Middlebrook’s Extra Element theorem

-September 15, 2013

Following historically from the earlier work of Blackman, Gray and Searle, Cochrun and Grabel (and Rosenstark) are some newer circuit analysis methods developed by R. D. Middlebrook of Caltech. This article presents the Extra Element Theorem (EET), a powerful and simple method of problem reduction that reduces circuits so that they can be analyzed with one reactance at a time. Some variations are also presented (the impedance EET) along with the original, non-obsolete theorem from Blackman for feedback loops.

Extra Element Theorem (EET)

The extra element theorem (EET) was developed by R. D. Middlebrook as a refinement of a long history of related methods. As the “genealogy” chart of dynamics methods from the first article of this series, “Circuit Dynamics for Design”, shows, the EET combines ideas that are found in a less refined form in Gray and Searle’s MIT textbook on active circuits (Electronic Principles: Physics, Models, and Circuits, Wiley, 1969) and in Blackman’s Impedance Theorem (BZT). BZT shows the power of port-oriented methods of circuit analysis.

More can be inferred from port analysis than is at first apparent by subjecting the ports to different conditions, and this is in part a consequence of the properties of linear systems. (All these methods are based on linearized circuit variables that vary incrementally around a static operating-point.) The EET is the culmination of a century of development of port-oriented analytic techniques and is the featured method to master, though simple methods such as the OCTC and quadratic Cochrun-Grabel-Rosenstark methods are quite useful to know and apply.

The EET is based on the following diagram of a circuit (box) with input and output ports, xi and xo and an additional port somewhere in the circuit with external impedance Z across it having port voltage v and current i.

The Z port is that of a circuit element - the “extra” element. With Z attached to the circuit, the port v-i relationship is

The negative sign signifies that Z is external to the port. By port convention, the port driving-point impedance, v/i, is that looking into the circuit port. The current direction must be reversed (its polarity changed) to refer to Z. The port equations are

The four port parameters can be found:

Aoc is the gain from xi to xo with the Z port open-circuited. ZD is the Z-port driving-point impedance, the impedance of the circuit from the port without the external Z and with the condition on ZD that the input be set to zero.

From the port equations, substituting for v and solving for i,

Substituting for i in the port equation for xo,