Effective Resistance of a Two Layer Mesh

The effective resistance of a resistive mesh is a commonly used analogy for various scientific and engineering problems such as voltage drop estimation, distributed control including time synchronization and sensor localization, determining the chemical distance among multiple bonds, and finding the distance between two vertices in a graph. Resistive networks are a commonly used structure in electronics to model different elements of an integrated circuit, such as a physical substrate, an integrated circuit layout, and a power distribution network. On-chip power and ground networks are composed of orthogonal metal lines from different metal layers, and a resistive mesh is typically used to model these networks. A two layer mesh is therefore commonly used to analyze IR voltage drops and decoupling capacitor placement. A closed-form expression is described here for the effective resistance between the intersections of a two layer resistive mesh where the horizontal and vertical unit resistances are different. The physical distance between the nodes of interest and the ratio between the horizontal and vertical resistances are included in the expression. The maximum error of the closed-form expression, as compared with the exact solution, is less than 5% for a wide range of . The error further decreases with greater separation between the nodes of interest.