Tellegen’s Theorem and Electrical Networks

B. D. H. Tellegen was the first to point out (1952, 1953) the generality and wide-ranging usefulness of the theorem that bears his name. Nevertheless, the theorem is still not as widely known as its utility warrants. The authors of this monograph set out to correct this neglect, noting that "There is hardly a basic network theorem that cannot be proved by invoking Tellegen's theorem. The simplicity and generality of the theorem make it attractive pedagogically, and its ability to generalize known results and lead to new results indicates its research value. This theorem definitely should be in every circuit designer's kit of tools."

Tellegen's theorem is unusual in that it depends solely upon Kirchhoff's laws and the topology of the network. The theorem thus applies to all electrical networks that obey Kirchhoff's laws, whether linear or nonlinear, time-invariant or time-variant, reciprocal or nonreciprocal, passive or active, single-valued or multiple-valued, hysteretic or nonhysteretic. The excitation is arbitrary—it may be sinusoidal, exponential, periodic, transient, or random. Also, the initial conditions may be arbitrarily chosen. The modern interest in nonlinear and time-variant networks gives Tellegen's theorem a special new importance, because it is one of the very few general theorems that apply to such networks.

To demonstrate its range of applications and the theorem's great power in the derivation of other basic and important theorems about electrical networks (and the extent that these other theorems are special cases of Tellegen's), the authors have collected more than 100 such theorems and have shown that they can be proved from Tellegen's theorem. Most of these were known before; but some are extended in their range of validity, and a few are new. (Apart from Tellegen's theorem, this collection of theorems is valuable in its own right.) Applications are given to automated network synthesis and to nonlinear, time-varying, switching, nonreciprocal, and other networks—all the major areas of network theory are covered. In addition, extensions of the theorem to other physical systems are discussed, including applications to the electromagnetic field, electron beams and plasmas, and quantum mechanics.

The theorem is proved in its most general form thus far known. In addition, two weaker forms that have useful properties for certain applications are presented. In these weaker forms, the theorem applies to voltages, currents, and wave (or scattering) variables. The use of wave variables in Tellegen's theorem is believed to be new.

Oliver Heaviside used a version of the theorem in 1883 to establish a specific result, and others have used its equivalent in a limited range of applications. Others (Weyl, 1923; Bott, 1949) have derived highly abstract and mathematical versions without regard to applications. But Tellegen was the first to devote a full paper to the subject and the first to grasp the theorem's general importance and applicability. In a similar way, the authors of this monograph are the first to devote a book to the subject and the first to collect (or newly present) all of the most important applications of the theorem in the hope of bringing it into the common currency it deserves.