An axiomatization of circuit theory

This paper concerns the mathematical foundations of electric circuit theory. The conventional model of a circuit is a bottom-up or internal view: a circuit is an interconnection - represented by a linear graph - of elements whose voltage-current relations are given. The paper outlines the model corresponding to the top-down view: a circuit is a function, mapping terminal voltages to terminal currents, which satisfies two axioms. New circuits are created from old ones by means of two operations: interconnection and restriction (the suppression of terminals). The top-down or functional description of a circuit can of course be derived from the bottom-up view, but the point of the paper is that we can ignore the bottom-up view entirely and express the theory of circuits in terms of their external descriptions alone. The case for the top-down view is not made here, but evidence is given that this view enables the definitions of the theory to be stated precisely and the theorems to be proved rigorously. The benefits of the top-down point of view are in the simplicity and rigour of the theory and, probably, in the synthesis of circuits.