AN ADJUSTING APPROACH TO THE DETERMINATION OF THE PERMEABILITY FUNCTIONAL DEPENDENCIES OF ANISOTROPIC COLD-ROLLED ELECTROTECHNICAL STEELS
Keywords:magnetization characteristics, anisotropy, permeability, spline interpolation, functional basis, nonlinear regression method, electromagnetic field
Purpose. To develop an effective approach for the determination of mathematical functional relationships for the high-precision description of the nonlinear dependences of the permeability on the magnetic flux density, taking into account the anisotropy of modern cold-rolled electrotechnical steel, which are used in the production of new series of power transformers.
Methodology. The researches were carried out using interpolation methods, approximation, regression analysis, mathematical physics, electromagnetic field theory.
Findings. The methods of functional conception of nonlinear characteristics of magnetization of ferromagnetic materials, including cold-rolled anisotropic electrotechnical steels, are researched. An adjustment approach is proposed to increase the dimension of the initial data, based on the spline interpolation method on the Hermite polynomial basis for the digital data of the magnetization characteristics. This excludes an increase of the errors in determining the permeabilities in the sections between the nodal values of the magnetic flux density. For a mathematical description of the dependences of the permeability on the magnetic flux density, a special functional basis with Gaussian functions and additional error functions has been developed, that ensures the continuity of derivatives of various orders at fixed angular values between the rolling direction of anisotropic electrical steel and magnetic flux. The method of nonlinear regression in the structure of function package of the Mathcad software for the high-precision determination of regression coefficient vectors in functional descriptions of permeabilities is applied. With the help of verification and validation of the results, which adjusting nonlinear regression with computational data using the generic method and with experimental data, a significant decrease in the relative errors that do not exceed 1.62% for fixed angles between the direction of rolling of anisotropic electrical steel and magnetic flux is obtained. The application of the new regression equations substantially improves the conditions for the field simulation of electromagnetic fields in the open-circuited mode of the power transformer in the Comsol Multiphysics software, and ensures the stability of iterative computing processes.
Originality. A new approach to increasing the dimension of input arrays, based on spline interpolation of digital data arrays of magnetization characteristics for fixed angles between the directions of rolling of the electrical steel and magnetic flux is implemented. For nonlinear regression equations, a new functional basis with Gaussian functions and additional error functions for mathematical descriptions with continuous derivatives for the dependences of the permeabilities of anisotropic cold-rolled electrotechnical steels on the magnetic flux density with high accuracy in the range of angles variation between the rolling directions and magnetic flux from 0º to 90º.
Practical value. The approaches and techniques for the basis of Gaussian functions and additional error functions proposed in this paper make it possible to significantly improve the accuracy of determining the nonlinear characteristics of anisotropic electrotechnical steels. Also to reduce the relative errors to 1.62% when the angles between directions of rolling of the electrotechnical steels and magnetic flux vary from 0° to 90°.
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Copyright (c) 2018 T.E. Divchuk, D.S. Yarymbash, S.T. Yarymbash, I.M. Kylymnyk, M.I. Kotsur, Yu.S. Bezverkhnia
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