# Linear Circuit Transfer Functions: An Introduction to Fast Analytical Techniques

ISBN: 978-1-119-23637-5
460 pages
May 2016, Wiley-IEEE Press

## Description

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.

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.
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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

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

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

3.5 What Should I Retain from this Chapter? 180

References 182

3.6 Appendix 3A – Problems 183

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

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

References 431

Conclusion 433

Glossary of Terms 435

Index 439

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## Author Information

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.

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## Reviews

Of the skills needed to be an analog circuit engineer, one of them is the ability to construct from a circuit diagram a representation of the behavior of the circuit. For linear circuits, the well-established general scheme has been to express behavior in the complex frequency or s-domain. The cause-effect, or input-output behavior of a circuit is its transfer function, and when expressed as a function of s, essentially all that circuit engineers are interested in can be found from it (including the time-domain response) - hence the importance of transfer functions expressed in the s-domain.

Electronics engineers begin to acquire this skill in the undergraduate engineering course on passive circuits, and it becomes more complicated in the active-circuits course. In circuits with n independent inductances and capacitances, basic s-domain circuit analysis (which, by the way, requires little more than pre-university algebra, so that technicians having only introductory calculus can do it) results in nth-degree transfer function polynomials. When the polynomials are factored into real and complex pairs, the poles and zeros of the circuit are determined, and they determine the dynamic circuit behavior. Yet factoring a polynomial higher that a quadratic (that is, having s2 as the largest power in the polynomial) is difficult enough to drive most engineers to computer circuit simulation instead. So why this book?

Circuits can often be compartmentalized into stages with only one or two reactances in each stage. These circuits can be formidable to analyze for engineers unaccustomed to using much math, yet Basso’s book presents higher-level circuit theorems or methods that reduce their apparent complexity. Chapter one starts easy, explaining basic concepts such as a port, the four possible transfer functions (input-output combinations of voltage and current), voltage dividers, Thevenin’s and Norton’s theorems, and how by shorting and opening circuits at the reactances, time constants can be found. These concepts form the “building blocks” for finding the three parameters of greatest interest: the transfer functions and input and output impedances.

Chapter 2 shows, using simple examples, how the structure of circuits relates to the coefficients in the transfer function polynomials. It begins what is continued in Chapter 3 that was a forte of Robert David Middlebrook of Cal Tech, that of simplified methods for analyzing circuits. Middlebrook developed the Extra Element Theorem (EET), a refinement of previous methods that include those of Blackman, Mulligan, Cochrun and Grabel, and Paul E. Gray and Campbell Searle at MIT. (I published a ten-part article on EDN starting January 2013 called “Design-Oriented Circuit Dynamics” that gives more of the history and detailed development of these methods.) Chapter 3 includes, along with the EET, the important and basic superposition theorem.

The EET is explained step-by-step by Basso and should eventually make its way into undergraduate circuits courses. This book is suitable as a textbook for an advanced active-circuits course; it has an extensive set of problems at the end of each chapter, with a chapter summary and references. The EET is a clever way of finding the effect on a circuit with an existing transfer function of adding an additional circuit element, usually a reactance. It is a way of starting with a simplified circuit, such as a transistor amplifier stage with an infinitely fast transistor (no Ce or Cc), and incrementally developing its transfer function by adding one capacitance for each invocation of the EET. Beware however, that these successive increments of transfer function development can become as algebraically-intensive as straightforward circuit analysis using the node-voltage and loop-current methods. Yet the EET is an improvement because it offers greater insight into how circuit elements affect the overall circuit behavior.

In chapter 3, Basso does not leave the reader wondering how to apply the methods he is explaining because he gives detailed, step-by-step examples to illustrate them. Chapters 4 and 5 continue this trend with transfer functions which have second-degree (quadratic) polynomials. The EET procedure is the same only the examples become more complicated. Finally, in chapter 5, the EET is expanded to circuits with n independent reactances, and the nEET, a further development of the EET (mainly by Ali Hajimiri) and aided by the Cochrun-Grabel method (which I also cover in Transistor Amplifiers, at Innovatia, but to a far lesser extent than in Basso’s book).

Besides Middlebrook, who is known for his emphasis on conceptual simplification and clarification of circuit analysis, Basso also credits Vatché Vorpérian who also has a book on methods of simplified circuit analysis. Basso’s book continues the tradition of finding ways of simplifying both an understanding of and analytical procedures for circuits. The book includes a glossary of key expressions and an index.

Planet Analog (http://www.planetanalog.com) Article by Dennis Feucht, Electronics Engineer, 6/21/2016
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