A comprehensive framework for the Thévenin-Norton theorem using homogeneous circuit models

The homogeneous description of a linear, uncoupled circuit is based on the assignment to each device of a triad $(p: q: s)$, where the parameters are defined up to a nonzero multiplicative constant and characterize a voltage-current relation of the form $pv-qi=s$. Given a one-port, the open-circuit and short-circuit network determinants, to be denoted as $p_e$ and $q_e$, are polynomial functions of the $p$- and $q$-parameters of the individual devices. With this formalism, we may state the Thévenin-Norton theorem in a uniform manner by saying that, for any given set of parameter values, if at least one of the functions $p_e$ and $q_e$ does not vanish then the voltage-current behavior at the port is characterized by a homogeneous triad $(p_e: q_e: s_e)$. In particular, the assumptions $p_e
eq 0$ and $q_e
eq 0$, respectively, characterize the existence of the Thévenin and the Norton equivalents, but the formulation proposed above avoids the need to make an a priori distinction between one form and another. We also show that the excitation parameter $s_e$ can be computed by inserting any admissible load at the port, but also analytically, in terms of the topology of the underlying digraph. The results hold without the need to specify whether each circuit element is a source or a passive device, much less to assume whether they are voltage- or current-controlled.