A Comprehensive Framework for the Thévenin–Norton Theorem Using Homogeneous Circuit Models

Abstract

The homogeneous description of a linear, uncoupled circuit is based on the assignment to each device of a triad <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(p: q: s)$ </tex-math></inline-formula> , where the parameters are defined up to a nonzero multiplicative constant and characterize a voltage-current relation of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$pv-qi=s$ </tex-math></inline-formula> . Given a one-port, the open-circuit and short-circuit network determinants, to be denoted as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{e}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{e}$ </tex-math></inline-formula> , are polynomial functions of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> - and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{e}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{e}$ </tex-math></inline-formula> does not vanish then the voltage-current behavior at the port is characterized by a homogeneous triad <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(p_{e}: q_{e}: s_{e})$ </tex-math></inline-formula> . In particular, the assumptions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{e} \neq 0$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{e} \neq 0$ </tex-math></inline-formula> , 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. The excitation parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s_{e}$ </tex-math></inline-formula> 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.