lEEE TRANSACTIONS ON MAGNETICS, VOL. zyxwvutsrqponm 32, NO 3, zyxwvutsrqpon MAY 1996 1637 Magnetic Field Analysis and Leakage Inductance Calculation in Current Transformers by Means of 3-D Integral Methods KZakrzewski Institute of Electrical Machines and Transformers, Techcal University of Lodz, zyxwv d. Stefanowskiego 18,90924 Lodz, Poland B. Tomczuk Dept. of Electricad Engineering zyxwvutsrq & Autoinatic Control, Teclmical Uiliversity of Opole, ul.Luboszycka 7, 45 03 6 Opole, Poland zyx Abstract- This paper presents 3-D integral approach to the magnetic field and inductance calculations. A minimization of zyxwvut n. INTEGRAL APPROACH thekernel norm has been carried out for the integral equation governing the field. The software package TRACAL3, based on The Of 3-D ma@1etic 'Ias been the integral methods for field and inductance calculations, has Perfomed making the following assumptions: the m x h m been devekoped and iimplemented for personal computers. is isotropic, the mametic permeability zyx U of the considered zy - I - ... The application of the 3-D mathematical models has been made for the leakage field in a current transformer. The results of calculations were compared with the measured ones. The Introducing potentid defined by region is piecewise constant, the field is quas.-stationary comparison yields good agreement. Thus, the worked out software package seems to be one of the zyxwvuts CAD tools. I. INTRODUCTION In recent years there has been an increased interest in the use of the integral methods [3] to model electromagnetic fields. In this work we present an integral approach with ininimization of thle kernel norm i.e. regularization of the integral equation. An application for tlie magnetic field and inductance calculations is given as well. The major application area for 3-D numerical methods in power engineering relates to the modeling of electromag- netic fields in power transformers [3], [7]. Not many papers deal with the special transformers 3-D field calculatioiis [5] by means of integral methods. As the transformers are without ferromagnetic casing, the field calculations consist in the solution of .magnetically open problems [6], and the methods are particularly effective. Current transformers fall into a class of special (measuring) transformers. For the transformers, 3-D magnetic fields and integral parameters calculations, together with measuring verification, have not been presented so far. 'Ths paper shows some calculating and measuring results in such type of a transformer. The operating conditions of the current transfomiers are near short-circuit. Hence, the magnetic flux density value in the core iron is relatively low. It can be assumed that the relative magnetic permeability value p,. is more than lo4, and the iron is linear. U= 4 h,l(471.Y)dS, (1 1 SP we can denote the simple layer density hp , at the core surface Sp , as an algebraic sum [6] of tlie interior h, and exterior hZ derivatives 121 of the potential: h, + h, = hp. The derivatives are chscontinuous functions [2] at the close surface Sp, and the limits of them have been applied to include the condition for surface divergence [6] of magnetic flux density zyxw 2 at the Sp Div B = 0. (2) The magnetic field strength @ is related to the potential U and current by H= I ( vx Jljlnv))dV-VU; (3) V where: 9 - current density in the volume I/ of the windings, r - the distance between the source (with 9 density) and tlie field points, V - del operator (nabla). Talung into account the magnetic linearity of the medium B = p H, and includmg (1) and (3), from the condition (2) we obtain Fredholm integral equation of the second lund with the unknown density hp at a point X hJX) + W$h,(Y)N(X,Y) dY = g(X). (4) s, The parameter a = (p," - pL3) / (pIv + pZ) includes the magnetic permeabilities of the interior (,ulu), and exterior (pZ) regions to the S, [GI. The right-hand side g(x> is Manuscnpt received July 10, 1995 g(x>= Ja(3x~) 2/(27cr3)d~, zy (5) B Tomczuk, e-mall tomczuk@ss5 wsi opole pl, fax 0048-77-556080, V phone 0048-77-538447, K. Zakrzewsh, fax 0048-42-362309, phone where 1s the umt vector nomla1 to the S, 0048-42-312581, 00 18-9464/96$05 .OO 0 1996 IEEE