152 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 1, FEBRUARY 2002 A Two-Level Plane Wave Time-Domain Algorithm for Fast Analysis of EMC/EMI Problems Kemal Aygün, Student Member, IEEE, Balasubramaniam Shanker, Senior Member, IEEE, A. Arif Ergin, and Eric Michielssen, Fellow, IEEE Abstract—In this paper, a fast time-domain integral equation (IE)-based scheme for analyzing transient electromagnetic com- patibility and interference (EMC/EMI) problems involving printed circuit boards and cables/connectors that reside in shielding enclo- sures, is presented. The proposed algorithm hybridizes a classical marching-on-in-time (MOT) solver for analyzing radiation from perfect electrically conducting (PEC) surface/wire geometries with the recently introduced two-level plane wave time-domain (PWTD) algorithm. The accuracy and efficacy of the resulting MOT-PWTD algorithm are validated via analysis of the radiation character- istics of a number of structures including a loaded motherboard that resides in a chassis. For a problem with spatial and temporal unknowns, the computational complexity of the two-level MOT-PWTD algorithm scales as as opposed to for a classical MOT scheme. Index Terms—Computational complexity, electromagnetic compatibility, electromagnetic interference, integral equations, PWTD. I. INTRODUCTION C URRENT electronic systems pose significant challenges for electromagnetic modeling. Increasing clock and edge rates require an accurate system characterization over ever-widening frequency bands. Simultaneously, advances in system integration call for precise representation of increas- ingly fine geometric features. All of this translates into a need for a simulation capability that permits the analysis of systems modeled in terms of large numbers of unknowns. Full-wave transient solvers for analyzing electromagnetic compatibility and interference (EMC/EMI) problems are either differential equation (DE)-based or integral equation (IE)-based. DE-based methods [e.g., the finite difference time-domain (FDTD) technique, the transmission line matrix method, and the finite element method (FEM)] are highly suit- able for analyzing problems involving inhomogeneous media. Manuscript received March 28, 2000; revised September 8, 2001. This work was supported in part by DARPA under Grant F49620-01-1-0228, in part by the U.S. Department of Defense under MURI under Grant F49620-01-1-0436, and in part by the Computational Science and Engineering Graduate Option Program at the University of Illinois at Urbana-Champaign. K. Aygün and E. Michielssen are with the Center for Computational Elec- tromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: aygun@ uiuc.edu; emichiel@uiuc.edu). B. Shanker is with the Department of Electrical and Computer Engineering, Materials Assessment Research Group, Iowa State University of Science and Technology, Ames, IA 50010 USA (e-mail: shanker@iastate.edu). A. A. Ergin is with the Department of Electronics Engineering, Gebze Institute of Technology, 41400 Gebze, Kocaeli, Turkey (e-mail: aergin@ penta.gyte.edu.tr). Publisher Item Identifier S 0018-9375(02)01441-2. However, they require a volumetric discretization of the com- putational domain (for both homogeneous and inhomogeneous regions) and approximate absorbing boundary conditions to truncate the latter. When analyzing homogenous media/surface structures, IE-based methods are often preferred over their DE counterparts since they only require the discretization of body surfaces, which renders them very useful for modeling complex geometries. Furthermore, solutions obtained using IE-based solvers implicitly satisfy the radiation condition and are (vir- tually) devoid of grid dispersion errors. A very good review and comparison of the aforementioned methods is given in [1] along with a large number of references with regard to their specific applications in the field of EMC. Additionally, a more recent description of the state-of-the-art in, and application to EMC/EMI analysis of, IE-based methods, the FDTD method, and the FEM, can be found in [2]–[4]. IE-based solvers operate either in the frequency or in the time domain. Frequency domain, method of moments (MOM)-based IE solvers have enjoyed widespread acceptance throughout the computational electromagnetics community. In comparison, time-domain IE solvers—most of them based on marching-on-in-time (MOT) principles—have not been widely used. However, MOT methods such as the time-domain partial equivalent element circuit method [5] are highly suited for analyzing EMC/EMI problems, because they permit the characterization of nonlinear, time-varying, and broad-band phenomena. Historically, two stumbling blocks have pre- vented the widespread use of these transient IE solvers. First, time-domain IE solvers have long been plagued by late time in- stabilities. However, within the last decade, noticeable progress in procedures for stabilizing MOT solvers has been reported [6]–[9]; as a result, MOT solvers now can be considered “stable for all practical purposes.” The second stumbling block is the high computational complexity of the MOT solvers. For a problem with spatial and temporal unknowns, the computational complexity of a classical MOT algorithm scales as . This complexity renders impractical the analysis of realistic problems with fine features and high clock rates. To lower this high computational cost, the plane wave time-domain (PWTD) algorithm was introduced [10], [11]. This algorithm constitutes the extension to the time domain (wave equation) of the frequency domain (Helmholtz equation) fast multipole method. A two-level implementation of the PWTD algorithm reduces the complexity of a classical MOT solver to . Previously, the two-level PWTD method has successfully been used for large-scale scattering analysis [12]. In this paper, this two-level PWTD algorithm is suitably 0018–9375/02$17.00 © 2002 IEEE