IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 7, 2008 421 An Equivalence Theorem Based on the Use of Electric Currents Radiating in Free Space Enrica Martini, Member, IEEE, Giacomo Carli, and Stefano Maci, Fellow, IEEE Abstract—The most common formulations of the equivalence principle use both electric and magnetic currents radiating in free space or only electric (magnetic) currents radiating in presence of a perfectly magnetic (electric) conductor. In this paper, a formula- tion is provided for equivalent currents of only electric (magnetic) type as a function of the tangential total fields of the original problem over the equivalence surface. The formulation of these equivalent currents becomes simple for some canonical surfaces, like a plane, a cylinder and a sphere. Explicit expressions of the equivalent electric currents on a spherical surface are provided in terms of spherical wave functions. Index Terms—Electromagnetic theory, equivalence principle, spherical wave expansion. I. INTRODUCTION T HE field equivalence principle, which is one of the fundamental concepts in electromagnetics, allows one to replace an arbitrary set of sources by equivalent currents distributed over a virtual closed surface enclosing the real sources, and providing the same field outside [1]. The value of these equivalent currents is related to the tangential components of the fields over the equivalence surface . The most common formulation uses both electric and magnetic currents radiating in free space. Formulations with only one kind of currents can be obtained by filling the equivalence surface with a perfect electric or magnetic conductor. However, the price to be paid to halve the typology of currents is that the residual currents radiate in presence of the conductor. In the case of an infinite planar equivalence surface, the conductor can be eliminated through the application of the image theorem; however, this approach can not be extended to arbitrarily shaped surfaces. It is however desirable in some applications to use only one kind of equivalent currents radiating in free space. An integral equation formulation with only electric currents has been developed in [2] for problems involving electromagnetic scattering from homogeneous dielectric bodies. In this case, the use of only one kind of currents allows one to halve the number of unknowns. In other practical cases it would be useful to have an equivalent description in terms of electric currents only. For instance, in [3], it has been shown that an arbitrary electromagnetic field can Manuscript received April 04, 2008; revised May 16, 2008. First published June 27, 2008; current version published November 14, 2008. This work was supported in part by the IDS, Ingegeneria dei sistemi, S. Piero a Grado (Pisa), Italy. The authors are with the Department of Information Engineering, University of Siena, 53100 Siena, Italy (e-mail: martini@dii.unisi.it; carligiac@dii.unisi.it; macis@dii.unisi.it). Digital Object Identifier 10.1109/LAWP.2008.2001764 be expanded in terms of complex source point (CSP) beams of only electric type launched from an arbitrary surface enclosing the real sources and radiating in free space. The weights of these sources are determined numerically by imposing the matching with the original field over a test surface. In this case, the use of both electric and magnetic type sources would result in a strongly ill-conditioned linear system, since the contribution to radiation of orthogonal electric and magnetic CSP beams is practically indistinguishable. This paper provides a rigorous justification for the represen- tation of a generic source in terms of only electric currents radi- ating in free space. The case of equivalent currents of magnetic type only is easily obtained by duality. In Section II, it is shown that the equivalent electric cur- rents that by themselves reconstruct the original field outside the equivalence surface consist of two contributions; the first one is calculated in the conventional way as , where is the outward normal to and is the total magnetic field radi- ated by the original sources. The second contribution is found by solving an equivalent internal problem with boundary con- ditions dictated by the tangential electric field radiated by the original sources. Following this approach for a planar surface leads to the same result as that provided by the application of the image theory. Moreover, it is shown that for certain shapes of the equiva- lence surfaces (cylinder, sphere), a simple explicit expression of the equivalent currents can be obtained by expanding the in- ternal and external electromagnetic field in series of orthogonal vector wave functions and applying a mode matching technique to guarantee the continuity of the tangential electric field at the interface. In Section III, the expression of the equivalent cur- rents is explicitly obtained for a spherical surface, as a function of the spherical wave coefficients of the electric field of the orig- inal problem. II. FORMULATION FOR ARBITRARY CLOSED SURFACE In the following, we address the problem relevant to equiv- alent electric currents; the case with only equivalent magnetic currents can be obtained by duality. Let us start by summa- rizing the formulation of the equivalence theorem in the general form presented in [1]. Consider an arbitrary set of impressed cur- rents and scatterers enclosed by a fictitious surface with outward normal , surrounded by a homogeneous medium, that for the sake of simplicity we will assume to be free space (Fig. (1a)). Let us denote by , the electromagnetic field radiated by this system. Consider the equivalent problem in Fig. (1b), where the in- terior of is source-free and filled by free space and a set of 1536-1225/$25.00 © 2008 IEEE