Effective resistances of two dimensional resistor networks

TL;DR

This paper explores how the effective resistance in two-dimensional resistor networks varies with size and geometry, revealing a dependence on the aspect ratio across different lattice types through numerical analysis.

Contribution

It provides a numerical study of effective resistance in various 2D resistor networks with different geometries, highlighting the role of aspect ratio in resistance behavior.

Findings

Effective resistance depends on the ratio L_x/L_y in all studied networks.

Rectangular networks exhibit simple resistance behavior, while other geometries are more complex.

Numerical results are validated experimentally for small network sizes.

Abstract

We investigate the behavior of two dimensional resistor networks, with finite sizes and different kinds (rectangular, hexagonal, and triangular) of lattice geometry. We construct the network by having a network-element repeat itself $L_x$ times in $x$-direction and $L_y$ times in the $y$-direction. We study the relationship between the effective resistance ($R_\mathrm{eff}$) of the network on dimensions $L_x$ and $L_y$. The behavior is simple and intuitive for a network with rectangular geometry, however, it becomes non-trivial for other geometries which are solved numerically. We find that $R_\mathrm{eff}$ depends on the ratio $L_x/L_y$ in all the three studied networks. We also check the consistency of our numerical results experimentally for small network sizes.