Operating points in n-dimensional nonlinear nonuniform infinite and finite grounded grids

An existence and uniqueness theorem for the operating point of a resistive, infinite, n-dimensional nonlinear nonuniform grounded grid is established. A similar result is established for finite grids of arbitrary size. The grid may be triangular (also called hexagonal), rectangular, or more generally automorphic under every shift mapping of the nodes, but it is locally finite. The analysis is accomplished under conditions restricting the nonlinearity and nonuniformity sufficiently to allow the operator arising from a nodal analysis to be decomposed into the sum of a Laurent operator and a nonlinear operator, which in turn can be rearranged into a contraction mapping. A similar analysis, wherein the Laurent operator is replaced by a circulant matrix, works for a finite grid. The grounding elements must have monotone characteristics, while the floating elements can be nonmonotone as well as active.