Information from the Nyquist Plot	p. 91
3.2.5 Modulus Margin	p. 93
3.3 Transient Response, Quality Factor, and Phase Margin	p. 97
3.3.1 A Second-Order System, the RLC Circuit	p. 97
3.3.2 Transient Response of a Second-Order System	p. 101
3.3.3 Phase Margin and Quality Factor	p. 110
3.3.4 Opening the Loop to Measure the Phase Margin	p. 117
3.3.5 The Phase Margin of a Switching Converter	p. 120
3.3.6 Considering a Delay in the Conversion Process	p. 122
3.3.7 The Delay in the Laplace Domain	p. 127
3.3.8 Delay Margin versus Phase Margin	p. 130
3.4 Selecting the Crossover Frequency	p. 133
3.4.1 A Simplified Buck Converter	p. 135
3.4.2 The Output Impedance in Closed-Loop Conditions	p. 138
3.4.3 The Closed-Loop Output Impedance at Crossover	p. 142
3.4.4 Scaling the Reference to Obtain the Desired Output	p. 143
3.4.5 Increasing the Crossover Frequency Further	p. 149
3.5 Conclusion	p. 150
References	p. 151
Chapter 4 Compensation	p. 153
4.1 The PID Compensator	p. 153
4.1.1 The Pip Expressions in the Laplace Domain	p. 155
4.1.2 Practical Implementation of a PID Compensator	p. 157
4.1.3 Practical Implementation of a PI Compensator	p. 161
4.1.4 The PID at Work in a Buck Convener	p. 163
4.1.5 The Buck Converter Transient Response with the PID Compensation	p. 170
4.1.6 The Setpoint Is Fixed: We Have a Regulator!	p. 171
4.1.7 A Peaky Output Impedance Plot	p. 174
4.2 Stabilizing the Converter with Poles-Zeros Placement	p. 176
4.2.1 A Simple Step-by-Step Technique	p. 177
4.2.2 The Plant Transfer Function	p. 178
4.2.3 Canceling the Static Error with an Integrator	p. 179
4.2.4 Adjusting the Gain with the Integrator: The Type 1	p. 182
4.2.5 Locally Boosting the Phase at Crossover	p. 183
4.2.6 Placing Poles and Zeros to Create Phase Boost	p. 185
4.2.7 Create Phase Boost up to 90° with a Single Pole/Zero Pair	p. 189
4.2.8 Mid-Band Gain Adjustment with the Single Pole/Zero Pair: The Type 2	p. 191
4.2.9 Design Example with a Type 2	p. 192
4.2.10 Create Phase Boost up to 180° with a Double Pole/Zero Pair	p. 194
4.2.11 Mid-Band Gain Adjustment with the Double Pole/Zero Pair: The Type 3	p. 197
4.2.12 Design Example with a Type 3	p. 199
4.2.13 Selecting the Right Compensator Type	p. 200
4.2.14 The Type 3 at Work with a Buck Converter	p. 201
4.3 Output Impedance Shaping	p. 210
4.3.1 Making the Output Impedance Resistive	p. 212
4.4 Conclusion	p. 221
References	p. 222
Appendix 4A The Buck Output Impedance with Fast Analytical Techniques	p. 222
Reference	p. 227
Appendix 4B The Quality Factor from a Bode Plot with Group Delay	p. 227
Appendix 4C The Phase Display in Simulators or Mathematical Solvers	p. 230
Calculating the Tangent	p. 232
Accounting for the Quadrant	p. 234
Improving the Arctangent Function	p. 236
Phase Display in a SPICE Simulator	p. 237
Conclusion	p. 242
Reference	p. 243
Appendix 4D Impact of Open-Loop Gain and Origin Pole on Op Amp-Based Transfer Functions	p. 243
The Integrator Case	p. 248
Appendix 4E Summary of Compensator Configurations	p. 252
Chapter 5 Operational Amplifiers-Based Compensators	p. 253
5.1 Type 1: An Origin Pole	p. 253
5.1.1 A Design Example	p. 255
5.2 Type 2: An Origin Pole, plus a Pole/Zero Pair	p. 257
5.2.1 A Design Example	p. 260
5.3 Type 2a: An Origin Pole plus a Zero	p. 262
5.3.1 A Design Example	p. 263
5.4 Type 2b: Some Static Gain plus a Pole	p. 264
5.4.1 A Design Example	p. 266
5.5 Type 2: Isolation with an Optocoupler	p. 267
5.5.1 Optocoupler and Op Amp: the Direct Connection, Common Emitter	p. 269
5.5.2 A Design Example	p. 271
5.5.3 Optocoupler and Op Amp: The Direct Connection, Common Collector	p. 273
5.5.4 Optocoupler and Op Amp: The Direct Connection Common Emitter and UC384X	p. 275
5.5.5 Optocoupler and Op Amp: Pull Down with Fast Lane	p. 276
5.5.6 A Design Example	p. 279
5.5.7 Optocoupler and Op Amp: Pull-Down with Fast Lane, Common Emitter, and UC384X	p. 280
5.5.8 Optocoupler and Op Amp: Pull Down Without Fast Lane	p. 283
5.5.9 A Design Example	p. 285
5.5.10 Optocoupler and Op Amp: A Dual-Loop Approach in CC-CV Applications	p. 288
5.5.11 A Design Example	p. 293
5.6 The Type 2: Pole and Zero are Coincident to Create an Isolated Type 1	p. 299
5.6.1 A Design Example	p. 301
5.7 The Type 2: A Slightly Different Arrangement	p. 303
5.8 The Type 3: An Origin Pole, a Pole/Zero Pair	p. 308
5.8.1 A Design Example	p. 313
5.9 The Type 3: Isolation with an Optocoupler	p. 315
5.9.1 Optocoupler and Op Amp: The Direct Connection, Common Collector	p. 315
5.9.2 A Design Example	p. 317
5.9.3 Optocoupler and Op Amp: The Direct Connection, Common Emitter	p. 319
5.9.4 Optocoupler and Op Amp: The Direct Connection, Common Emitter, and UC384X	p. 321
5.9.5 Optocoupler and Op Amp: Pull-Down with Fast Lane	p. 322
5.9.6 A Design Example	p. 326
5.9.7 Optocoupler and Op Amp: Pull Down without Fast Lane	p. 328
5.9.8 A Design Example	p. 332
5.10 Conclusion	p. 335
References	p. 335
Appendix 5A Summary Pictures	p. 335
Appendix 5B Automating Components Calculations with k Factor	p. 340
Type 1 p. 340
Type 2 p. 341
Type 3 p. 342
Reference	p. 344
Appendix 5C The Optocoupler	p. 346
Transmitting Light	p. 346
Current Transfer Ratio	p. 347
The Optocoupler Pole	p. 348
Extracting the Optocoupler Pole	p. 350
Watch for the LED Dynamic Resistance	p. 351
Good Design Practices	p. 354
References	p. 355
Chapter 6 Operational Transconductance Amplifier-Based Compensators	p. 357
6.1 The Type 1: An Origin Pole	p. 358
6.1.1 A Design Example	p. 359
6.2 The Type 2: An Origin Pole plus a Pole/Zero Pair	p. 360
6.2.1 A Design Example	p. 364
6.3 Optocoupler and OTA: A Buffered Connection	p. 365
6.3.1 A Design Example	p. 368
6.4 The Type 3: An Origin Pole and a Pole/Zero Pair	p. 370
6.4.1 A Design Example	p. 377
6.5 Conclusion	p. 380
Appendix 6A Summary Pictures	p. 380
References	p. 381
Chapter 7 TL431-Based Compensators	p. 383
7.1 A Bandgap-Based Component	p. 383
7.1.1 The Reference Voltage	p. 385
7.1.2 The Need for Bias Current	p. 387
7.2 Biasing the TL431: The Impact on the Gain	p. 390
7.3 Biasing the TL431: A Different Arrangement	p. 392
7.4 Biasing the TL431: Component Limits	p. 395
7.5 The Fast Lane Is the Problem	p. 396
7.6 Disabling the Fast Lane	p. 397
7.7 The Type 1: An Origin Pole, Common-Emitter Configuration	p. 399
7.7.1 A Design Example	p. 402
7.8 The Type 1: Common-Collector Configuration	p. 403
7.9 The Type 2: An Origin Pole plus a Pole/Zero Pair	p. 403
7.9.1 A Design Example	p. 407
7.10 The Type 2: Common-Emitter Configuration and UC384X	p. 408
7.11 The Type 2: Common-Collector Configuration and UC384X	p. 411
7.12 The Type 2: Disabling the Fast Lane	p. 411
7.12.1 A Design Example	p. 413
7.13 The Type 3: An Origin Pole plus a Double Pole/Zero Pair	p. 415
7.13.1 A Design Example	p. 423
7.14 The Type 3: An Origin Pole plus a Double Pole/Zero Pair-No Fast Lane	p. 424
7.14.1 A Design Example	p. 429
7.15 Testing the Ac Responses on a Bench	p. 431
7.16 Isolated Zener-Based Compensator	p. 434
7.16.1 A Design Example	p. 436
7.17 Nonisolated Zener-Based Compensator	p. 441
7.18 Nonisolated Zener-Based Compensator: A Lower Cost Version	p. 443
7.19 Conclusion	p. 445
References	p. 445
Appendix 7A Summary Pictures	p. 445
Appendix 7B Second Stage LC Filter	p. 448
A Simplified Approach	p. 449
Simulation at Work	p. 450
References	p. 454
Chapter 8 Shunt Regulator-Based Compensators	p. 455
8.1 The Type 2: An Origin Pole plus a Pole/Zero Pair	p. 456
8.1.1 A Design Example	p. 460
8.2 The Type 3: An Origin Pole plus a Double Pole/Zero Pair	p. 466
8.2.1 A Design Example	p. 468
8.3 The Type 3: An Origin Pole plus a Double Pole/Zero Pair-No Fast Lane	p. 471
8.3.1 A Design Example	p. 474
8.4 Isolated Zener-Based Compensator	p. 476
8.4.1 A Design Example	p. 480
8.5 Conclusion	p. 483
References	p. 483
Appendix 8A Summary Pictures	p. 484
Chapter 9 Measurements and Design Examples	p. 487
9.1 Measuring the Control System Transfer Function	p. 487
9.1.1 Opening the Loop with Bias Point Loss	p. 488
9.1.2 Power Stage Transfer Function without Bias Point Loss	p. 492
9.1.3 Opening the Loop in ac Only	p. 493
9.1.4 Voltage Variations at the Injection Points	p. 496
9.1.5 Impedances at the Injection Points	p. 504
9.1.6 Buffering the Data	p. 505
9.2 Design Example 1: A Forward dc-dc Converter	p. 509
9.2.1 Moving Parameters	p. 509
9.2.2 The Electrical Schematic	p. 511
9.2.3 Extracting the Power Stage Transfer Response	p. 514
9.2.4 Compensating the Converter	p. 515
9.3 Design Example 2: A Linear Regulator	p. 519
9.3.1 Extracting the Power Stage Transfer Function	p. 520
9.3.2 Crossover Frequency Selection and Compensation	p. 521
9.3.3 Testing the Transient Response	p. 527
9.4 Design Example 3: A CCM Voltage-Mode Boost Converter	p. 528
9.4.1 The Power Stage Transfer Function	p. 529
9.4.2 Compensating the Converter	p. 533
Strategy 1 p. 535
Strategy 2 p. 535
9.4.3 Plotting the Loop Gain	p. 537
9.5 Design Example 4: A Primary-Regulated Flyback Converter	p. 539
9.5.1 Deriving the Transfer Function	p. 540
9.5.2 Verifying the Equations	p. 544
9.5.3 Stabilizing the Converter	p. 545
9.6 Design Example 5: Input Filter Compensation	p. 552
9.6.1 A Negative Incremental Resistance	p. 553
9.6.2 Building an Oscillator	p. 554
9.6.3 Taming the Oscillations	p. 556
9.7 Conclusion	p. 562
References	p. 562
Conclusion	p. 565
Appendix	p. 567
About the Author	p. 571

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