Homological and Cohomological Invariants of Electric Circuits

Abstract

An attempt was made to substantiate strictly the tensor point of view on the electric circuit that was first introduced in the classical works of G. Kron. Geometrical structure of the circuit was shown to generate groups of homologies and cohomologies to which two pairs of vector spaces are assigned isomorphically. Here, invariance of the input and output powers turns out to be a natural consequence of the topological nature of the circuit, which enables one to construct a tensor model of electric circuits: currents and voltages of the all-loop and all-node circuits can be regarded as the countervariant and covariant vectors of the conjugate spaces, and the passage from the primitive circuit to the all-loop (all-node) one is done by transforming the bases of the homological and cohomological spaces; the matrices of inductances, capacities, and conductances are of tensor nature, i.e., are the coordinate representations of the covariant and countervariant circuit invariants, and the kinetic and potential energies of the circuit are represented as the corresponding bilinear functionals.