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

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1 Author(s)
Elfadel, I.M. ; IBM Corp., Yorktown Heights, NY, USA

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.

Published in:

Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on  (Volume:28 ,  Issue: 8 )