The theory of electrical networks became fully launched, it seems fair to say, when Gustav Kirchhoff published his voltage and current laws in 1847 [72]. Since then, a massive literature on electrical networks has accumulated, but almost all of it is devoted to finite networks. Infinite networks received scant attention, and what they did receive was devoted primarily to ladders, grids, and other infinite networks having periodic graphs and uniform element values. Only during the past two decades has a general theory for infinite electrical networks with unrestricted graphs and variable element values been developing. The simpler case of purely resistive networks possesses the larger body of results. Nonetheless, much has also been achieved with regard to RLC networks. Enough now exists in the research literature to warrant a book that gathers the salient features of the subject into a coherent exposition.

As might well be expected, the jump in complexity from finite electrical networks to infinite ones is comparable to the jump in complexity from finite-dimensional spaces to infinite-dimensional spaces. Many of the questions we conventionally ask and answer about finite networks are unanswerable for infinite networks – at least at the present time.

As was indicated in Example 1.6-3, the total power dissipated in the resistances by a voltage-current regime, satisfying Ohm's law, Kirchhoff's current law at finite nodes, and Kirchhoff's voltage law around finite loops, need not be finite. Moreover, these laws need not by themselves determine the regime uniquely. However, if voltage-current pairs are assigned to certain branches, the infinite-power regime may become uniquely determined. The latter result requires in addition the “nonbalancing” of various subnetworks, as is explained in the next section. In which branches the voltage-current pairs can be arbitrarily chosen and how the nonbalancing criterion can be specified are the issues resolved in this chapter. The discussion is based on a graph-theoretic decomposition of the countably infinite network into a chainlike structure, which was first discovered by Halin for locally finite graphs [63]. That result has been extended to graphs having infinite nodes [166]. The chainlike structure implies a partitioning of the network into a sequence of finite subnetworks, which can be analyzed recursively to determine the voltage-current pair for every branch. We call this a limb analysis.

As was mentioned before, in most of this book we restrict our attention to resistive networks. However, a limb analysis can just as readily embrace complex-valued voltages, currents, and branch parameters. In short, a limb analysis can be used for a phasor representation of an AC regime or for the complex representation of a Laplace-transformed transient regime in a linear RLC network [166].

Perhaps the most important infinite electrical networks with respect to physical phenomena – putting aside the finite networks – are the infinite grids. This is because finite-difference approximations of various partial differential equations have realizations as electrical networks whose nodes are located at the sample points of the approximation. Those sample points are distributed in accordance with increments in each of the coordinates, hence the gridlike structure. Moreover, if the phenomenon exists throughout an infinite domain, it is natural (but, to be sure, not always necessary) to choose an infinity of sample points. In this way, one is led to infinite electrical grids as models for the so-called “exterior problems” of certain partial differential equations. Two cases of this were presented in Section 1.7, and more will be discussed in the next chapter. The grids we examine are of two general types: the grounded grids, wherein a resistor connects each node to a common ground node, and the ungrounded grids, wherein those grounding resistors are entirely absent. Grounded grids are readily analyzed, but ungrounded grids are more problematic because of a singularity in a certain function that characterizes the network.

This chapter is devoted to rectangular grids, the natural finite difference model for Cartesian coordinates, but the analysis can be extended to other coordinate systems such as cylindrical and spherical ones [180], [183], [184], [188].

The adjective “nonlinear” will be used inclusively by taking “linear” to be a special case of “nonlinear.” As promised, we present in this chapter two different theories for nonlinear infinite networks. The first one is due to Dolezal and is very general in scope – except that it is restricted to 0-networks. It is an infinite-dimensional extension of the fundamental theory for scalar, finite, linear networks [67], [115], [127]. In particular, it examines nonlinear operator networks, whose voltages and currents are members of a Hilbert space ℋ; in fact, infinite networks whose parameters can be nonlinear, multivalued mappings restricted perhaps to subsets of ℋ are encompassed. As a result, virtually all the different kinds of parameters encountered in circuit theory – resistors, inductors, capacitors, gyrators, transformers, diodes, transistors, and so forth – are allowed. However, there is a price to be paid for such generality: Its existence and uniqueness theorems are more conceptual than applicable, because their hypotheses may not be verifiable for particular infinite networks. (In the absence of coupling between branches, the theory is easy enough to apply; see Corollary 4.1-7 below.) Nonetheless, with regard to the kinds of parameters encompassed, this is the most powerful theory of infinite networks presently available. Dolezal has given a thorough exposition of it in his two books [40], [41]. However, since no book on infinite electrical networks would be complete without some coverage of Doleza's work, we shall present a simplified version of his theory.

The purposes of this initial chapter are to present some basic definitions about infinite electrical networks, to show by examples that their behaviors can be quite different from that of finite networks, and to indicate how they approximately represent various partial differential equations in infinite domains. Finally, we explain how the transient responses of linear RLC networks can be derived from the theory of purely resistive networks; this is of interest because most of the results of this book are established in the context of resistive networks.

Notations and Terminology

Let us start by reviewing some symbols and phraseology so as to dispel possible ambiguities in our subsequent discussions. We follow customary usage; hence, this section may be skipped and referred to only if the need arises. Also, an Index of Symbols is appended for the more commonly occurring notations in this book; it cites the pages on which they are defined.

Let X be a set. X is called denumerably infinite or just denumerable if its members can be placed in a one-to-one correspondence with all the natural numbers: 0, 1, 2,. … X is called countable if it is either finite or denumerable. In this book the set of branches of any network will always be countable.

The notation {x ∈ X: P(x)}, or simply {x: P(x)} if X is understood, denotes the set of all x ∈ X for which the proposition P(x) concerning x is true.