Chapter 1 Basics of Loop Control 1

1.1 Open-Loop Systems 1

1.1.1 Perturbations 3

1.2 The Necessity of Control-Closed-Loop Systems 4

1.3 Notions of Time Constants 6

1.3.1 Working with Time Constants 7

1.3.2 The Proportional Term 9

1.3.3 The Derivative Term 10

1.3.4 The Integral Term 11

1.3.5 Combining the Factors 12

1.4 Performance of a Feedback Control System 12

1.4.1 Transient or Steady State? 13

1.4.2 The Step 15

1.4.3 The Sinusoidal Sweep 16

1.4.4 The Bode Plot 17

1.5 Transfer Functions 19

1.5.1 The Laplace Transform 20

1.5.2 Excitation and Response Signals 22

1.5.3 A Quick Example 23

1.5.4 Combining Transfer Functions with Bode Plots 25

1.6 Conclusion 27

Selected Bibliography 27

Chapter 2 Transfer Functions 29

2.1 Expressing Transfer Functions 29

2.1.1 Writing Transfer Functions the Right Way 31

2.1.2 The 0-db Crossover Pole 32

2.2 Solving for the Roots 32

2.2.1 Poles and Zeros Found by Inspection 35

2.2.2 Poles, Zeros, and Time Constants 36

2.3 Transient Response and Roots 39

2.3.1 When the Roots Are Moving 43

2.4 S-Plane and Transient Response 49

2.4.1 Roots Trajectories in the Complex Plane 54

2.5 Zeros in the Right Half Plane 56

2.5.1 A Two-Step Conversion Process 56

2.5.2 The Inductor Current Slew-Rate Is the Limit 58

2.5.3 An Average Model to Visualize RHP Zero Effects 60

2.5.4 The Right Half Plane Zero in the Boost Converter 62

2.6 Conclusion 66

References 66

Appendix 2A Determining a Bridge Input Impedance 67

Reference 69

Appendix 2B Plotting Evans Loci with Mathcad 70

Appendix 2C Heaviside Expansion Formulas 71

Reference 74

Appendix 2D Plotting a Right Half Plane Zero with SPICE 74

Chapter 3 Stability Criteria of a Control System 77

3.1 Building An Oscillator 77

3.1.1 Theory at Work 79

3.2 Stability Criteria 82

3.2.1 Gain Margin and Conditional Stability 84

3.2.2 Minimum Versus Nonminimum-Phase Functions 87

3.2.3 Nyquist Plots 89

3.2.4 Extracting the Basic Information from the Nyquist Plot 91

3.2.5 Modulus Margin 93

3.3 Transient Response, Quality Factor, and Phase Margin 97

3.3.1 A Second-Order System, the RLC Circuit 97

3.3.2 Transient Response of a Second-Order System 101

3.3.3 Phase Margin and Quality Factor 110

3.3.4 Opening the Loop to Measure the Phase Margin 117

3.3.5 The Phase Margin of a Switching Converter 120

3.3.6 Considering a Delay in the Conversion Process 122

3.3.7 The Delay in the Laplace Domain 127

3.3.8 Delay Margin versus Phase Margin 130

3.4 Selecting the Crossover Frequency 133

3.4.1 A Simplified Buck Converter 135

3.4.2 The Output Impedance in Closed-Loop Conditions 138

3.4.3 The Closed-Loop Output Impedance at Crossover 142

3.4.4 Scaling the Reference to Obtain the Desired Output 143

3.4.5 Increasing the Crossover Frequency Further 149

3.5 Conclusion 150

References 151

Chapter 4 Compensation 153

4.1 The PID Compensator 153

4.1.1 The Pip Expressions in the Laplace Domain 155

4.1.2 Practical Implementation of a PID Compensator 157

4.1.3 Practical Implementation of a PI Compensator 161

4.1.4 The PID at Work in a Buck Convener 163

4.1.5 The Buck Converter Transient Response with the PID Compensation 170

4.1.6 The Setpoint Is Fixed: We Have a Regulator! 171

4.1.7 A Peaky Output Impedance Plot 174

4.2 Stabilizing the Converter with Poles-Zeros Placement 176

4.2.1 A Simple Step-by-Step Technique 177

4.2.2 The Plant Transfer Function 178

4.2.3 Canceling the Static Error with an Integrator 179

4.2.4 Adjusting the Gain with the Integrator: The Type 1 182

4.2.5 Locally Boosting the Phase at Crossover 183

4.2.6 Placing Poles and Zeros to Create Phase Boost 185

4.2.7 Create Phase Boost up to 90° with a Single Pole/Zero Pair 189

4.2.8 Mid-Band Gain Adjustment with the Single Pole/Zero Pair: The Type 2 191

4.2.9 Design Example with a Type 2 192

4.2.10 Create Phase Boost up to 180° with a Double Pole/Zero Pair 194

4.2.11 Mid-Band Gain Adjustment with the Double Pole/Zero Pair: The Type 3 197

4.2.12 Design Example with a Type 3 199

4.2.13 Selecting the Right Compensator Type 200

4.2.14 The Type 3 at Work with a Buck Converter 201

4.3 Output Impedance Shaping 210

4.3.1 Making the Output Impedance Resistive 212

4.4 Conclusion 221