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T-Comm_Article 6_6_2020

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CALCULATION OF MAGNETIZATION OF PERMANENT MAGNETS AND OPTIMIZATION OF THE LOCATION OF INDUCTION MEASUREMENT POINTS USING THE METHOD
OF INTEGRAL EQUATIONS WITH HYSTERESIS

DOI: 10.36724/2072-8735-2020-14-6-39-47

Robert V. Harutyunyan, Moscow State Technical University named after NE Bauman, Moscow, Russia, rob57@mail.ru

Abstract
The article proposes the development of the method for estimating the magnetization of permanent magnets by the known distribution of the magnetic field in the surrounding space, which differs from the known methods in that it allows to take into account the presence of soft magnetic ferromagnetic materials with known characteristics. The approach applied in the article to the problem of identification of the magnetic state of the system of permanent magnets and ferromagnets is based on the use of the corresponding integral equation of magnetostatics. The main objective of the study is to determine the conditions for the applicability of the modification of the previously developed numerical-experimental method for assessing the magnetization of permanent magnets for the diagnosis of electrical devices with permanent magnets in practice. For this purpose, the influence of neglect in the mathematical model of the magnetic field by the magnetic hysteresis of the frame material of the electrical device on the accuracy of the result of the inverse problem of identification of the magnetization of permanent magnets is investigated. We performed a series of numerical experiments, intended to establish the deviation of the calculated characteristics of the field in the magnetic material depending on the choice of the curves of magnetization within the area of the hysteresis loop of the material of the frame of the diagnosed electrical devices. In mathematical modeling, the presence of wind-ings with current was not taken into account. It is assumed that the magnetic system consists of permanent magnets and structural parts of ferromagnetic materials with known characteristics. For the regularization operator SLOUGH, used the method of A. N. Tikhonov based on the minimization of the stabilizing functional.
The calculation of integrals in the spatial integral equation is carried out exactly according to analytical formulas. Derivatives in the kernel of the equation were calculated both numerically and analytically, taking into account the new relations. The presence and influence on the computational process of special points of the func-tions — derivatives of the kernel at the origin, where the continuity of the functions is violated and there is no single limit value. The calculation of induction, intensity and study of the influence of the shape of the magnetization characteristics. At the first stage of the solution by known experimental values of induction outside the volume of magnetic material the solution of the inverse problem is carried out and approximate values of magnetization in this volume are found. At the second stage, ac-cording to the known characteristic of magnetization in the volume, the values of induction and magnetic field strength are found in various ways. The main magnetization curve is calculated by the Langevin formula. The hysteresis loop is calculated from the selected range of magnetic in-tensity values according to the jils-Atterton model. The implementation of the method in terms of magnetization with different ratio of the number of cells of the magnet region and measurement points is considered. For the selected hysteresis loop, envelope loops are plotted at the top and bottom of the line. As a result of solv-ing the equations, the corresponding minimum, maximum and average values of the intensity, as well as the induction of the field are found. This makes it possible to estimate the error when re-placing the hysteresis curve with the main characteristic of magnetization. The identification of magnetization of a rectangular permanent magnet on a ferromagnetic plate is considered as a model problem. It is stated that for small numbers of measurements the detailed picture of the field is found with a large error, which requires caution in the inter-pretation of the experimental data. The influence of the choice of measurement points location at random measurement errors is also significant, especially at a small number of measurement points. The dependence of the solution accuracy on the regularization parameter values is inves-tigated. Shows the influence of the location of measurement points on the accuracy of the identi-fication of the magnetic field. The influence of the hysteresis loop width on the corresponding range of magnetic field intensity values in solving the problem of magnetization identification is studied. The obtained results can also be used in solving the inverse problem for the system of ferromagnetic bodies and in test problems using other methods.

Keywords: permanent magnets, magnetization, scalar magnetic potential, integral equation of magnetostatics, inverse problem, identification.

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Information about author:

Robert V. Harutyunyan, Moscow State Technical University named after NE Bauman, Ph.D., associate professor of the Department of Computation Mathematics and Mathematical Physics, Moscow, Russia