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].