On equivalent resistance of electrical circuits

TL;DR

This paper reviews the method of nodal potentials for calculating equivalent resistance in electrical circuits, demonstrating its simplicity over Kirchhoff's rules and deriving a general formula with applications to graph theory and computer science.

Contribution

It introduces a simplified, mathematically elegant approach to compute equivalent resistance and derives a closed-form solution applicable to complex circuits, linking electrical engineering with graph theory.

Findings

Derivation of a closed-form formula for equivalent resistance.

Illustration of the method using the Wheatstone bridge.

Connection between circuit analysis, matrix algebra, and graph theory.

Abstract

While the standard (introductory physics) way of computing the equvalent resistance of non-trivial electrical ciruits is based on Kirchhoff's rules, there is a mathematically and conceptually simpler approach, called the method of nodal potentials, whose basic variables are the values of electric potential at the circuit's nodes. In this paper, we review the method of nodal potentials and illustrate it using the Wheatstone bridge as an example. At the end, we derive - in a closed form - the equivalent resistance of a generic circuit, which we apply to a few sample circuits. The final result unveils a curious interplay between electrical circuits, matrix algebra, and graph theory and its applications to computer science. The paper is written at a level accessible by undergraduate students who are familiar with matrix arithmetic. For the more inquisitive reader, additional proofs and technical details are provided in the appendix.