Mixed-type topological formulas for general linear networks

"Topological formulas" are established for the most general type of linear electrical networks in which the characteristics of branches are expressed in terms partly of impedances and partly of admittances. We shall call this type of topological formulas "mixedtype topological formulas." Moreover, the networks under consideration may contain any type of mutual coupling such as active elements, controlled sources, ideal transformers, etc. The formulas are derived directly from the fundamental laws in network theory. Existing topological formulas are particular cases of those presented in this paper, and our way of derivation will be one of the most straightforward and can be applied to derive any known topological formula. Use is made only of the topological properties of original networks but not of auxiliary graphs such as voltage graphs, current graphs, and (signal-) flow graphs. The denominators and numerators of the formulas are expressed as the sum of the immittance products corresponding to the tree-cotree pairs of special kinds, and the sign of each term in the sum is characterized in topological terms.