The paper deals with a proposal of a novel computational method of magnetic field distribution in a transformer
window. The method reduces demands on computational time and computer memory whereas it increases accuracy of the results.
The principle of the method is described and discussed on a simple case of planar 2D arrangement

Kůs, Pavel

Melichar, Jan

Musil, Ladislav

English

DSpace at University of West Bohemia

II-21Novel Way of Computation of Magnetic Field  in Transformer Window Ladislav Musil*, Jan Melichar*, Pavel Kůs† *Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic  E-mail: musill@fel.cvut.cz, melicja6@fel.cvut.cz †Academy of Sciences of the Czech Republic, Dolejškova 5, 182 00 Praha 8, Czech Republic,  E-mail: pavel.kus@gmail.com  Abstract The paper deals with a proposal of a novel computational method of magnetic field distribution in a transformer window. The method reduces demands on computational time and computer memory whereas it increases accuracy of the results. The principle of the method is described and discussed on a simple case of planar 2D arrangement. Keywords transformer window, electromagnetic field computation.  I. INTRODUCTION The paper deals with a novel method of computing of magnetic field distribution in a transformer window.  When computing short-circuit parameters of a transformer, knowledge of the magnetic field distribution in the transformer window is an essential prerequisite. When a FEM-based method was employed without any geometrical simplification, each single wire would have to be included into its geometry and into the corresponding mesh. It would lead to a great number of degrees of freedom and, subsequently, to time costs and memory demanding code. Therefore, major geometry simplifications are usually done in the magnetic field computation. This leads to errors which are still acceptable when an integral value is computed. However, when local values are needed, for example, for calculation of forces in the windings, the reliability of such results is much lower. Inclusion of details into the computational geometry can also improve integral value computations. Therefore. a novel, low demanding method was suggested for computation of the magnetic field in a transformer window. The novel method is based on basic characteristic properties of computation of magnetic fields in transformer windows. The first one is linearity of such a computation and the second is the fact that the geometry consists of a great number of identical and periodically repeated patterns.  II. DESCRIPTION OF THE METHOD The method will be explained on a very simple example. Let’s assume a transformer with two identical windings, each with the same number of coils. Assume further that the coils carry the same, but opposite currents WI , so that the total number of amperturns through the transformer window is zero. This is almost completely satisfied for a transformer in the short-circuit state.  We will utilize geometrical symmetry and compute magnetic field only in the left half of the transformer window. For simplification, we will use Cartesian coordinate system and compute planar 2D problem for the purpose of explanation of the method. Later we will discuss another situation, when an axisymmetric problem is solved.   Fig. 1. Original geometry (transformer window with two coils) The complete geometry of one half of the transformer window is depicted in Fig. 1. Only half of the window is computed because of the geometrical symmetry. The equation that describes the corresponding magnetic field distribution reads  W1curl curlµ =  A J ,                     (1) where A  is the magnetic vector potential and WJ  is the current density in windings. The skin-effect in the wires is neglected.  The Neumann boundary conditions are used on the boundaries representing the magnetic core surface and Dirichlet boundary condition is used along the axis of symmetry. As the first step of the novel method we must describe the periodicity of geometry by enclosing each winding into regions uniformly divided to blocks with identical geometries.  Than a coarse field calculation is carried out on geometry which consists only of the artificial regions without any division (Fig. 2). All the repeated blocks are removed from this geometry. The current density rJ  in each region is set so, that the total current in the region is equal to the total current WI∑  through the winding, so that II-22WrrIJA=∑,           (2)  where rA  is the cross-section area of the region. Such geometry with appropriate boundary conditions can be solved on a substantially rougher mesh.   Fig. 2. Geometry for the computation on rougher mesh  Next a corrective field will be computed. This field is computed on geometry comprising the shape of the block repeated in region and of the shape of the winding held by the block (Fig. 3). The computational area should appropriately overreach the block geometry. The current density in the area of the block (but not in the winding) should be set to the value rJ−  and in the area with the winding to the value W rWIJA− r, where WA  is the cross-section area of the winding.  Fig. 3. Geometries for calculation of corrective fields  The resultant field is computed as a superposition of the coarse field computation and the corrective computations. The corrective geometries for corresponding blocks differ only in space shift and, therefore, it can be computed only for one case and the other can corrective fields can be obtained by a simple linear transformation.  When an axi-symmetrical calculation is carried out, only the geometrical repetition in the z - direction can be used for transformation of one result of corrective field into another. The method is then somewhat less effective, but the computational savings can be significant as well. III. CONCLUSION A novel method for computing of magnetic field distribution in a transformer window was suggested. The method should save computational time and memory demands and increase the accuracy of the computation at the same time.  IV. ACKNOWLEDGEMENTS This work has been supported from grant .SGS11/121/OHK3/2T/13.  V. REFERENCES [1] Sratton, J. A.: Theory of electromagnetic field (in Czech) SNTL,1961 [2] Sadiku, M.N.O.: Numerical Techniques in Electromagnetics. CRC Press, Florida, 2000, ISBN 0-8493-1395-3                                                 

Novel way of computation of magnetic field in transformer window

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Novel way of computation of magnetic field in transformer window

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