Passive Operator Networks

The concept of an RLC network is introduced, where each R, L, and C-instead of being a positive number or matrix-is a positive bounded linear operator on a Hibert space H. Actually, this idea of a network of operators is generalized still further since in the frequency domain the branch impedances or admittances are only required to be certain kinds of operator-valued positive-real functions. Such operator networks can be used to study various types of infinite networks that are natural models for integrated and distributed networks. The procedure is to decompose a given infinite RLC network into infinite one-element-kind subnetworks in such a way that each subnetwork is characterized by either , or , where now R, L, and C are positive invertible operators on Hilbert\´s coordinate space . Some fairly general conditions on the subnetworks are presented, which insure the validity of the operator representations. At this point, essential use is made of some recently published results by Flanders concerning the uniqueness of the behavior of infinite networks. In the general case, where is replaced by H, it is shown that the fundamental theorem of electrical networks continues to hold, so that H-valued voltage and current sources produce unique current distributions throughout the network. Moreover, the driving-point impedances of such networks are shown to be operator-valued forms of positive-real functions. A more special characterization for the driving-point impedances of RL and RC-type operator networks is also established. Mutual coupling can also be taken into account.