Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries

We develop a general recursion-transform (R-T) method for a two-dimensional resistor network with a zero resistor boundary. As applications of the R-T method, we consider a significant example to illuminate the usefulness for calculating resistance of a rectangular $m\times n$ resistor network with a null resistor and three arbitrary boundaries, a problem never solved before, since Green's function techniques and Laplacian matrix approaches are invalid in this case. Looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of an arbitrary boundary since the boundary is like a wall or trap which affects the behavior of finite network. In this paper we obtain several general formulas of resistance between any two nodes in a nonregular $m\times n$ resistor network in both finite and infinite cases. In particular, 12 special cases are given by reducing one of the general formulas to understand its applications and meanings, and an integral identity is found when we compare the equivalent resistance of two different structures of the same problem in a resistor network.

Figure 1

An $m\times n$ resistor network with a null resistor and three arbitrary boundaries. Bonds in the horizontal and vertical directions are resistors $r$ and $r_0$ except for four boundaries.

Figure 2

Segment of resistor with current directions and parameters.

Figure 3

The $1\times n$ resistor network with two arbitrary boundaries.