Linear Circuit Transfer Functions

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 Vatche Vorperian 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