IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 56, NO. 6, DECEMBER 2014 1313 Efficient Modeling of Interactions Between Radiating Devices With Arbitrary Relative Positions and Orientations Gert-Jan Stockman, Hendrik Rogier, Senior Member, IEEE, and Dries Vande Ginste, Senior Member, IEEE Abstract—A novel method to efficiently compute the interac- tion between two devices is proposed with the aim of accurately reproducing radiated immunity and emission tests in simulations. The technique allows an arbitrary relative position and orientation between the two devices. It relies on a single simulation (or mea- surement) of the radiation pattern of each device. To take rotation of the devices into account, a spherical harmonics decomposition is applied together with Wigner-D rotation matrices. The resulting procedure is practical, has a low computational cost, and shows good agreement with measurements and full-wave simulations. Index Terms—Electromagnetic interference (EMI), emission, radiated immunity, spherical harmonics, wigner D-matrix. I. INTRODUCTION A NALYSIS of the electromagnetic compatibility (EMC) behavior of novel devices and systems is of critical im- portance, especially given the presence of the vast amount of electronic products with ever increasing operating frequencies in our society. When studying EMI, both emission and suscepti- bility aspects require careful examination. Compliance tests for the assessment of radiated emission or immunity are often car- ried out in an anechoic chamber. The device under test (DUT) is then rotated and a new measurement is performed for ev- ery angular position. However, also during the design phase (or precompliance phase), it is beneficial to take radiated emission and immunity into account. In the ideal case, this is done via simulations, although often not a trivial task. A single simula- tion requires large computational resources in order to achieve accurate results. Moreover, every angular position calls for a separate simulation. To relax these high computational requirements, many meth- ods have been developed to mimic (aspects of) the large elec- tromagnetic problem. In [1], emission is studied and devices are modeled as equivalent sources using the source reconstruction method. A printed circuit board (PCB), for example, is replaced by an equivalent source containing both the electric and mag- netic currents. Another emission model is developed in [2] for integrated circuits (ICs), where the electromagnetic field is ex- panded in multipoles, reducing the number of required problem Manuscript received February 7, 2014; revised June 18, 2014; accepted July 24, 2014. Date of publication September 4, 2014; date of current version De- cember 11, 2014. The authors are with the IBCN/Electromagnetics Group, Department of Infor- mation Technology, Ghent University/iMinds, B-9000 Gent, Belgium (e-mail: gertjan.stockman@intec.ugent.be; hendrik.rogier@intec.ugent.be; dries.vande. ginste@intec.ugent.be). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2014.2345127 parameters. When looking at susceptibility, various hybrid tech- niques have been proposed. In [3], model-reduction techniques are combined with full-wave solvers. In [4], a hybrid method, based on the finite-difference time-domain (FDTD) algorithm and a field coupling model of transmission lines, is used for the analysis of coupled interconnects on dispersive dielectrics. Fur- thermore, [5] and [6] discuss susceptibility of transmission line networks with linear/nonlinear load terminations, subject to ar- bitrary excitations, and PCBs subject to near-zone illuminations via an extended S-parameter model. Another way to reduce computational resources is by the so- called domain decomposition methods (DDMs). Nonconformal finite element (FE)-based DDMs to simulate high-speed inter- connects and multiscale electromagnetic scattering problems have been reported in [7] and [8], respectively. Furthermore, the interaction between an antenna and a perfect electrically con- ducting (PEC) scatterer has been described via a hybrid physi- cal optics (PO)/generalized-scattering-matrix approach [9]. Re- cently, in [10], a DDM-like approach was suggested to model the interaction between different segments of complex radio fre- quency (RF) structures by using eigenmodal expansions. In [11], multilayer high-speed interconnects are modeled using modal ports. The latter methods do not include radiation. In this paper, we will not consider emission or susceptibility separately, but the interaction between radiating devices. The novelty of our method lies in the fact that it is based purely on a single simulation (or measurement) of the radiation patterns of each device. Rotation of the devices is performed via spherical harmonics expansions and Wigner-D matrices. The advantage is that the devices may be moved and rotated over various an- gles without requiring new simulations (or measurements). As such, the technique is very tractable and opens the way to the efficient reproduction of radiated emission and immunity tests in simulations. To validate the novel method, simulation and measurement results are presented, demonstrating the accuracy and efficiency of our approach. This paper is organized as follows. Section II explains the for- malism. The electromagnetic interaction between two devices is described, as well as how these devices can be rotated over ar- bitrary angles. Section III contains numerical and experimental results to thoroughly validate and illustrate the novel method. The conclusions and an outline for future research are presented in Section IV. In the sequel, all sources and fields are assumed to be time harmonic with angular frequency ω and time depen- dencies e jωt are suppressed. Unit vectors are denoted with a “hat,” e.g., ˆ v. 0018-9375 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.