Browse Prior Art Database

# Time Integration Algorithms for Circuit Analysis

IP.com Disclosure Number: IPCOM000050038D
Original Publication Date: 1982-Aug-01
Included in the Prior Art Database: 2005-Feb-09
Document File: 4 page(s) / 44K

IBM

## Related People

Liniger, W: AUTHOR [+2]

## Abstract

Introduction. The purpose of this article is to give a new formulation for the integration of circuit equations in the time domain. It offers advantages over existing, widely used methods (like the backward differentiation methods) in the following areas:

This text was extracted from a PDF file.
At least one non-text object (such as an image or picture) has been suppressed.
This is the abbreviated version, containing approximately 53% of the total text.

Page 1 of 4

Time Integration Algorithms for Circuit Analysis

1. Introduction.

The purpose of this article is to give a new formulation for the integration of circuit equations in the time domain. It offers advantages over existing, widely used methods (like the backward differentiation methods) in the following areas: a) Larger, strongly variable time steps can be taken

without loss of stability.

b) In computing oscillatory responses, the "numerical

damping" caused by the integration method is greatly

reduced.

c) A family of second-order methods is offered in the

algorithm (rather than a single one) so that we can

conveniently adapt the integration technique to the

problem at hand.

d) The methods can be implemented in the TAA (Tableau

Approach) as well as in the MNA (Modified Nodal Approach)

circuit formulation.

The formulas used here and the two basic modes of implementation, as well as the underlying theory, are given in [1-5. 2. Integration Methods.

The methods mentioned above are implementations of the two-step "A- contractive Arc" (ACA) formulas (see original) whose coefficients are specified, for the general variable-step case, by (see original). 3. Local Error Criterion.

A local error must be computed for control of accuracy and of the variable time step. In addition to A and E the following auxiliary quantities are needed: B:=r(n)r(n-1), C:=1+(n-1), and D:=1+ r(n-1)+r(n)r(n-1) equals B+C. The error is (see original). 4. Extrapolation.

For solving the nonlinear difference equations we require a predicted solution value based on extrapolating the backward data to the now-time. The predicted solution, based on Lagrange interpolation at t(n-1), t(n-2)and t(n-3), is (see original). 5. Time Integration for Linear Circuits.

For linear time-variant circuits one can implement the methods listed under section 2 above in the familiar multistep (MS) form (1) which, for this type of circuit, is as stable as the one-leg implementation discussed hereafter. Either the TAA or the MNA formulation can be used.

We are interested in the changes which the ACA methods introduce, as compared to the widely used two-step backward differentiation method (BD2). If for simplicity in writing we drop the subscript n of the coefficients and solve (1) w.r. to x(n), we obtain (see original) which is used to replace the derivatives (5). We find, for example, that the stamp for a capacitor between nodes A and B is (see original) where RHS=Right Hand Side. Other elements are treated similarly. We note that, except for the changes in the coefficients of stored elements, the

1

Page 2 of 4

MS network formulations are similar to those of the BD2 method (for which b(0) =1,b(1)=b(2)=0). 6. Time Solution for Nonlinear Circuits.

For circuits containing nonlinear or time-variant elements, the one-leg (OL) implementation of the ACA methods should be used in order to assure the strong stability properties mentioned under section 1 above. The OL implementation also requires less sto...