A Topological Method of Generating Constant Resistance Networks

Zadeh has shown that any self-dual network, fixed or linear time varying, is a constant resistance network. To date, the only known constant resistance networks with self-dual structures are the classical lattice and bridged-T networks. In this paper, we investigate the topological aspect of the problem, with the aim of obtaining new constant resistance network configurations. Let
G_{\rho}
be a self-dual one-terminal-pair graph with respect to vertices (
i, j
), and with the degrees of (
i, j
) both equal to
\rho
. It is proved that for
\rho \geqq 2, G_\rho
can be realized with
8 \rho - 11
edges, but not with fewer edges, if the union of
G_\rho
and an edge joining (
i, j
) is to be 3-connected. Using these graphs as the basis, a class of constant resistance networks are generated, which include the classical lattice and bridged-T networks as special cases for
\rho = 2
. The generation of a constant resistance network for
\rho = 3
is shown in detail, with a numerical example illustrating its application in transfer function synthesis.