IEEE TRANSACTIONS ON zyxwvutsrq MAGNETICS, VOL. 30, NO. 6, NOVEMBER 1994 4299 Eddy Current Losses in Saturable Magnetic Materials zyx Vitor Ma16 Machado Centro de Electrotecnia da UTL, DEEC, IST, 1096 Lisboa Codex, Portugal zyxwv Abstract-In this paper, a finite element method, based on a variational formulation, is used to evaluate eddy current losses inside a nonlinear, saturable ferromagnetic material, for a two dimensional field problem. Boundary conditions are im- posed in an integral form combined with Neumann conditions. The dynamic field is evaluated by using an explicit integration method. I. INTRODUCTION This paper is a contribution to the evaluation of the eddy cur- rent losses inside a nonlinear, saturable, ferromagnetic material. A variational formulation [3]-[6] is adopted by using an energy bctional involving the magnetic flux density, B, and the mag- netic field strength, H, which are related by a nonlinear zyxwvutsrq B-H char- acteristic, taking into account the magnetic saturation. The finite element method zyxwvutsrqpo [ 11-[ 61, corresponding to the space discretization, is applied giving an ordinary time differential matrix equation. The numerical integration of the nonlinear equation may be evalu- ated by using either explicit [ 11, [2] or implicit 161methods. In this paper, an explicit method is adopted, corresponding to a finite difference approximation to describe the time dependence for the field. zyxwvutsrqpo An adequate discretization criterion is chosen for the space-time domain [ 11, [2] ensuring a stable, nonoscillating itera- tive procedure. A closed boundary problem is considered for two dimensional field configurations by assuming the magnetic field with the axial direction. A new method is developed by imposing boundary con- ditions in an integral form (corresponding to the induced e.1n.f. along the boundary) combined with homogeneous Neumann con- ditions (corresponding to the surfaces of symmetry). The induced e.1n.f. along the boundary, being the field source, IS assumed to be sinusoidally time dependent. In addition, this approach is appropriate to evaluate eddy cur- rent losses by using the Poynting's theorem. For a linear magnetic field problem, eddy current losses have been determined in [4]. 11. FORMULATION OF THE FIELD PROBLEM The field problem is formulated by using the variational principle. Consider the functional zyxwvutsrq d, given by: Manuscript received April zyxwvutsrqpo 4, 1994. V. Ma16 Machado, phone 351-1-8486676, fax 351-1-8482987. This work was supported by the Centro de Electrotecnia da IJ.T.L.. B 0 (I being the conductivity of the conducting medium and H, B be- ing the magnetic field strength and the magnetic flux density, re- spectively, whose time domain is the interval [to, t,] and the space domain is the volume zyxwv v. The field is subject to the initial and final conditions: H is known at all points of the space domain v at the instants zyxwvut to and t,, as well as to the boundary conditions: H is known on Si for to I t I tl (3) (4) --f &(rot Hxn,) is known on Si for to I t I t 1 , where Si and Sz are nonoverlapping parts of the boundary sur- face Sv of the volume v satisfying Si U Si = Sv. The functional 7 ( 1 ) , subject to the relevant boundary, initial and final conditions is stationary about the correct field solution. The stationarity condition applied to solve the variational problem allows the corresponding Euler equation to be obtained [5], which is equivalent to the fundamental field equation: (5) aB rot^=-- at , rot^, where E is the electric field strength. energy in the conducting medium during the time interval [t, t,]. In addition, note that 3 is numerically equal to the dissipated 111. NUMERICAL SOLUTION FOR THE 2D FIELD PROBLEM A two dimensional field configuration is considered for a ferro- magnetic sample whose 1/8 of its square cross section is repre- sented in Fig. 1. The magnetic field is assumed to be in the axial direction: (6) -3 H=H uz , H=H(x,y,t) . The field satisfies the following boundary conditions: (7) 1 4 -+ &wtHxne) =-(Eh) ut = 0 on s1 , Fig. 1. Cross section of a square ferromagnetic sample. S corresponds to 1/8 of the sample. s2 is an electric field line; the electric field is orthogonal to s,. 0018-9464/94$4.00 0 1994 IEEE