# MATHEMATICAL APPARATUS FOR MODELING OF THE PROPAGATION THE MAGNETIC FIELD ELECTRIC MACHINES WITH A GIVEN ACCURACY

## Main Article Content

## Abstract

**. **The problem of modeling the propagation local magnetic fields and spatially dispersed sources is large errors compared to field measurements. An important aspect of adequate modeling is the use of the correct mathematical apparatus. It is shown that in order to obtain reliable models of the propagation magnetic fields around electrical machines (generators, electric motors of different power, geometric dimensions and poles), it is advisable to apply the Gauss equation for a scalar potential. The solution of the equation in polar coordinates makes it possible to take into account not only the fundamental, but also other harmonics of the magnetic field (dipole, quadrupole, octupole). This allows, depending on the number of spatial harmonics taken into account, to obtain a model with the required accuracy (error) for predicting the magnetic field strength at any point around the machine. It is considered in the paper that an electronic machine is an object of base radius R_{0}. The presented approach makes it possible to unambiguously determine the location of zero field points at a distance from the source (for a quadrupole source and zero field lines, for an octupole source). The results of modeling and their verification by full-scale measurements for the most common four-pole machines (quadrupole source) are presented. The main task of modeling the propagation the magnetic field of such sources is to ensure the required accuracy based on the goals of modeling. It is shown that the modeling accuracy and the presence of zero field points are due to different field levels near the electrical machine housing for different harmonics. The dipole harmonic at the cabinet is 20% of its own harmonic. But it falls more slowly with distance. This necessitates taking into account a different number of harmonics depending on the value of the ratio R_{0}/R, R is the distance to the point of determining the field strength from the source. Therefore, with the ratio R_{0}/R=2/3, the eighth harmonic is essential. At R0/R=1/5, already the fourth spatial harmonic can be neglected. Such data allow you to choose a rational number of harmonics. This reduces the amount of calculations and simplifies the process of modeling the propagation of the magnetic field around the source.

## Article Details

*Advanced Information Systems*,

*6*(2), 5–9. https://doi.org/10.20998/2522-9052.2022.2.01

## References

Podoltsev, O.D. and Kucheriava I.N. (2015), ”Muljtyfyzycheskye processы v oblasty vkljuchenyja polyэtylenovoj yzoljacyy sylovogho kabelja [Multiphysical processes in the field of inclusion of polyethylene insulation of a power cable]”, Tekhnichna Elektrodynamika, Iss. 3, pp. 3–9 (In Russian).

Sukach, S., Riznik, D., Zachepa, N. and Chenchevoy, V. (2020), “Normalization of the Magnetic Fields of Electrical Equipment in Case of Unauthorized Influence on Critical Information Infrastructure Facilities”, Soft Target Protection, NATO Science for Peace and Security Series C: Environmental Security, Theoretical Basis and Practical Measures, pp. 337–349, URL: https://www.springerprofessional.de/normalization-of-the-magnetic-fields-of-electrical-equipment-in-/17766772/.

Levchenko, L.O., Sukach ,S.V. and Konovalova, O.V. (2014), ”Modeljuvannja prostorovykh rozpodiliv maghnitnykh poliv elektrychnykh mashyn dlja vyznachennja zon bezpechnogho perebuvannja personalu [Modeling of spatial distributions magnetic fields electric machines to determine areas of safe stay of personnel]”, Transactions of Kremenchuk Mykhailo Ostrohradskyi National University, Is. 6 (89), Part. 1, pp. 27–31 (In Ukrainian).

Primin, M. A. and Nedaivoda, I.V. (2015), ”Alghorytm anatomycheskogho reshenyja obratnoj zadachy maghnytostatyky dlja ystochnyka polja dypoljnogho typa [Algorithm for the anatomical solution of the inverse problem magnetostatics for a dipole-type field source]”, Computing tools, networks and systems, Is. 14, pp. 5−15 (In Russian).

Khodakovsyi, O., Levchenko, L., Kolumbet, V. and Kozachuk, A. (2021), ”Rozrakhunkovyj aparat modeljuvannja poshyrennja elektromaghnitnykh poliv riznoridnykh dzherel [Computational apparatus for modeling the propagation of electromagnetic fields dissimilar sources]”, Advanced information systems, Vol. 5, Iss. 1, pp. 34–38, DOI: https://doi.org/10.20998/2522-9052.2021.1.04 (In Ukrainian).

Volokhov, S. A. and Dobrodeev, P.N. (2006), ”Закономерности распределения внешнего магнитного поля электрооборудований [Patterns of distribution of the external magnetic field of electrical equipment]”, Electrical Engineering, Is. 4, pp. 28–33 (In Russian).

Getman, A. V, (2011), ”O normyrovanyy urovnja maghnytnogho polja s pomoshhjju muljtypoljnykh maghnytnykh momentov [On normalization of the magnetic field level using multipole magnetic]”, Eastern European Journal of Advanced Technology, Vol. 4, No 5., pp. 7–10 (In Russian).

Getman, A. (2018), ”Cylindrychnyj gharmonichnyj analiz maghnitnogho polja v aperturi nadprovidnoji obmotky elektromaghnita [Development of a method for reducing the structure of the magnetic field in the aperture of a quadrupole electromagnet with an overwire winding]”, Eastern-European Journal of Enterprise Technologies, 1(5 (91), pp. 4–9, DOI: https://doi.org/10.15587/1729-4061.2018.123607 (In Ukrainian).

Getman, A. (2018), ”Rozrobka sposobu pokrashhennja struktury maghnitnogho polja v aperturi kvadrupoljnogho elektromaghnitu z nadprovidnoju obmotkoju [Development of a method for improving the structure of the magnetic field in the aperture of a quadrupole electromagnet with a superconducting winding]”, Eastern-European Journal of Enterprise Technologies, 5(5 (95), pp. 6–12, DOI: https://doi.org/10.15587/1729-4061.2018.142163 (In Ukrainian).