RESISTANCE CALCULATION OF AN INFINITE NETWORK OF RESISTORS- APPLICATION ON GREEN’S FUNCTION

Abstract

The resistance of an infinite network of identical resistors is calculated in two- and three-dimensions, using the Lattice Green’s function (LGF). This work deals with two cases: the perfect lattice and the perturbed lattice (i.e. a bond between two lattice points is removed). It is shown how to derive the basic formula which relates the resistance to the LGF. In calculating the resistance we make use of the values of the LGF at arbitrary sites and we use some recurrence formulae. Comparison of calculated values is carried out with experimental results for finite square and simple cubic lattices. The asymptotic behavior of the resistance in a square and simple cubic (SC) lattices for both the perfect and perturbed cases is studied. The study resulted in finding that for a perfect lattice (i.e. square or SC) the resistance is symmetric along the low-index directions, whereas for the perturbed case the symmetry is broken. We demonstrate that the resistance in a square lattice diverges as the separation between the sites increases, while in the SC lattice it tends to a finite 19 value. Finally, the measured bulk values are in good agreement with those calculated, but as approaching the edge or the surface of the lattice the measured values exceed those calculated.