Infinite electrical networks with finite sources at infinity

Abstract

Various physical situations leading to the exterior problem for partial differential equations are modeled by infinite electrical networks having finite-valued sources connected to extremities of the network at infinity. Up to now, there was no theory for such a network. The present work establishes one. An existence and uniqueness theorem is proven for a countably infinite resistive network that has an infinity of voltage and current sources, some of which connect to different "parts of infinity," as well as certain finite and infinite nodes that are shorted to different "parts of infinity" as well. The phrase in quotes is made precise by introducing the new concepts of extended nodes, extended branches, pathlike extremities, and extremities for the network. Moreover, short circuits between different pathlike extremities are allowed. The hypothesis of the theorem requires that the maximum power available from all the sources in the network be finite. Some examples are given, relating to petroleum flow in an oil well and to well-logging for geophysical exploration, to show how the considered infinite electrical networks can arise from practical applications.