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IDENTIFICATION OF MAGNETIZATION IN A PERMAMEMT MAGNET ON A FERROMAGNETIC BASIS BY THE METHOD OF INTEGRAL EQUATIONS WITH ALLOWANCE FOR THE WIDTH OF THE HYSTERESIS LOOP
Peter A. Denisov, South-Russian State Polytechnic University, Novocherkassk, Russia, soff_novoch@mail.ru
Dmitry N. Chernoivan, South-Russian State Polytechnic University, Novocherkassk, Russia, npi_pm@mail.ru
Polina B. Seredina, South Russian State Polytechnic University, Novocherkassk, Russia
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 windings 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 stabiliziruyushchego functionality.
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 functions – 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, according 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 intensity 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 solving 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 replacing 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 interpretation 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 investigated. Shows the influence of the location of measurement points on the accuracy of the identification 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
References
1. Denisov P.A. (2016). The solution of direct and inverse problems of analyzing the magnetic field of electrical devices with permanent magnets during their local demagnetization. Thesis for the degree of candidate of technical sciences. Novocherkassk. 119 p.
2. Cherkasova O.A. Study of the magnetic field of a permanent magnet using computer simulation.
http://hmm.sgu.ru/sites/default/files/issledovanie_magnitnogo_polya_postoyannogo_magnita_s_pomoshchyu_kompyuternogo_modelirovaniya.pdf
3. Matyuk V.F., Churilo V.R., Strelyukhin A.B. (2003). Numerical simulation of the magnetic state of a ferromagnet in an inhomogeneous constant field by the method of spatial integral equations. Defectoscopy. No. 8, pp. 71–84. http://naukarus.com/chislennoe-modelirovanie-magnitnogo-sostoyaniya-ferromagnetika-v-neodnorodnom-postoyannom-pole-metodom-prostranstvennyh-i.
4. Ignatiev V.K., Orlov A.A. (2014). The inverse magnetostatic problem for ferromagnets. Science and education. January 1, pp. 300-324. http://technomagelpub.elpub.ru/jour/article/viewFile/467/469.
5. Zhirkov V.F., Novikov K.V., Sushkova L.T. Solving the inverse problem of magnetostatics using the Tikhonov regularization method. http://autex.spb.su/download/dsp/dspa/dspa2005/t1/74.pdf.
6. Dyakin V.V., Kudryashova O.V., Raevsky V.Ya. (2017). On the question of the correctness of the direct and inverse problems of magnetostatics. Part 1. / Flaw. No.7, pp. 35-45. https://www.libnauka.ru/journal/defektoskopiya/defektoskopiya-2017-7/voprosu-korrektnosti-pryamoy-i-obratnoy-zadach-magnitostatiki-chast-1-defektoskopiya.
7. Pechenkov A.N. (2007). Algorithms for calculating and modeling direct and inverse problems of magnetostatic flaw detection and technical magnetostatic devices. Doctor of Technical Sciences. Abstract of the candidate of technical sciences. 02/05/11 — Methods of control and diagnostics in mechanical engineering. Yekaterinburg. 42 p.
8. Zhidkov E.P., Perepelkin E.E. (2003). Behavior of the solution of a nonlinear magnetostatic problem in the vicinity of a corner point of a ferromagnet. Mat. Modeling. No. 15 (4). 2003, pp. 77-84 http://www.mathnet.ru/links/8987cc5434c5f5d6fd5efd33fe58812b/mm471.pdf
9. Solution of a nonlinear inverse problem of magnetostatics using the regularization method. E. P. Zhidkov, I. V. Kuts, R. V. Polyakova et al. Dubna: JINR, 1988. 10 p. (Prepr. Combined Institute of Nuclei. Research; R11-88-335). https://search.rsl.ru/ru/record/01001422197
10. Shur M.L., Novoslugina A.P., Smorodinskii Ya.G. (2013). On the Inverse Problem of Magnetostatics. Russian Journal of Nondestructive Testing. Vol. 49, No. 8, pp. 465-473. http://elar.urfu.ru/bitstream/10995/27292/1/scopus-2013-0456.pdf.
11. Shary S.P. Interval methods for regularization of ill-conditioned and ill-posed problems. Institute of Computational Technologies SB RAS Novosibirsk State University. XVIII All-Russian Conference of Young Scientists on Mathematical Modeling and Information Technologies. http://conf.nsc.ru/files/conferences/ym2017/415381/Sharyi_SP-YM2017.pdf.
12. Denisov A.M. (1984). A method for solving equations of the first kind in a Hilbert space. Dokl. Academy of Sciences of the USSR. 1984. Vol. 274, no. 3, pp. 528-530.
15. Verlan A.F., Sizikov V.S. (1986). Integral equations: methods, algorithms, programs. Kiev: Naukova Dumka. 128 p.
16. Trenogin V.A. (1980). Functional analysis. Moscow: Science. 495 p.
17. Dyakin V.V., Kudryashova O.V., Raevskii V.Y. (2015). The problem of the magnetic field is that it is dependent on coordinates. Russian Journal of Nondestructive Testing. V. 51. No 9, pp. 554-562.
18. Krotov L.N. Simulation of the inverse geometric problem of magnetostatics in magnetic control. Author’s abstract diss. on the competition uch. Art. Dr. Phys.-Math. Of science Permian. 2004. http://www.science.by/upload/iblock/915/ 915879eb2bc1677d493a2262187274b0.pdf.
19. Akishin P.G., Sapozhnikov A.A. (2014). The method of volume integral equations in the problem of magnetostatics. Journal RUDN Bulletin. Series: Mathematics, Computer Science, Physics. No.2, pp. 310-315. https://cyberleninka.ru/article/n/metod-obyomnyh-integralnyh-uravneniy-v-zadachah-magnitostatiki.
20. Arutyunyan R.V., Nekrasov S.A., Seredina P.B.(2018). Identification of the magnetization of permanent magnets based on the scalar magnetic potential method. Izv. universities. Electromechanics. No.6.