# The Waveform Relaxation Method for Time Domain Analysis of Large Scale Integrated Circuits: Theory and Applications

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/1982/ERL-m-82-40.pdf

The Waveform Relaxation (WR) method is a new decomposition method for solving a system of mixed implicit algebraic-differential equations over a given time interval. This method essentially uses an iterative relaxation scheme such as the Gauss-Seidel relaxation or the Gauss-Jacobi relaxation in which the elements of the relaxation are waveforms of unknown variables. The decomposed system obtained through the relaxation scheme is also a system of mixed implicit algebraic-differential equations but is much easier to solve than the original system.

The application of this method in the area of time domain simulation of integrated circuits is emphasized. Although the WR method has a theoretical basis, it can be given a simple physical interpretation when applied to the analysis of integrated circuits. In particular, the convergence conditions of the method can be given either in terms of the numerical properties of the circuit equations or in terms of the properties of the circuit components. This method is shown to be a viable alternative to the conventional techniques for simulating large scale integrated circuits since sufficient conditions for its convergence are quite mild and are always satisfied by a large class of practical circuits.

The performance of the WR method when applied to a particular class of circuits, i.e., MOS digital integrated circuits, are studied and evaluated via a prototype simulator called RELAX. The repetivity and directionality of digital subcircuits as well as the digital nature of the signals are exploited in the simulator to increase the speed of computation and to utilize the storage efficiently. Preliminary comparisons between RELAX and the standard circuit simulator SPICE have shown that RELAX is fast and reliable for simulating MOS digital integrated circuits.

BibTeX citation:

```@phdthesis{Lelarasmee:M82/40,
Author = {Lelarasmee, Ekachai},
Title = {The Waveform Relaxation Method for Time Domain Analysis of Large Scale Integrated Circuits: Theory and Applications},
School = {EECS Department, University of California, Berkeley},
Year = {1982},
Month = {May},
URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/1982/9614.html},
Number = {UCB/ERL M82/40},
Abstract = {The Waveform Relaxation (WR) method is a new decomposition method
for solving a system of mixed implicit algebraic-differential
equations over a given time interval.  This method essentially uses
an iterative relaxation scheme such as the Gauss-Seidel relaxation or
the Gauss-Jacobi relaxation in which the elements of the relaxation
are waveforms of unknown variables.  The decomposed system obtained
through the relaxation scheme is also a system of mixed implicit
algebraic-differential equations but is much easier to solve than
the original system.

The application of this method in the area of time domain
simulation of integrated circuits is emphasized.  Although the
WR method has a theoretical basis, it can be given a
simple physical interpretation when applied to the analysis
of integrated circuits.  In particular, the convergence conditions
of the method can be given either in terms of the numerical
properties of the circuit equations or in terms of the properties
of the circuit components.  This method is shown to be a viable
alternative to the conventional techniques for simulating large
scale integrated circuits since sufficient conditions for its
convergence are quite mild and are always satisfied by a large
class of practical circuits.

The performance of the WR method when applied to a particular class
of circuits, i.e., MOS digital integrated circuits, are studied and
evaluated via a prototype simulator called RELAX. The repetivity
and directionality of digital subcircuits as well as the digital
nature of the signals are exploited in the simulator to increase
the speed of computation and to utilize the storage efficiently.
Preliminary comparisons between RELAX and the standard circuit
simulator SPICE have shown that RELAX is fast and reliable for
simulating MOS digital integrated circuits.}
}
```

EndNote citation:

```%0 Thesis
%A Lelarasmee, Ekachai
%T The Waveform Relaxation Method for Time Domain Analysis of Large Scale Integrated Circuits: Theory and Applications
%I EECS Department, University of California, Berkeley
%D 1982
%@ UCB/ERL M82/40
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/1982/9614.html
%F Lelarasmee:M82/40
```