# Fast Analytical Techniques for Electrical and Electronic Circuits by Vatche Vorperian

PREFACE:

The title of this book could easily have been called Variations on a Theme by Middlebrook, or Applications of The Extra Element Theorem and its Extensions. Neither title, however, would have captured the unique message of this book that one can solve very complicated linear circuits in symbolic form almost by inspection and obtain more than one meaningful analytical answer for any transfer function or impedance. The well-known and universally practiced method of nodal or loop analysis not only becomes intractable when applied to a complicated linear circuit in symbolic form, but also yields unintelligible answers consisting of a massive collection of symbols.

In a meaningful analytical answer, the symbols must be grouped together in low-entropy form — a term coined by R. D. Middlebrook — clearly indicating series and parallel combination of circuit elements, and sums and products of time constants. The illustrative examples in Chapter 1 serve as a quick and informal introduction to the basic concepts behind the radically different approach to network analysis presented in this book. Today, the only method of circuit analysis known to most engineers, students and professors is the method of nodal or loop analysis.

Although this method is an excellent general tool for obtaining numerical solutions, it is almost useless for obtaining analytical solutions in all but the simplest cases. Anyone who has attempted inverting a matrix with symbolic entries — sometimes as low as secondorder — knows how tedious the algebra can get and how ridiculous the resulting high-entropy expressions can look. The purpose of this book is not to eliminate the linear algebra approach to network analysis, but instead to provide additional new and efficient tools for obtaining analytical solutions with great ease and without letting the algebra run into a brick wall.

Among the most important techniques discussed in this book are the extra element theorem (EET) and its extension the N-extra element theorem (NEET). These two theorems are discussed in Chapters 3 and 4 after a brief and essential review of transfer functions given in Chapter 2. The EET and its proof were given by R. D. Middlebrook. The NEET was given without proof by Sarabjit Sabharwal, an undergraduate at Caltech in 1979. In Chapter 4, a completely original treatment of the NEET is given, where it is stated in its most general form using a new compact notation and, for the first time, proven directly using matrix analysis.

The subject of electronic feedback is treated in Chapter 5 using the EET for dependent sources, and another theorem by R. D. Middlebrook called simply ‘‘the feedback theorem’’. Both methods lead to a much more natural formulation of electronic feedback than the well-known block diagram approach found in most textbooks. Block diagrams are useful tools in linear system theory to help visualize abstract concepts, but they tend to be very awkward tools in network analysis.

For instance, in an electronic feedback circuit neither the impedance loading nor the bi-directional transmission of the feedback network are easily captured by the single-loop feedback block diagram unless the feedback network and the amplifier circuit are both manipulated and forced to fit the block diagram. The fact is block diagrams bear little resemblance to circuits and their use in network analysis mainly results in loss of time and insight.

The examples presented in Chapters 6 and 7 are a tour de force in analysis of complicated circuits which demonstrate the efficacy of the fast analytical techniques developed in the previous chapters. Among the examples discussed in these chapters are higher-order passive filters and a MESFET amplifier. Some infinite networks, including fractal networks, are discussed in Chapter 7 where an interesting, and possibly new, result is presented. It is shown that a resistor, an inductor and a capacitor are all special cases of a single, two-terminal, linear element whose voltage and current are related by a fractional derivative or its inverse, the Riemann—Liouville fractional integral.