Effective Resistance of Two-Dimensional Truncated Infinite Mesh Structures

Abstract

Determining the effective resistance of a mesh is a powerful tool for simplifying the analysis of complex electrical structures, ranging from transistors to power delivery systems. A common structure in science and engineering is a two-dimensional resistive grid. Applications of this structure include IR drop analysis and decoupling capacitor allocation in on-chip power and ground networks in VLSI systems, and the analysis of electrical and thermal conductive media, such as a semiconductor substrate. In the case where the effective resistance is evaluated far from the grid edges, an infinite resistive lattice can be used to simplify the analysis of a grid. If however the target nodes are located close to the grid edges, the assumption of infinite dimensions becomes invalid, producing inaccurate results. To bridge this gap, a resistive mesh truncated along a single or two dimensions is discussed here. Integral expressions for the effective resistance within a truncated infinite mesh structure are provided, as well as a closed-form approximation, which exhibits good agreement with the exact integral equation, exhibiting an average error of 0.27% and a maximum error of 4.77%. These expressions significantly improve the accuracy of the effective resistance estimation near the edges and corner of a resistive mesh, providing a tenfold reduction in error. In case studies, the expressions provide four to six orders of magnitude speedup in IR analysis of a $10^{4} \times 10^{4}$ grid, while providing accuracy comparable to nodal analysis.