Capacitance matrix revisited

TL;DR

This paper provides a comprehensive review and generalization of the capacitance matrix, exploring its mathematical properties, regularization methods, and applications to electric circuit approximations.

Contribution

It offers a rigorous generalization of the capacitance matrix's properties and discusses alternative regularizations and their implications for circuit analysis.

Findings

Generalized properties and inequalities of the capacitance matrix

Introduced alternative regularization methods

Connected capacitance matrix theory to standard circuit formulas

Abstract

The capacitance matrix relates potentials and charges on a system of conductors. We review and rigorously generalize its properties, block-diagonal structure and inequalities, deduced from the geometry of system of conductors and analytic properties of the permittivity tensor. Furthermore, we discuss alternative choices of regularization of the capacitance matrix, which allow us to find the charge exchanged between the conductors having been brought to an equal potential. Finally, we discuss the tacit approximations used in standard treatments of the electric circuits, demonstrating how the formulae for the capacitance of capacitors connected in parallel and series may be recovered from the capacitance matrix.