Asymptotic Waveform Evaluation

The intense drive for signal integrity has been at the forefront ofrapid and new developments in CAD algorithms. Thousands ofengineers, intent on achieving the best design possible, use SPICE on a daily basis for analog simulation and general circuit analysis. But the strained demand for high data speeds, coupled with miniaturizationon an unprecedented scale, has highlighted the previously negligible effects of interconnects; effects which are not always handled appro priately by the present levels of SPICE. Signals at these higher speeds may be degraded by long interconnect lengths compared to the increasingly shorter sig nal rise times. Interconnect structures can be diverse (pins, connectors, leads, microstrips, striplines, etc. ) and present at any of the hierarchical packaging levels: integrated circuits, printed circuit boards, multi-chip modules or sys tem backplanes. Analysis of these effects in any CAD package has become a necessity. Asymptotic waveform evaluation (AWE) and other moment matching tech niques have recently proven useful in the analysis of interconnect structures and various networks containing large linear structures with nonlinear termi nations. Previously, all that was available to the designer was a full SPICE simulation or a quick but uncertain timing estimation. Moment matching, used in linear systems analysis as a method of model reduction, describes a method to extract a small set of dominant poles from a large network. The information is obtained from the Taylor series coefficients (moments) of that system.

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The intense drive for signal integrity has been at the forefront of rapid and new developments in CAD algorithms. With increasing demands for high signal speeds coupled with a decrease in feature size, interconnect effects such as signal delay, distortion and crosstalk become the dominant factor limiting overall performance of VLSI systems. Although SPICE is used on a daily basis by many engineers for analog simulation and general circuit analysis, current versions of SPICE do not handle adequately the new emerging challenges of interconnect effects. Moment-matching techniques, such as asymptotic waveform evaluation, have recently proven useful in the analysis of large interconnect structures containing elements such as lossy coupled transmission lines with linear or nonlinear terminations. At a CPU cost of a little more than one DC analysis, these techniques are 2--3 orders of magnitude faster than full simulation techniques such as FFT. Asymptotic Waveform Evaluation presents an overview of the diverse algorithms and applications of moment matching techniques. The material is presented systematically and is supported by many examples. Issues such as sensitivity analysis and three-dimensional analysis are also covered. Asymptotic Waveform Evaluation will be of interest to engineers, students and researchers involved in the development and study of circuit simulation as well as interconnect analysis. It will also interest design engineers who are involved in dealing with high-speed issues, and graduate students who are active in the development of CAD tools for electronic systems.

Contenu
1 Introduction.- 1.1 Motivation.- 1.2 Overview.- 1.3 Notation.- 2 Asymptotic Waveform Evaluation.- 2.1 State space formulation.- 2.2 Generating an approximate response.- 2.3 MNA formulation.- 2.4 Examples.- 3 Transmission Lines.- 3.1 Linear subnetwork formulation.- 3.2 Moments of distributed networks.- 3.3 Uniform lossy coupled transmission lines.- 3.4 Nonuniform transmission lines.- 3.5 Miscellaneous combinations.- 4 Padé Approximations.- 4.1 The rational approximation.- 4.2 Accuracy.- 4.3 Laurent expansion about o0.- 5 Accuracy Improvements.- 5.1 Higher precision programming.- 5.2 Moment scaling.- 5.3 Padæ table analysis.- 5.4 Optimal pole selection.- 5.5 Partitioning.- 5.6 Method of characteristics.- 5.7 Frequency shifting.- 6 Complex Frequency Hopping.- 6.1 Complex expansions.- 6.2 Pole convergence.- 6.3 Pole selection.- 6.4 Search strategy.- 6.5 Extra CPU requirements.- 7 Nonlinear Analysis.- 7.1 Notation.- 7.2 Differential equations (SPICE) approach.- 7.3 Recursive convolution approach.- 8 Sensitivity Analysis.- 8.1 Pole and zero sensitivity.- 8.2 Coefficient sensitivity.- 8.3 Moment sensitivity: lumped elements.- 8.4 Moment sensitivity: distributed elements.- 8.5 Sensitivity of multi-conductor transmission line stamp.- 9 Other Applications.- 9.1 PEEC simulation.- 9.2 3-D RC mesh analysis.- 9.3 Symbolic analysis.- A Appendix.- A.1 Transmission line equations.