Convergence of Transverse Waveform Relaxation for the Electrical Analysis of Very Wide Transmission Line Buses

Abstract

In this paper, we study the convergence and approximation error of the transverse waveform relaxation (TWR) method for the analysis of very wide on-chip multiconductor transmission line systems. Significant notational simplicity is achieved in the analysis using a splitting framework for the per-unit-length matrix parameters of the transmission lines. This splitting enables us to show that the state-transition matrix of the coupled lines satisfies a linear Volterra integral equation of the second kind, whose solution is generated by the TWR method as a summable series of iterated kernels with decreasing norms. The upper bounds on these norms are proved to be O(k r/r !), where r is the number of iterations and k is a measure of the electromagnetic couplings between the lines. Very fast convergence is guaranteed in the case of weak coupling (k Lt 1). These favorable convergence properties are illustrated using a test suite of industrial very large scale integration global buses in a modern 65-nm CMOS process, where it is shown that few ( ap 3) Gauss-Jacobi iterations are sufficient for convergence to the exact solution.