Time-Domain Analysis of Retarded Partial Element Equivalent Circuit Models using Numerical Inversion of Laplace Transform


Full-wave time-domain computational electromagnetic (CEM) solvers, which are integral equation (IE)-based, may suffer from what is called 'late-time instability' problems. This unstable behavior occurs for CEM solvers for very fast rise-time input signals. A multitude of techniques has been devised by researchers over the years to solve the problem. In this article, we pursue an approach for the stable solution of full-wave partial element equivalent circuit (PEEC) models for fast-rising input waveforms. In particular, step and impulse response will be considered that are the most challenging from a stability point of view. For the solver part, a conventional full-wave PEEC code is used that requires one to use retarded partial elements. Unfortunately, a PEEC, as well as impedance Z-(method of moment) solvers using suitable numerical time-stepping methods have stability problems, especially for fast rising impulse or step inputs. An important step forward is achieved in this work by providing a larger class of stable solutions well above the stability achieved for time-stepping methods in the last 50 years. The time-domain stability is achieved by replacing the stepping integration methods with a numerical inversion of Laplace transform (NILT) technique. The NILT transform starts out by applying it to a frequency-domain PEEC solution. The surprising result is that the NILT-based method has a variable time-dependent bandwidth that is advantageous for the full-wave IE solution stability. In this article, we give several examples that show that a PEEC-NILT solution provides accurate and stable results for impulse, step- and piece-wise linear input waveforms.


Electrical and Computer Engineering

Research Center/Lab(s)

Electromagnetic Compatibility (EMC) Laboratory


Natural Sciences and Engineering Research Council of Canada, Grant None

Keywords and Phrases

Impulse Response; Partial Element Equivalent Circuit (PEEC) Method; Transient Analysis

International Standard Serial Number (ISSN)

1558-187X; 0018-9375

Document Type

Article - Journal

Document Version


File Type





© 2021 Institute of Electrical and Electronics Engineers (IEEE), All rights reserved.

Publication Date

01 Jun 2021