General passive networks-Solvability, degeneracies, and order of complexity

A topological expansion of the network determinant is developed which makes possible the investigation of some properties of general passive networks modeled by two-terminal elements and gyrators. Necessary and sufficient conditions are obtained for the unique solvability of the network equations. These conditions are expressed in terms of the network graph and certain gyrator-only networks, called
G
-networks, which are derived from the original network by removing and contracting two-terminal elements. The solvability of
G
-networks is also shown to be fully related to the degeneracies existing in the state-variable characterization of the network. A procedure is given for determining the actual number of state equations (order of complexity) needed for describing an arbitrarily degenerate passive network. The theoretical results obtained are illustrated with some examples.