IDENTIFICATION OF VOLTAGE-CURRENT OPERATORS SATISFYING TELLEGEN'S THEOREM

Abstract

Let [Formula: see text] be a directed nonseparable network composed of a finite set ℬ of branches and F(u,i), an operator defined on the set of all ordered pairs of voltages and currents. It is proved that, if [Formula: see text] is neither a cutset nor a loop, and contains both a loop and a cutset with unequal numbers of opposite directed branches, and [Formula: see text] for all branch voltages [Formula: see text] and currents [Formula: see text] satisfying Kirchhoff's voltage and current laws respectively in every loop and cutset of [Formula: see text], then F is biadditive. This substantially strengthens an old theorem of Kishi and Kida in which, [Formula: see text] was allowed to vary over the set of all networks and physically reveals that the power is essentially the only continuous voltage-current operator which, when associated with a branch, yields Tellegen's famous theorem. It is also proved that, conversely, if F is biadditive, then the above sum is zero for an arbitrary network [Formula: see text]. The topological assumptions imposed on [Formula: see text] are thoroughly examined and shown to be necessary. Finally, similar analytic characterizations of eulerian and bipartite networks, respectively in terms of voltage and current operators with zero sums of values over all network branches are given.