2284 IEEE TRANSACTIONS zyxwvut ON MAGNETICS, VOL. MAG-21, NO. zyxw 6, NOVEMBER zy 1985 z A FINITE ELEMENTCOMPUTATION PROCEDURE FOR ELECTROMAGNETIC FIELDS UNDER DIFFERENT SUPPLY CONDITIONS P. Belforte, M. Chiampi zyxwvut and M. Tartaglia Abstract zyxwvutsrqpon - A computational procedure is developed to analyze two-dimensional nonlinear electromagnetic fields in steady-state alternating conditions both when the currents in the conductors are specified and when the voltage sources and the electrical connec- tions are assigned; furthermore the effects of eddy currents can be included. In every case the finite element and Fourier approximations and the iterative methodemployedtohandlemagneticnonlinearities lead to a set of linear algebraic systems having the same block structure. The application of the proce- dure 1s illustrated by analyzing a no-load three- phasetransformerneglectingeddycurrentsanda high-frequency single-phase transformer including the skin effect. 1. INTRODUCTION Theversatility ofthefiniteelementmethod makes this method attractive for studying electromag- netic field problems in different magnetic devices. Of course, depending on the applicatlon purposes, different simplifying assumptions are imposed case by case, in order to reduce the storage and processlng requirements, so usuallydifferentproceduresand computer programs must be developed. This paper presents a computational scheme suitable to study a large category of electromagnetic nonli- near problems. In particular two-dimensional fields in alternating steady-state conditions can be an- alyzed both when the currents in the conductors are specified and when the terminal voltages are known; moreover the effects of eddy currents within the conductors can be taken into account. The proposed procedure easily imposes the constraints introduced bytheconnectionsoftheconductorsand by the topology of the external network. The computational approach essentially consists of discretlzing the field problem by means of the finite element and Fourier approximations and zyxwvutsrq of employing an iterative technique to handle iron saturation. Moreover, the circuit equations are used to express the current denslties in terms of the unknown vector potential. and the assigned supply quantities, as proposed by the same authors in [l]; in such a way a set of linear algebraic block systems is obtained, whichmustbesolvedateachiteration step on nonlinearity. The analysis of combined field and circuit problems has been already dealth with in the literature accor- ding iterative and direct techniques. In particular some authors [2,3] solve separately the field and the circuit equations by introduclng an iterative proce- dure; this technique requires the successive solution of several nonlinear field problems. Some direct methods [4,5] simultaneously solve the field and the circuit equations by regarding both the vector poten- tlalandthecurrentsasunknowns;inthiscase ______-__________--_--------------------------------- This work has been partially supported by the Italian Ministry of Education and CSI Piemonte. P. Belforte is with CSS - Centro Stsdi sui Sistemi, via Vela 27, 10128 Torino, Italy; M. Chiampi is with Istituto Elettrotecnico Nazionale Galileo Ferraris, corso M. D'Azeglio 42, 10125 Torino, Italy; M. Tar- taglia is with Dipartimento di Elettrotecnica, Poli- tecnico di Torlno, corsoDucadegliAbruzzi 24, 10128 Torino, Italy. asymmetrical coefficient matrices must be solved. The direct method presented in this paper leads to suitable block systems which are conveniently solved using the SOR technique suggested in [6,7] . This procedure shows the following advantages : - the memory requirements are strongly reduced, - the iterative technique to tackle iron saturation [a] .is improved when it is employed together with the SOR procedure; - in spite of the differences arising when diverse supply conditionsare imposed, the structureof the linearsystemsisthe same, so thataunified computational scheme is employed. The basic assumptions and the field equations for a wide category of applications are summarized in Sec- tion 2. The computational procedure is dealt with in Section 3, where the more interesting ways to improve the proposed method are remarked. Section zyx 4 reports the no-load analysis, neglecting eddy currents, of a star neutral-delta three-phase transformer supplied by voltages containing a 3rd harmonic homopolar com- ponent; moreover the study of a high-frequency sin- gle-phase transformer, including the skin effect, is described. 2. FIELD EQUATIONS The field problem formulation is developed ac- cording to the following simplifying assumptions. - Two-dimensional field distributions are considered where the flux density B is parallel to the x-y plane and the current density J flows along the z axis; these quantities are independent of the z coordinate. - The end-turn effectsare negligible. - An alternating time evolution of the electromag- - The displacement currents are neglected. - The magnetic hysteresis is neglected; the conduc- tivity is assumed constant within each conductor. - Dirichlet or Neumann homogeneous boundary condi- tions are imposed. Under these assumptions and representing the flux density by a unidirectional vector potential A, one has to solve the well known field equation netic quantities is considered. curl [[*,curl A(t;)] = J(t,') (1) being 5 the nonlinear magnetization curve H = 5 (B). The right-hand side J takes into account the contri- bution of all N conductors included in the domain, that zyxwvu is: N J = X, Jl ." where XI 15 the characteristic function such that XI = 1 in the 1-th conductor and XI = 0 elsewhere. In order to linearize (l), the technique proposed in [8] is employed, whlch decomposes the function as: 1=1 <(',B(t,*)) = l'(')S(t,.) + M(t;) zyxwvu (3) where w is a known coefficient and where the re- sidual nonlinearity M(t, * ) is evaluated by an itera- tive procedure. The supply conditions and the effects of eddy cur- rents lead to individualize the following classes of problems: Problem a - assigned currents without eddy currents 0018-9464/85/1100-2284$01.00@1985 IEEE