IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 823 Surface Impedance Boundary Conditions for the Finite Integration Technique Sergey Yuferev , Luca Di Rienzo , and Nathan Ida , Fellow, IEEE Nokia Corporation, Tampere FIN-34101, Finland Dipartimento di Elettrotecnica, Politecnico di Milano, 32-20133 Milano, Italy Department of Electrical Engineering, The University of Akron, Akron, OH 44325-3904 USA Approximate time-domain relations between the electric field integrated along the edge and the magnetic flux density integrated over the facet of the computational cell at the dielectric/conductor interface are derived and implemented into the finite integration tech- nique to accurately eliminate the conducting region from the computational mesh. Both Cartesian and tetrahedral grids are considered. A numerical example is included to illustrate the method. Index Terms—Differential forms, finite integration technique (FIT), surface impedance boundary condition. I. INTRODUCTION T HE surface impedance concept has proven an efficient tool in computational electromagnetics. It provides approxi- mate relations between the parameters of the electromagnetic field at the surface of the conductor. Thus, the conducting region does not need to be included in the mesh and can be “replaced” by surface impedance boundary conditions (SIBCs) in the nu- merical procedure. Originally SIBCs were developed in terms of the electric and magnetic field intensities, so they have been naturally implemented and widely used with the method of mo- ments, the finite-difference time-domain (FDTD) method, and the node-based finite-element method. In the past ten years, alternative formulations employing cir- culations and fluxes as state variables have gained acceptance. In particular, the “sister” method to the FDTD method is the finite integration technique (FIT) [1], and in both methods stag- gered dual grids are used for approximation of the electric- and magnetic-related parameters. Both the FDTD method and FIT are currently widely used to model high-frequency electromag- netic problems. FDTD-SIBC formulations have also become very popular [2]–[5], but the coupling of SIBCs and FIT does not seem to have been done. One possible reason is that the FIT requires SIBCs being represented in terms of the electric field integrated along the edge of the computational cell and the magnetic flux density integrated over the facet (differential 1- and 2-forms, respectively [6]). Thus, the purpose of this paper is derivation of time-domain SIBCs in terms of state variables used in FIT for Cartesian and tetrahedral grids. II. TIME-DOMAIN SURFACE IMPEDANCE CONCEPT Consider a homogeneous body of finite conductivity sur- rounded by a nonconductive medium and illuminated by a pulsed electromagnetic field. Let the time variation of the inci- dent field be such that the electromagnetic penetration depth Digital Object Identifier 10.1109/TMAG.2006.871987 into the body remains small compared with the characteristic dimension of the surface of the body (1) where is the incident pulse duration, and and are con- ductivity and permeability of the body, respectively. It means that the conducting region is so large that the wave attenuates completely inside the region. Then, the electromagnetic field distribution in the conductor’s skin layer can be described as a damped plane wave propagating in the bulk of the conductor normal to its surface. In other words, the behavior of the electro- magnetic field in the conducting region may be assumed to be known a priori. The electromagnetic field is continuous across the real conductor’s surface so the intrinsic impedance of the wave remains the same at the interface. Therefore, the relations between tangential ( - and -) components of the electric field and magnetic flux density or normal ( -) and tangential ( - and -) components of the magnetic flux density at any point of the conductor/dielectric interface can be written in the form [7] (2a) (2b) (3) where time-domain functions and are defined as (4) (5) and where is the -order modified Bessel function. The conditions in (2) and (3) are of the Leontovich order of approximation. Note that they are the first nonzero terms in the asymptotic expansions representing high-order SIBCs [7]. III. FIT The example of the orthogonal dual mesh used in FDTD and FIT is shown in Fig. 1. In FDTD, the nodes, where electric and magnetic fields are calculated, are located at the middle of edges 0018-9464/$20.00 © 2006 IEEE