Effective Resistance of Finite Two-Dimensional Grids Based on Infinity Mirror Technique
Rassul Bairamkulov, Eby G. Friedman
Abstract
Conventional numerical circuit analysis tools typically scale superlinearly with the number of nodes. With the rapid increase in nodes in modern VLSI systems, alternative methods are required. The effective resistance is an important characteristic of electrical systems, which is used to simplify the circuit analysis process. An infinite resistive rectangular mesh is commonly assumed in the analysis of grid structures to determine the effective resistance of a grid. The assumption of infinity provides a useful approximation when a large grid is analyzed far from the boundaries. If however the grid is analyzed in close proximity to a boundary or if the grid dimensions are small, the assumption of infinity may lead to significant error. To address this issue, the infinity mirror technique is proposed to determine the effective resistance of a two-dimensional structure, where one or both dimensions are finite. The method exhibits good agreement with nodal analysis, achieving an error below 1% in case studies. The proposed expressions enhance the speed of static grid analysis by several orders of magnitude by replacing computationally expensive nodal analysis with an equivalent reduced grid analysis. A 1,400 fold speedup is achieved in the analysis of 100 nodes within a 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> grid.