Infinite Electrical Networks Research Articles

Countably Infinite Nonlinear Time-Varying Active Electrical Networks

This work presents existence and uniqueness theorems for the currents and voltages in a countably infinite RLC electrical network for which the total power dissipated in all the resistors or stored in all the capacitors and inductors is allowed to be infinite. This relaxation of the finite-power condition prevents the use of a number of Hilbert-space techniques and requires instead a more graph-theoretical approach. The latter has previously been used to analyze linear time-invariant networks. The main contribution of the present work is that it encompasses, under certain conditions, time-varying active networks with nonlinear resistors, inductors, and capacitors.

Nonlinear, resistive, countably infinite electrical networks †

Nonlinear infinite electrical networks can be analysed by using Hilbert-space techniques when the total power in the network is finite. This work attacks the case where the total power is not finite. Graph-theoretic methods are used to show that a unique current flow occurs in a countably infinite, nonlinear, resistive network after the voltage-current pairs in certain specified branches are arbitrarily assigned. The currents and voltages throughout the entire network can be determined by computing them recursively in a sequence of finite subnetworks that partition the network. The main theorem requires that the resistances and conductances in the network satisfy sufficiently strong Lipschitz conditions.

The limb analysis of countably infinite electrical networks

A method, called “limb analysis,” for analyzing certain countably infinite electrical networks was developed in a prior publication. It consisted of two parts, a graph-theoretic decomposition of the network and an analysis wherein Kirchhoff's node and loop laws were applied. The present paper is primarily devoted to proving that every countable network possesses the required graphtheoretic decomposition. In addition, it also establishes a sufficient set of conditions under which the analytic portion of limb analysis has a solution.

The Complete Behavior of Certain Infinite Networks Under Kirchhoff’s Node and Loop Laws

The objective of this work is the investigation of all possible current distributions in an infinite electrical network subject only to Kirchhoff’s node and loop laws. These laws are in general not strong enough to yield a unique set of branch currents. All prior investigations of infinite networks imposed additional requirements, such as finiteness of the total power dissipation, in order to force uniqueness, but in doing so the other possible responses of a network were discarded.The main result of this work holds not for all infinite, locally finite networks but for some fairly general classes of such networks as well as for certain countably infinite networks that need not be locally finite. It states that if the currents in certain branches, called “joints,” are arbitrarily chosen, then all other branch currents are uniquely determined. Moreover, all possible sets of branch currents satisfying only the node and loop laws are encompassed in this result; one need merely choose the joint currents proper...

Infinite electrical networks

The investigation of infinite electrical networks having no restrictions on their graphs other than countability and connectedness is a recent occurrence. The first rigorous analysis of locally finite purely resistive networks was published in 1971 by Flanders, and the subject has developed apace since then. This article summarizes the current status of the subject and points out some open questions and areas for future research. First of all, various approaches to infinite purely resistive networks are described. They use some ideas from algebntic topology and Hilbert-space theory. These methods no longer work when the network is allowed to have a variety of electrical parameters. Two approaches to the latter situation ate then presented. In one of these, the network is decomposed into an interconnection of a finite number of infinite subnetworks, each of which contains parameters of only one kind. If each subnetwork can be characterized as an operator on Hilbert's coordinate space l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , it is possible under certain circumstances to achieve a solution to the network by extending the customary analyses for finite scalar networks to finite operator networks. The second approach, which is inherently very general and is due to Dolezal, treats an incidence matrix a as an operator on a certain Hilbert space and characterizes the solution of the network by means of the null space of a and its orthogonal complement. All these approaches achieve a unique solution by imposing various conditions in addition to Kirchhoff's node and loop laws. The last part of the paper investigates the whole class of all possible solutions when only Kirchhoff's node and loop laws are imposed.

Infinite electrical networks

The investigation of infinite electrical networks having no restrictions on their graphs other than countability and connectedness is a recent occurrence. The first rigorous analysis of locally finite purely resistive networks was published in 1971 by Flanders, and the subject has developed apace since then. This article summarizes the current status of the subject and points out some open questions and areas for future research. First of all, various approaches to infinite purely resistive networks are described. They use some ideas from algebntic topology and Hilbert-space theory. These methods no longer work when the network is allowed to have a variety of electrical parameters. Two approaches to the latter situation ate then presented. In one of these, the network is decomposed into an interconnection of a finite number of infinite subnetworks, each of which contains parameters of only one kind. If each subnetwork can be characterized as an operator on Hilbert's coordinate space l 2 , it is possible under certain circumstances to achieve a solution to the network by extending the customary analyses for finite scalar networks to finite operator networks. The second approach, which is inherently very general and is due to Dolezal, treats an incidence matrix a as an operator on a certain Hilbert space and characterizes the solution of the network by means of the null space of a and its orthogonal complement. All these approaches achieve a unique solution by imposing various conditions in addition to Kirchhoff's node and loop laws. The last part of the paper investigates the whole class of all possible solutions when only Kirchhoff's node and loop laws are imposed.