IEEE Circuits and Systems Magazine - Q1 2020 - 32

Feature

An Alternative Path
to Foster's Reactance
Theorem and Its
Relation to Narrow-Band
Equivalent Circuits
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Bernhard Jakoby, Senior Member, IEEE

Abstract
Foster's seminal treatise on lossless networks has been published almost 100 years ago and a particularly notable conclusion
drawn therein, i.e. that the reactance (and susceptance) functions
are always monotonically increasing with frequency, is frequently referred to as Foster's theorem. In this paper we present two
variants for an alternative simple derivation of a stronger form of
this theorem, which holds for the driving point reactance (susceptance) of general lossless devices, i.e. also configurations without
lumped elements. One version introduces a realizable lumped element equivalent circuit approximating the considered circuit in a
narrow band around a particularly considered frequency. It turns
out that this avenue of proof also facilitates an alternative validation of the realizability of the so called Foster 1 and 2 realizations.

I. Introduction
n 1924 Foster stated conditions for realizable immittances
(i.e. impedances Z or admittances Y ) of lossless networks (one-ports) composed of (potentially coupled)
inductances L and capacitances C [1]. The impedance
Z = 1/Y of such a network is purely imaginary Z (~) =

Digital Object Identifier 10.1109/MCAS.2019.2961725
Date of current version: 11 February 2020

IEEE CIRCUITS AND SYSTEMS MAGAZINE

jX (~) = 1/ ( jB (~)), where X and B denote the real-valued
reactance and susceptance, respectively, which are functions of the (radian) frequency ~. Foster showed that the immittances are rational functions in s = j~ and that the associated partial fraction decomposition can be translated into
realizable equivalent circuits, where terms corresponding to
poles at particular frequencies ! ~ x can be represented by
a resonant LC (parallel or series) circuits. Poles at infinity
or zero are represented by single inductances L or capacitances C and all these individual branches are connected in
series or parallel when considering the partial fraction decompositions for Z and Y, respectively, as shown in Fig. 1.
These canonical realizations are termed "Foster 1" and "Foster 2" circuits, respectively, and they represent equivalent
circuits for the considered lossless network, which are valid
in the entire frequency range. Example 1 (in a separate box)
illustrates this procedure for a particular example.
While establishing a partial fraction decomposition
for a rational function is a trivial step, the fact that the
so derived circuit is actually realizable, i.e. that the resulting values for L and C are positive and real, is not
self-evident and was proven by Foster. In literature, the
term "Foster's theorem" is frequently associated with a

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IEEE Circuits and Systems Magazine - Q1 2020

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