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Convergence of Transverse Waveform Relaxation for the Electrical Analysis of Very Wide Transmission Line Buses | IEEE Journals & Magazine | IEEE Xplore

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 t...Show More

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 $(k \ll 1)$. 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 $(\approx 3)$ Gauss–Jacobi iterations are sufficient for convergence to the exact solution.
Page(s): 1150 - 1161
Date of Publication: 17 July 2009

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