1578 IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY2003 A Novel Integral Formulation for the Solution of Maxwell Equations Guglielmo Rubinacci, Antonello Tamburrino, and Fabio Villone Abstract—This paper presents a novel integral formulation for the solution of the full Maxwell equations in the frequency domain. The solenoidality of the current density inside a ho- mogeneous conductor is exploited to introduce a two-component electric vector potential, which is numerically expanded in terms of edge elements. Index Terms—Electromagnetic compatibility, electromagnetic coupling, equivalent circuits, finite-element method, integral equations, interconnections. I. INTRODUCTION T HE CONTINUING increase of clock frequency and chip size makes the electric design of interconnect structures very challenging. The very high performances required for these interconnect circuits can be achieved only by means of very so- phisticated electromagnetic modeling methodologies [1]. One of the most widely used approaches for the intercon- nect analysis is the partial element equivalent circuit (PEEC) technique [2]. The electric field integral equation is solved in PEEC using pulse (i.e., piecewise constant) basis and test func- tions defined over capacitance and inductance cells. Such cells are suitably staggered in order to conveniently impose charge conservation. The PEEC method has the advantage of naturally leading to an equivalent circuit schematization where any addi- tional lumped circuit element can be easily added. On the other hand, complex three-dimensional (3-D) structures require very accurate modeling techniques and do not allow an explicit split- ting of inductive and capacitive effects. Moreover, the study of complex 3-D configurations requires special care when using piecewise constant basis functions. In this frame, a generaliza- tion of PEEC using quadrilateral and hexahedral nonorthogonal cells [3] or triangular cells and prisms [4] as the building blocks for modeling conductor surfaces and conductor/dielectric vol- umes has been recently proposed. The formulation presented in this paper is in the stream of PEEC approach, as far as the use of integral equations is con- cerned. The main numerical difference is in the use of edge-el- ement basis functions to represent the current density vector, together with the electric vector potential. This allows us to au- tomatically guarantee the solenoidality of the current density in Manuscript received June 18, 2002. This work was supported in part by MIUR. The authors are with DAEIMI Laboratory of Computational Electromag- netics, Università degli Studi di Cassino, I-03043 Cassino, Italy (e-mail: rubinacci@unicas.it; tamburrino@ unicas.it; villone@unicas.it). Digital Object Identifier 10.1109/TMAG.2003.810362 homogeneous conductors. Since the link with lumped parame- ters is not so evident as in PEEC, an additional effort is required in order to get an equivalent multiport representation of complex 3-D structures, in the stream of what the authors did in the low frequency cases [5], [6]. Moreover, since in many situations the electromagnetic system may be split into several subsystems in- teracting essentially through the terminals, this approach allows to reduce the initial problem to the proper modeling of each of such subsystems, regarded as a multiport element. The paper is organized as follows. In Section II, the mathe- matical formulation is discussed, while in Section III, a brief de- scription of the numerical formulation is given. Finally, in Sec- tion IV, simple preliminary numerical results are presented and discussed. In Section V, we draw our conclusions. II. MATHEMATICAL FORMULATION We solve Maxwell equations in the frequency domain. At all points of space we have (1a) (1b) (1c) (1d) where ( ) is an impressed current density (charge), with the addition of the radiation condition at infinity, and of constitutive relations in (2a) in (2b) in in (2c) where is a homogeneous conducting domain fed via a number of equipotential superficial electrodes . Posing (3) and using the Lorenz gauge (4) 0018-9464/03$17.00 © 2003 IEEE