On extendibility of voltage and current regimes from subnetworks Original Research Article

Let N be a finite directed network and M its nonempty subnetwork. It is proved that if uM and iM are vectors of branch voltages and currents in M satisfying Kirchhoffʹs laws in every loop and cutset of N contained in M, then they can be extended to vectors u and i of branch voltages and currents in N satisfying Kirchhoffʹs laws in every loop and cutset of N. A constructive procedure leading to such extensions is given. Moreover, u and i are uniquely determined if and only if there exist, respectively, a forest and a coforest of N contained in M. These results are considered also in the case of infinite networks. The proof is set-theoretic and employs the axiom of choice. Finally, the extendibility of nonnegative voltage and current regimes consistent with Kirchhoffʹs laws from subnetworks is discussed. Then sufficient topological conditions which make it possible are specified and both kinds of extendibility are shown to be mutually exclusive already in the case when M consists of a single branch. All this is motivated by the possibility of Tellegenʹs theorem on the total power consumption in subnetworks investigated recently by the author in the case of finite electrical networks, Mintyʹs classical paper and by various technical problems.