Circuit theory in projective space and homogeneous circuit models

TL;DR

This paper introduces a projective geometry-based framework for linear circuit analysis that generalizes classical methods by avoiding predefined electrical assignments and allowing for a fully homogeneous, dimension-reducible model of circuits.

Contribution

It develops a novel, fully homogeneous circuit modeling approach using projective geometry, enabling dimension reduction and general device descriptions without classical restrictions.

Findings

Provides a general $m$-dimensional reduction of circuits.

Characterizes non-degenerate configurations via multihomogeneous Kirchhoff polynomial.

Includes methods to incorporate controlled sources and coupled devices.

Abstract

This paper presents a general framework for linear circuit analysis based on elementary aspects of projective geometry. We use a flexible approach in which no a priori assignment of an electrical nature to the circuit branches is necessary. Such an assignment is eventually done just by setting certain model parameters, in a way which avoids the need for a distinction between voltage and current sources and, additionally, makes it possible to get rid of voltage- or current-control assumptions on the impedances. This paves the way for a completely general $m$-dimensional reduction of any circuit defined by $m$ two-terminal, uncoupled linear elements, contrary to most classical methods which at one step or another impose certain restrictions on the allowed devices. The reduction has the form $$\begin{pmatrix} AP \\ BQ \end{pmatrix} u = \begin{pmatrix} AQ \\ -BP \end{pmatrix} \bar{s}.$$ Here, $A$ and $B$ capture the graph topology, whereas $P$, $Q$, $\bar{s}$ comprise homogeneous descriptions of all the circuit elements; the unknown $u$ is an $m$-dimensional vector of (say) ``seed'' variables from which currents and voltages are obtained as $i=Pu -Q\bar{s}$, $v=Qu + P\bar{s}$. Computational implementations are straightforward. These models allow for a general characterization of non-degenerate configurations in terms of the multihomogeneous Kirchhoff polynomial, and in this direction we present some results of independent interest involving the matrix-tree theorem. Our approach can be easily combined with classical methods by using homogeneous descriptions only for certain branches, yielding partially homogeneous models. We also indicate how to accommodate controlled sources and coupled devices in the homogeneous framework. Several examples illustrate the results.