Recursion-transform method on computing the complex resistor network with three arbitrary boundaries

TL;DR

This paper advances the recursion-transform method to derive exact resistances in complex resistor networks with arbitrary boundaries, providing new formulas and solutions previously unattainable with traditional techniques.

Contribution

The paper develops a complete recursion-transform theory capable of calculating exact resistances in resistor networks with arbitrary boundaries, including cases where previous methods fail.

Findings

Seven general formulas for resistance between any two nodes in nearly m×n networks.

Eight specific case studies illustrating the application of the formulas.

Successful derivation of resistance solutions where Greens function and Laplacian methods are invalid.

Abstract

We perfect the recursion-transform method to be a complete theory, which can derive the general exact resistance between any two nodes in a resistor network with several arbitrary boundaries. As application of the method, we give a profound example to illuminate the usefulness on calculating resistance of a nearly $m\times n$ resistor network with a null resistor and three arbitrary boundaries, which has never been solved before since the Greens function technique and the Laplacian matrix approach are invalid in this case. Looking for the exact solutions of resistance is important but difficult in the case of the arbitrary boundary since the boundary is a wall or trap which affects the behavior of finite network. For the first time, seven general formulae of resistance between any two nodes in a nearly $m\times n$ resistor network in both finite and infinite cases are given by our theory. In particular, we give eight special cases by reducing one of general formulae to understand its application and meaning.