IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 4,JULY 2000 795 Magneto-Dynamic Formulation to Solve Capacitive Effect Problems in an Axi-Symmetrical Coil Fabiano L. S. Garcia, Gerard Meunier, Vincent Leconte, and Vincent Mazauric Abstract—This paper presents a magneto-dynamic finite ele- ment formulation coupled with circuit equations using the time-in- tegrated electric potential. This formulation in high frequency, in- cluding displacement currents, permits the study of electric–mag- netic interactions in electric coils. The massive conductors in which eddy currents can develop are considered. This formulation can resolve the parasitic capacitive effect problems in massive compo- nents for an axi-symmetrical domain. Index Terms—Coils, finite element methods, frequency response, impedance measurement. I. INTRODUCTION I N HIGH frequency, the parasitic elements have to be taken into account in wound components like inductors and trans- formers [1]. To solve electric and magnetic coupled problems in 2D or axi-symmetrical domain, a large number of papers have been re- cently published. There are different methods to study the mag- netic and the electric phenomena: analytical, experimental and the field calculation by the finite element method (FEM) [2], [3]. In most papers the electric and magnetic phenomena are studied separately. The purpose of this paper is to solve simul- taneously the equations by an axi-symmetrical formulation coupled with electrical network equations in high frequency. Such a coupling was developed by Z. Ren in an earlier paper [4] using an – formulation. It is proposed here an formulation, where is the time-dependent electric potential. The studied electromagnetic structure is an axi-symmetrical massive coil which is connected to an external voltage source. The presented formulation takes into account eddy currents in the massive conductors as well as parasitic effects between the strands of the coil. The results are compared to a de-coupled method. II. BASIC EQUATIONS In a sinusoidal magneto-dynamic mode, if the displacement currents are not neglected, the Maxwell’s equations can be written as: (1) Manuscript received October 25, 1999. F. L. S. Garcia and G. Meunier are with the Laboratoire d’Electrotechnique de Grenoble, (INPG/UJF - CNRS, UMR 5529), ENSIEG BP46,38402 St.-Martin d’ Hères, France (e-mail: meunier@leg.ensieg.inpg.fr). V. Leconte and V. Mazauric are with Schneider Electric - Research Center/A2, 38050 Grenoble Cedex 9, France (e-mail: vincent_leconte@mail.schneider.fr). Publisher Item Identifier S 0018-9464(00)06885-0. (2) It is also used the following constitutive relations: (3) where is the magnetic excitation, the magnetic induction, the electric field, the electric displacement and the current density. And is the reluctivity, the electric conductivity and the per- mittivity . When introducing the magnetic vector potential such that , the traditional field equation is obtained: (4) where is the total current density which is the sum of the conductive current density and the displacement current den- sity : (5) Equation (2), can be rewritten as: (6) From which the expression of the electric field is derived: grad (7) where is the scalar electric potential. III. FORMULATION HYPOTHESIS This formulation is based on the hypothesis that the displace- ment current can be separated from the current of conduction. The two currents can then be studied in the two directions, in squaring: the plane, and along the conductors (azimuth direction). In order to lead to the formulation, the following hypothesis are made: 1) In an axi-symmetrical coil, one considers the conducting current has only one component along the conductors (azimuth direction); 0018–9464/00$10.00 © 2000 IEEE