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

Source abstract

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 properly in order to obtain any given permissible set of branch currents. Another result concerns the idea of a homogeneous current flow; this is a set of branch currents satisfying the node and loop laws when all sources are set equal to zero. The dimension of the linear space of all homogeneous current flows is shown to be equal to the cardinal number of the set of joints. Finally, it is worth noting that our analysis provides a method for calculating the current in any given branch through a finite number of computational steps.