Geometric Characterization of Series-Parallel Variable Resistor Networks

The range of operating conditions for a series-parallel network of variable linear resistors, voltage sources, and current sources can be represented as a convex polygon in a Thevenin or Norton half-plane. For a network with n elements of which k are variable, these polygons have at most 2k vertices and can be computed in O(nk) time. These half planes are embedded in the real projective plane to represent circuits with potentially infinite Thevenin resistance or Norton conductance. For circuits that have an acyclic structure once all branches to ground are removed, the characteristic polygons for all nodes with respect to ground can be computed simultaneously by an algorithm of complexity O(nk).