ON APPROACHES TO THE CALCULATION OF THE MAGNETIC CIRCUIT OF AN ELECTRIC MACHINE BASED ON THE USE OF MAGNETIZATIONS OF FERROMAGNETIC SECTIONS OF ITS CORES

DOI: 10.47026/1810-1909-2025-2-5-16

УДК 621.313-523.2-047.72

ББК З26-028

Aleksandr A. AFANASYEV, Nadezhda N. IVANOVA,
Vladimir A. VATKIN, Dmitry A. TOKMAKOV

Key words

intensities, lengths, magnetic stresses of circuit sections, parallelepipeds, ferromagnetic media, magnetizations, Laplace equation, method of separation of variables, Fourier constants.

Abstract

Despite the fact that modern electric motors have a fairly high efficiency, it is far from ideal. This coefficient can only be increased based on accurate calculations at the design stage of electrical machines. New approaches to the calculation of magnetic circuits of electric motors make it possible to clarify and supplement the classical theory used for these purposes.

The purpose of the study is a comparative analysis of the traditional approach to the electromagnetic calculation of an electric machine, based on the use of the strength and magnetic voltage of the magnetic field in ferromagnetic elements of a magnetic circuit, and a new approach related to the use of solutions to boundary problems of mathematical physics for volumetric fragments of a machine with an air and ferromagnetic magnetized medium.

Materials and methods. Theoretical and empirical methods were used in the study. Calculations were made for a 12DVM250 magnet motor with a capacity of 150 kW. To automate the modeling process, the Mathcad computer mathematics system was used.

Results. The paper compares two approaches to the electromagnetic calculation of electrical machines. The first one is based on the traditional calculation of a nonlinear magnetic circuit, which is considered from the standpoint of a two-dimensional nonlinear electric circuit: magnetic flux and resistance are in the first approximation, they are analogues of the electric current and nonlinear resistor. In the topological sense, a magnetic circuit is fixed as the path of closing the lines of force of a magnetic field is within its period. As a result of the calculation, the maximum values of magnetic induction in the elements of the magnetic circuit are found: in the air gap, yokes and teeth of the stator and rotor. The equilibrium of magnetic field sources (magnetomotive force of windings and magnets) and magnetic voltage drops in the above elements of the magnetic circuit is also recorded. The second approach is more rigorous from the mathematical point of view: the active part of an electric machine is represented as a set of two- or three-dimensional geometric spaces with a ferromagnetic and air medium, the magnetic field in each of which can be calculated as a solution to the Dirichlet or Neumann boundary problem. The means of adaptation to the boundary problems of mathematical physics are the boundary conditions of the magnetic field for the active ferromagnetic zones of the machine: scalar magnetic potentials and magnetic inductions do not undergo a jump (discontinuity). If the magnetic sheets of the windings are located at the boundaries for reasons of convenience of calculation, the magnetic potentials have a jump equal to the value of the total current of the magnetic sheet. As a result of the calculation in accordance with the second approach, two- or three-dimensional graphs of the distribution of magnetic induction (or magnetization of ferromagnetic media) in the active zones were obtained: in the air gap, yokes, tooth layers of the stator and rotor of an electric machine.

Conclusion. The introduction of magnetization of bulk structures with a ferromagnetic medium into the calculation allows us to take into account the nonlinear properties and geometric features of these elements of the magnetic circuit of an electric machine. At standard levels of magnetic induction in the machine’s active media, dictated by permissible magnetic and electrical losses, the relationship between induction and magnetization of a ferromagnetic medium can be considered linear. The use of the second approach to the electromagnetic calculation of an electric machine makes it possible to obtain more detailed and mathematically sufficiently rigorous information on the nature of the magnetic field distribution in its active parts.

References

1. Avtaev S.N., Kashtanov I.V., Kuzina E.A. Povyshenie KPD asinkhronnogo elektricheskogo privoda izmeneniem sposobov regulirovaniya [Increasing the efficiency of an induction electric drive by changing the methods of regulation]. In: L Ogarevskie chteniya: materialy Vseros. s mezhdunar. uchastiem nauch. konf.: v 3 ch. [Proc. of Russ. Sci. Conf. «Ogarev Readings». 3 parts]. Saransk, 2022, part 1, pp. 12–17.
2. Afanasyev A.A. Matematicheskie modeli elektricheskikh mashin v dvukh- i trekhmernykh formatakh [Mathematical Models of Electric Machines in Two- and Three-Dimensional Formats]. Cheboksary, 2024, 278 p.
3. Bogolyubov A.N., Kravtsov V.V. Zadachi po matematicheskoi fizike [Problems in mathematical physics]. Moscow, 1998, 350 p.
4. Budak B.M., Fomin S.V. Kratnye integraly i ryady [Multiple Integrals and Series]. Moscow, Nauka Publ., 1965, 608 p.
5. Govorkov V.A. Elektricheskie i magnitnye polya [Electric and magnetic fields]. Moscow, Energia Publ., 1958, 488 p.
6. Katsman M.M. Elektricheskie mashiny. Spravochnik [Electrical machines. Handbook]. Moscow, KnoRus Publ., 2022, 480 p.
7. Lebedev A.I., Radaev A.V. Puti povysheniya energeticheskoi effektivnosti elektroenergeticheskikh sistem sudov s elektrodvizheniem [Ways to improve energy efficiency of electric power systems of ships with electric propulsion]. Vestnik Astrakhanskogo gosudarstvennogo tekhnicheskogo universiteta. Ser. Morskaya tekhnika i tekhnologiya, 2022, no. 1, pp. 58–66. DOI: 10.24143/2073-1574-2022-1-58-66.
8. Moskalev Yu.V., Milyutin A.Yu. Sposoby i tekhnicheskie sredstva povysheniya energeticheskoi effektivnosti sistem elektroprivoda [Methods and Technical Means of Improving the Energy Efficiency of Electric Drive Systems]. In: Innovatsionnye proekty i tekhnologii v obrazovanii, promyshlennosti i na transporte: materialy XVI nauch. konf., posvyashch. Dnyu Rossiiskoi nauki (Omsk, 8 fevralya 2022 g.) [Proc. of 16^th Conf. «Innovative Projects and Technologies in Education, Industry and Transport»]. Omsk, Omsk State University of Communications, 2022, pp. 313–317.
9. Petrov G.N. Elektricheskie mashiny. 2. Asinkhronnye i sinkhronnye elektricheskie mashiny [Electrical machines. Part 2. Asynchronous and synchronous electrical machines]. Moscow, Leningrad, Gosenergoizdat Publ., 1963, 416 p.
10. Maksimov N.M., Golovan’ I.N., Kushnarev V.A., Bukhtoyarov V.F. Povyshenie energoeffektivnosti asinkhronnoi elektricheskoi mashiny za schet optimizatsii udel’nykh pokazatelei [Increasing the energy efficiency of an induction motor by optimizing specific indicators]. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Ser. Energetika, 2023, vol. 23, no. 4, pp. 56–65. DOI: 10.14529/power230406.
11. Polivanov K.M. Teoreticheskie osnovy elektrotekhniki. 3. Teoriya elektromagnitnogo polya [Theoretical foundations of electrical engineering. Part 3. Theory of the electromagnetic field]. Moscow, Energia Publ., 1969, 352 p.
12. Polyanin A.D. Spravochnik po lineinym uravneniyam matematicheskoi fiziki [A Handbook of Linear Equations in Mathematical Physics]. Moscow, Fizmatlit Publ., 2001, 576 p.
13. Kopylov I.P. ed., Kopylov I.P., Goryainov F.A., Klokov B.K. et al. Proektirovanie elektricheskikh mashin [Electrical Machine Design]. Moscow, Energia Publ., 1980, 496 p.
14. Tamm I.E. Osnovy teorii elektrichestva [Fundamentals of the Theory of Electricity]. Moscow, Nauka Publ., 1989, 504 p.
15. Farlow St.J. Partial differential egnations for scientists and engineers. Wileu, 1982 (Russ. ed.: Uravneniya s chastnymi proizvodnymi dlya nauchnykh rabotnikov i inzhenerov. Moscow, Mir Publ., 1985, 384 p.).
16. Molotilov B.V., ed., Molotilov B.V., Petrenko A.G. et al. Kholodnokatanye elektrotekhnicheskie stali [Cold-rolled electrical steels]. Moscow, Metallurgiya Publ., 1989, 168 p.
17. Kholodova S.E., Peregudin S.I. Dopolnitel’nye razdely vysshei matematiki [Additional Sections of Higher Mathematics]. St. Petersburg, ITMO University Publ., 2020, 89 p.
18. Gundabattini E., Kuppan R., Solomon D.G. et al. A review on methods of finding losses and cooling methods to increase efficiency of electric machines. Ain Shams Engineering Journal, 2021, vol. 12(1), pp. 497–505.
19. El Hadraoui H., Zegrari M., Chebak A. et al. A Multi-Criteria Analysis and Trends of Electric Motors for Electric Vehicles. World Electric Vehicle Journal, 2022, vol. 13(4), 65. DOI: 10.3390/wevj13040065.

Information about the authors

Aleksandr A. Afanasyev – Doctor of Technical Sciences, Professor, Department of Automation and Control in Technical Systems, Chuvash State University, Russia, Cheboksary (afan39@mail.ru).

Nadezhda N. Ivanova – Candidate of Technical Sciences, Associate Professor, Department of Mathematical and Hardware Support of Information Systems, Chuvash State University, Russia, Cheboksary (niva_mail@mail.ru; ORCID: https://orcid.org/0000-0001-7130-8588).

Vladimir A. Vatkin – Candidate of Technical Sciences, Chief Designer, Department of Electrical Machines, JSC «ChEAZ», Russia, Cheboksary (vatkinv@yandex.ru).

Dmitry A. Tokmakov – Director for Development, JSC «ChEAZ», Russia, Cheboksary (tokmakov_da@mail.ru).

For citations

Afanasyev A.A., Ivanova N.N., Vatkin V.A., Tokmakov D.A. On approaches to the calculation of the magnetic circuit of an electric machine based on the use of magnetizations of ferromagnetic sections of its cores. Vestnik Chuvashskogo universiteta, 2025, no. 2, pp. 5–16. DOI: 10.47026/1810-1909-2025-2-5-16 (in Russian).

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