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 $p_e \neq
0$ and $q_e \neq 0$, 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.

Apr 2023