Topological Rules for Linear Networks

Topological rules are provided for the expansion of admittance functions of linear networks which may include reciprocal and nonreciprocal branches, infinite-gain operational amplifiers, dependent current or voltage sources, vacuum tubes, transistors, and ideal transformers. Kirchhoff's rules for reciprocal networks, on the one hand, and signal-flow graph rules, on the other, appear as special cases. The rules are purely topological and, in contradistinction to some prior work, do not include sign rules dependent upon an arbitrary assignment of labels to nodes, or directions to reciprocal branches. Any term in an expansion corresponds to a subgraph of the network. The subgraphs are "loop-woods," formed by trees comprised of reciprocal branches and disjoint loops which include at least one nonreciprocal branch each. Transformers are represented by a set of weighted short circuits. The rules are proved by means of "polar" incidence matrices A+, A- whose properties are explored.