Linear Circuit Transfer Functions An Introduction to Fast Analytical Techniques

Linear Circuit Transfer Functions: An introduction to Fast Analytical Techniques teaches readers how to determine transfer functions of linear passive and active circuits by applying Fast Analytical Circuits Techniques. Building on their existing knowledge of classical loop/nodal analysis, the book improves and expands their skills to unveil transfer functions in a swift and efficient manner.

Starting with simple examples, the author explains step-by-step how expressing circuits time constants in different configurations leads to writing transfer functions in a compact and insightful way. By learning how to organize numerators and denominators in the fastest possible way, readers will speed-up analysis and predict the frequency response of simple to complex circuits. In some cases, they will be able to derive the final expression by inspection, without writing a line of algebra.

Key features:

• Emphasizes analysis through employing time constant-based methods discussed in other text books but not widely used or explained.
• Develops current techniques on transfer functions, to fast analytical techniques leading to low-entropy transfer functions immediately exploitable for analysis purposes.
• Covers calculation techniques pertinent to different fields, electrical, electronics, signal processing etc.
• Describes how a technique is applied and demonstrates this through real design examples.
• All Mathcad^® files used in examples and problems are freely available for download.

An ideal reference for electronics or electrical engineering professionals as well as BSEE and MSEE students, this book will help teach them how to: become skilled in the art of determining transfer function by using less algebra and obtaining results in a more effectual way; gain insight into a circuit’s operation by understanding how time constants rule dynamic responses; apply Fast Analytical Techniques to simple and complicated circuits, passive or active and be more efficient at solving problems.

Christophe Basso, Engineering Director, ON Semiconductor, Toulouse, France
Christophe Basso holds a BSEE equivalent from Montpellier University (France) and an MSEE from the Institut National Polytechnique of Toulouse. He has over 20 years of power supply industry experience. His recent research interests focus on developing new offline PWM controller specifications. On top of his 3 published books on Switch mode power supplies, Basso also has 30 patents on power conversion and has authored numerous conference papers and trade magazines.

Preface xi

Acknowledgement xiii

1 Electrical Analysis – Terminology and Theorems 1

1.1 Transfer Functions, an Informal Approach 1

1.1.1 Input and Output Ports 3

1.1.2 Different Types of Transfer Function 6

1.2 The Few Tools and Theorems You Did Not Forget . . . 11

1.2.1 The Voltage Divider 11

1.2.2 The Current Divider 12

1.2.3 Thévenin’s Theorem at Work 14

1.2.4 Norton’s Theorem at Work 19

1.3 What Should I Retain from this Chapter? 25

1.4 Appendix 1A – Finding Output Impedance/Resistance 26

1.5 Appendix 1B – Problems 37

Answers 39

2 Transfer Functions 41

2.1 Linear Systems 41

2.1.1 A Linear Time-invariant System 43

2.1.2 The Need for Linearization 43

2.2 Time Constants 44

2.2.1 Time Constant Involving an Inductor 47

2.3 Transfer Functions 49

2.3.1 Low-entropy Expressions 54

2.3.2 Higher Order Expressions 59

2.3.3 Second-order Polynomial Forms 60

2.3.4 Low-Q Approximation for a 2nd-order Polynomial 62

2.3.5 Approximation for a 3rd-order Polynomial 68

2.3.6 How to Determine the Order of the System? 69

2.3.7 Zeros in the Network 76

2.4 First Step Towards a Generalized 1st-order Transfer Function 78

2.4.1 Solving 1st-order Circuits with Ease, Three Examples 82

2.4.2 Obtaining the Zero with the Null Double Injection 89

2.4.3 Checking Zeros Obtained in Null Double Injection with SPICE 94

2.4.4 Network Excitation 95

2.5 What Should I Retain from this Chapter? 100

References 101

2.6 Appendix 2A – Problems 102

Answers 105

3 Superposition and the Extra Element Theorem 116

3.1 The Superposition Theorem 116

3.1.1 A Two-input/Two-output System 120

3.2 The Extra Element Theorem 126

3.2.1 The EET at Work on Simple Circuits 130

3.2.2 The EET at Work – Example 2 132

3.2.3 The EET at Work – Example 3 137

3.2.4 The EET at Work – Example 4 138

3.2.5 The EET at Work – Example 5 140

3.2.6 The EET at Work – Example 6 146

3.2.7 Inverted Pole and Zero Notation 150

3.3 A Generalized Transfer Function for 1st-order Systems 153

3.3.1 Generalized Transfer Function – Example 1 156

3.3.2 Generalized Transfer Function – Example 2 159

3.3.3 Generalized Transfer Function – Example 3 163

3.3.4 Generalized Transfer Function – Example 4 170

3.3.5 Generalized Transfer Function – Example 5 174

3.4 Further Reading 180

3.5 What Should I Retain from this Chapter? 180

References 182

3.6 Appendix 3A – Problems 183

Answers 185

References 218

4 Second-order Transfer Functions 219

4.1 Applying the Extra Element Theorem Twice 219

4.1.1 Low-entropy 2nd-order Expressions 227

4.1.2 Determining the Zero Positions 231

4.1.3 Rearranging and Plotting Expressions 233

4.1.4 Example 1 – A Low-Pass Filter 235

4.1.5 Example 2 – A Two-capacitor Filter 241

4.1.6 Example 3 – A Two-capacitor Band-stop Filter 245

4.1.7 Example 4 – An LC Notch Filter 248

4.2 A Generalized Transfer Function for 2nd-Order Systems 255

4.2.1 Inferring the Presence of Zeros in the Circuit 256

4.2.2 Generalized 2nd–order Transfer Function – Example 1 257

4.2.3 Generalized 2nd–order Transfer Function – Example 2 262

4.2.4 Generalized 2nd–order Transfer Function – Example 3 266

4.2.5 Generalized 2nd–order Transfer Function – Example 4 273

4.3 What Should I Retain from this Chapter ? 277

References 279

4.4 Appendix 4A – Problems 279

Answers 282

References 311

5 Nth-order Transfer Functions 312

5.1 From the 2EET to the NEET 312

5.1.1 3rd-order Transfer Function Example 317

5.1.2 Transfer Functions with Zeros 320

5.1.3 A Generalized Nth-order Transfer Function 327

5.2 Five High-order Transfer Functions Examples 335

5.2.1 Example 2: A 3rd-order Active Notch Circuit 341

5.2.2 Example 3: A 4th-order LC Passive Filter 349

5.2.3 Example 4: A 4th-order Band-pass Active Filter 355

5.2.4 Example 5: A 3rd-order Low-pass Active GIC Filter 368

5.3 What Should I Retain from this Chapter ? 383

References 385

5.5 Appendix 5A – Problems 385

Answers 388

References 431

Conclusion 433

Glossary of Terms 435

Index 439

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