An Alternative Path to Foster's Reactance Theorem and Its Relation to Narrow-Band Equivalent Circuits

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.