Circuit Theory in Projective Space and Homogeneous Circuit Models

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 <italic>a priori</italic> 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, in addition, makes it possible to get rid of voltage- or current-control assumptions on the impedances. This paves the way for a completely general <inline-formula> <tex-math notation= LaTeX >$m$ </tex-math></inline-formula>-dimensional reduction of any circuit defined by <inline-formula> <tex-math notation= LaTeX >$m$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation= LaTeX >$\\left ({\\!\\begin{smallmatrix} AP \\\\ BQ \\end{smallmatrix}\\!}\\right) u = \\left ({\\!\\begin{smallmatrix} AQ \\\\ -BP \\end{smallmatrix}\\!}\\right)\\bar {s}$ </tex-math></inline-formula>. Here, <inline-formula> <tex-math notation= LaTeX >$A$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$B$ </tex-math></inline-formula> capture the graph topology, whereas <inline-formula> <tex-math notation= LaTeX >$P$ </tex-math></inline-formula>, <inline-formula> <tex-math notation= LaTeX >$Q$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation= LaTeX >$\\bar {s}$ </tex-math></inline-formula> comprise homogeneous descriptions of all the circuit elements; the unknown <inline-formula> <tex-math notation= LaTeX >$u$ </tex-math></inline-formula> is an <inline-formula> <tex-math notation= LaTeX >$m$ </tex-math></inline-formula>-dimensional vector of (say) “seed” variables from which currents and voltages are obtained as <inline-formula> <tex-math notation= LaTeX >$i=Pu -Q \\bar {s}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$v=Qu + P \\bar {s}$ </tex-math></inline-formula>. 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.