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

In this paper, we study the convergence and the 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 where is the number of iterations and is a measure of the electromagnetic couplings between the lines. Very fast convergence is guaranteed in the case of weak coupling . These favorable convergence properties are illustrated on a fully embedded implementation of TWR, called TWRTL, that uses the application programming interface of an industrial circuit analysis program (IBM’s PowerSpice). TWRTL is shown to converge with few Gauss-Jacobi iterations to the exact solution on a full test suite of industrial VLSI buses in a modern CMOS process.