Title :
Countably Infinite Networks that Need not be Locally Finite
fDate :
3/1/1974 12:00:00 AM
Abstract :
A network that is not locally finite is an infinite network with at least one node having an infinity of incident branches. Such networks arise naturally when the short circuits in a locally finite infinite network are coalesced into nodes. Although a number of existence and uniqueness theorems for the behavior of a locally finite network have been given in the literature, such is not the case for nonlocally finite networks. This paper provides such a theorem for the branch-voltage drops of a nonlocally finite resistive network having an infinity of current sources satisfying a certain restriction. The conditions under which this result holds are Kirchhoff´s loop law, the finiteness of the total power dissipation, and a generalized form of Tellegen´s theorem. However, Kirchhoff´s node law need not hold at those nodes having an infinity of incident branches. Voltage sources can be taken into account by making appropriate changes of sources.
Keywords :
Distributed and microwave networks and filters; Distributed networks; Infinite networks; Resistive networks; Circuit theory; Filters; H infinity control; Laboratories; Mathematics; Network topology; Power dissipation; Statistics; Voltage;
Journal_Title :
Circuits and Systems, IEEE Transactions on
DOI :
10.1109/TCS.1974.1083837