Waveform relaxation and transverse partitioning algorithms for simulation of massively coupled interconnects

Abstract

With the continually increasing operating frequencies, signal integrity and interconnect analysis in high-speed designs are becoming increasingly important. Interconnect effects such as ringing, signal delay, distortion and crosstalk can severely degrade signal integrity. As a result, these effects have become the dominant factors which limit the overall performance of VLSI systems. If not considered during the design stage, interconnect effects can render a circuit inoperable or cause it to fail in meeting the required specifications. In addition, at relatively higher frequencies, conventional lumped models are no longer adequate in describing the interconnect performance and distributed models become necessary. The major difficulty usually encountered while linking distributed transmission lines and nonlinear simulators is the problem of mixed frequency/time. This is because distributed elements are best characterized in the frequency-domain, whereas nonlinear components such as drivers and receivers are represented generally in the time-domain. To address the above difficulties, several algorithms were proposed in the literature for macromodeling and transient analysis of high-speed circuits and interconnects. The common goal of these techniques is to transform the Telegrapher's equations describing the transmission lines, into a set of ordinary differential equations with appropriate time-delayed controlled sources that can be integrated with circuit simulators. Most of these methods use some form of decoupling algorithm to convert the set of coupled partial differential equations describing the lines into a set of decoupled single modal equations. Subsequently, the line voltages and currents are obtained as a linear combination of modal variables. As a result, the interconnect stamp used by a time-domain circuit simulator is in the form of coupled ordinary differential equations. This coupling is one of the major reasons for the excessive computational cost of simulating large multiconductor structures. It has been shown that the average cost of simulating an N-coupled lines circuit is proportional to Nβ, where 3 < β < 4, which results in a prohibitively time-consuming simulation task compared to the simple case of simulating a single line. In this thesis, new algorithms which address the computational complexity associated with the time-domain simulation of massively coupled interconnect circuits are developed. The new methods reduce the coupled simulation problem into a series of simulation steps, where each step is of complexity equivalent to that of simulating a single line. Advantages are that the computational cost of the new algorithms grows only linearly with the number of lines. In addition, the methods are highly parallelizable, thus array processors can be used to provide further significant reductions in the computational cost. Various numerical examples are provided which validate the accuracy and efficiency of all the proposed algorithms.