Macromodel-Based Iterative Solvers for Simulation of High-Speed Links With Nonlinear Terminations

Abstract

Data transmission on high-speed channels may be affected by several undesired effects, including coupling from nearby interconnects, dispersion, losses, signal reflections from terminations and from internal discontinuities, and nonlinear/dynamic effects of drivers and receivers. The latter are often neglected, leading to very fast solvers, whose results may, however, be questionable when driver/receiver nonlinearities are important. This paper presents a framework for the transient analysis of complex high-speed channels with arbitrary nonlinear termination circuits. The approach is based on decoupling channel and terminations through a scattering-based waveform relaxation (WR) formulation. The channels are here cast as delay-rational macromodels, which are solved in discrete time domain through fast delayed recursive convolutions. The terminations can be either arbitrary circuits, solved by SPICE, or nonlinear behavioral macromodels, which are here formulated in discrete-time scattering representations. To overcome the known convergence issues of standard WR methods, we apply here more general iterative solution schemes, such as generalized minimal residual and biconjugate gradient stabilized, integrated into inexact Newton iterations, obtaining a set of numerical schemes with guaranteed convergence. The excellent performance of the proposed approach is illustrated on a large set of benchmarks.

A Waveform Relaxation Solver for Transient Simulation of Large-Scale Nonlinearly Loaded Shielding Structures

“…The presented WR-based solver starts from the results of [13], which provides a macromodeling-based decoupling scheme to solve the electromagnetic problem, and extends the stateof-the-art circuit solver [25] by addressing the problem of a poor matching condition at the interface ports. With respect to [25], a time-partitioning strategy [18] is embedded in the solver formulation to improve the convergence property of the WR-NGMRES scheme and to handle the large number of ports of the presented application. A different approach, still within the adopted decoupling and WR framework, was introduced in [14].…”

“…2) The proposed approach takes advantage of the WR-based iteration of the WR-NGMRES scheme [25], while Wendt et al [14] solve for the basic WR-LP iteration.…”

“…1) a time-partitioning approach is exploited, to improve convergence of the basic WR-LP scheme; 2) instead of a pure WR scheme, we adopt a more general Newton-Krylov solver [25]. As in basic WR iterations, a concurrent estimate of chuncks of time samples of the port signals is computed at each iteration.…”

“…The proposed solver draws the main idea of WR, which is however achieved through an inexact Newton-Krylov iteration. The approach is similar to [25]. This iteration is combined with a second (outer) iteration that processes sequentially different time subintervals, with an appropriate reinitialization of any required initial conditions.…”

Constructive Signal Approximations for Fast Transient Simulation of Coupled Channels

“…Similarly, conversion to single-ended and/or differential scattering representations in terms of incident and reflected transient wave signals can be considered too, leading to a pure scattering formulation as in [22]. In the following, we base our derivations to single-ended voltages and currents for simplicity of illustration, but any other representation that can be linearly related to such variables can be adopted as a starting point as well.…”