Group Theory and the Electric Circuit

Abstract

Electrical networks consisting of inductances, resistances, and capacitances form a group with the impedance function as an absolute invariant. That is, to a given impedance function there corresponds an infinite number of networks, any one of which can be obtained from any other by a special linear transformation of the instantaneous mesh currents and changes of the network. In this manner one may arrive at the complete infinite set of networks equivalent to a given network of any number of meshes. This is done by writing down the three fundamental quadratic forms of the network. Then a linear affine transformation of the instantaneous mesh currents and charges of the network results in the formation of new quadratic forms, the matrices of the coefficients of which represent a member of the group, i.e., an equivalent network. Instead of performing the substitutions, the three matrix multiplications $C^{\prime }AC$ are used, one for each quadratic form, where $A$ represents the original matrix, $C$ the transformation matrix, and $C^{\prime }$ its conjugate. It may be possible to extend this theory to include continuous systems where the quadratic forms become integrals or infinite series and one deals with infinite matrices and infinite transformations.