EEE TRANSACTIONS ON MAGNETICS, VOL. 28, N0.2, MARCH 1992 zyxwvuts Periodic Solutions of Nonlinear Eddy Current Problems in Three-Dimensional Geometries R. Albanesel, E. Coccorese’, R. Martone’, G. Miano3, G. Rubinacci4 Istituto di Ingegneria Elettronica, Universith di Salemo, Baronissi (SA), Italy Facolth di Ingegneria, Universid di Reggio Calabria, Italy Dipartimento di Ingegneria Elettrica, Universith di Napoli “Federico11”, Italy Dipartimento di Ingegneria Industriale, Universitk di Cassino, Italy zyxwvu Abstract In this paper numerical procedures for the finite element computation of periodic steady-state nonlinear eddy currents in three dimensional geometry are presented and compared. The periodic solutions are computed by using two “static methods” and are compared with the solution obtained by the “brute force” method. I. INTRODUCTION Most electric devices working in time periodic regimes make use of magnetic materials. The main aspects characterizing the electromagnetic fields in such devices are given by the presence of eddy currents, saturation, hysteresis, anysotropy and motion. A comprehensive analysis of the whole problem in three-dimensional geometry would present considerable numerical difficulties due to the complexity of the interacting phenomena from both computational and physical points of view. As far as motion is concerned, upwind, moving meshes and mixed FEMIBIEM procedures have been proposed. For the treatment of the hysteresis, well tested numerical models taking into account minor loops and rotational effects are not yet available for the computation of two or three-dimensional fields [l]. In this paper we will focus our attention on the aspect of saturation, i.e. on the nonlinear magnetic properties. It should be stressed that the assumption of neglecting the hysteresis is far from being satisfied in most of the practical cases (especially in zyxwvutsr ax. regime). The results achieved in this study can then be applied only to a limited class of motionless devices (i.e. transformers) which make use of magnetic materials characterized by a narrow hysteresis loop. However, we feel that an analysis of nonlinear periodic problems is still worth while and, in any case, is a necessary step towards the analysis of the whole problem. Having disregarded anysotropy, motion and hysteresis, the problem has been simplified. Yet, as clarified in [2] and [3] we will also- make a number of additional assumptions on the material properties in order to guarantee uniqueness and stability of the steady- siate solution (thus avoiding the possibility of having multiple or chaotic solutions). The aim of this paper is to propose and compare a number of numerical procedures for the finite element computation of periodic steady-state nonlinear eddy currents in three-dimensional geometry based on “static methods”. In static methods, the solution is expanded by means of a finite number of periodic (usually, harmonic) functions of the time, and the unknown coefficients are determined by weighted residual methods zyxwvutsrq [4], [5]. These solutions are compared with that obtained by means of the “brute force” method. U. FINITE ELEMENT FORMULATION The problem is described by the quasi-stationary Maxwell’s equations: and by the constitutive relations: aB VXE=-~ VXH=J (1) J=oE+J,, H=HqB (2) in the domain V zyxwvutsrqpo ; r~ is the conductivity of the material, H(.) is a non- linear function supposed to be strictly increasing with upper and * Manuscript received July 7, 1991. This work was supported in part by the Minister0 dell’universithe della Ricerca Scientificae Tecnologica lli8 lower bounds for &/dB and J, is a source term. The conductivity (J and the nonlinear mapping H(.) may depend on position. The boundary conditions should specify the value Exn (or Hxn) on dV ; moreover Exn and Hxn should be continuous at any interfaces. In particular, if V is the whole space R3, then the fields must verify the regularity conditions at infinity. The problem is closed by adding initial conditions for B which must verify V.B=O. The problem can be formulated in terms of the two-component magnetic vector potential A and the scalar potential V (whose initial condition can be arbitrarily set to zero), defined by: B=VxA, A.w=O, E = - %- V( zyx x) (3) where \v is an arbitrary vector field without closed field lines. By using a finite edge-element discretization and exploiting a treelco-tree decomposition [a, A and V can be expanded as : av Na NV A(r,t) = Z ai(t)Wi(r), V(r,t) =Z vi(t)cpi(r) zyxw (4) i=l i=l where (Wi) is the set of vector shape functions associated with the N, edges of the co-tree, and [ ‘pi) is the set of isoparametric shape functions related to the Nv nodes. Assuming homogeneous boundary conditions on aV and Applying the Galerkin method, we have: 1 W ,.a x(A+VV)dv a = - I(VxWi).H(VxA)dv VC zyxwvutsrqponm V + I Wi. J,dv VWi i=l,N, I (Vq,).ox(A+VV)dv = I J,.V’p,dv VC VC V a whereVC is the conducting region and JS(+) is the value of J, on the extemal side of the surface av,. The right hand side of Eq. (6) is zero if the source filed J, is solenoidal in the whole space. AssumingV,=V for the sake of simplicity, the system (5)-(6) can be rewritten as (with an obvious notation): S U + f(u) = g(t) where: (7) T u E (a,v) , S I 2 I, f(u) E (P(a),O)T and g(t) E (dw(t),dq(t))T. After the elimination of zyxw i, Eq.(7) yields: (8) where A is the NaxNa matrix given by A =Sa, - SavSv,-lSva and d(t) is a linear combination of dw(t) and d (t). The matrix A is symmetric and definite positive. Furthermore $e assumptions on the magnetic properties guarantee that the continuous and differentiable nonlinear mapping P is strongly uniformly increasing and its Jacobian is symmetric [2], [3]. The matrix S is in a sparse form while A is a full matrix. Therefore it is worth while to solve Eq.(7) A a = - P(a) + d(t) 0018-9464/92$03.00 zyxwvuts 0 1992 IEEE