Resistances of Infinite Electrical Networks

摘要

It is interesting to find the equivalent resistance between two nodes of an infinite electrical network. In this paper, we consider an infinite electrical network that can be described as a series of squares whose edges are resistors with resistance $R$ and whose corresponding vertices are joined successively by resistors with resistance $R$ as well. Our major work is to find the equivalent resistance between the diagonal vertices of the base square of this infinite network. First, we apply the techniques of balanced bridges and symmetry of voltages to convert each iteration of the network to a parallel circuit that includes the previous iteration. Then, we evaluate the equivalent resistance of each iteration of the network and derive a recursive sequence of equivalent resistances with iterations. After that, we prove that the recursive sequence is convergent using the contraction theorem in real analysis. Finally, we claim that the limit of the recursive sequence is the equivalent resistance of the infinite network.