Manyetik nüve içerisinde çok sarımlı bobin probleminin küresel koordinatlarda analitik çözümü ve FEA sonuçları ile karşılaştırılması

Öz

Teknolojinin gelişimi ile birlikte hareket ve kontrol mekanizmaları çok eksende hareket edebilen, daha hızlı ve hassas hareket sağlanabilecek cihazlara ihtiyaç duymaktadır. Bu nedenle; küresel koordinat sisteminde elektromanyetik sistemlerin parametrelerinin analitik veya yarı analitik yöntemlerle hesaplanması son yıllarda önemli araştırma konularından biri haline gelmiştir. Bu çalışmada manyetik bir nüve içerisine sarılmış çok sarımlı bobin yapısı incelenmiştir. Öncelikle tek sarımlı bobin yaklaşımından yararlanarak çok sarımlı bobinler için B, E ve A ifadeleri analitik olarak hesaplanmıştır. Aynı geometriler, sonlu elemanlar analizi (FEA) kullanılarak ANSYS Maxwell programında dairesel simetri kabulü ile hesaplanmıştır. Son olarak, biri r1 yarıçapına sahip manyetik nüve içerisinde olan, eş merkezli iki bobin geometrisi belirlenmiştir. Bu geometriler için öz indüktans (Lii) ve karşılıklı indüktans (Mij) katsayıları küresel koordinatlarda γ açısına bağlı olarak incelenmiştir. ANSYS Maxwell programı üzerinde bobin geometrilerinin 3 boyutlu modeli oluşturularak benzetim çalışmaları yapılmıştır. FEA ve analitik sonuçlar kıyaslanarak geçerliliği gösterilmiştir.

Anahtar Kelimeler

• Küresel koordinatlar
• Maxwell denklemleri
• Sonlu elemanlar analizi (FEA)
• Karşılıklı indüktans

Analytical solution of multi-winding coil problem in magnetic core in spherical coordinates and comparison with FEA results

Öz

With the development of technology, motion and control mechanisms need devices that can move in multiple axes and provide faster and more precise movement. So, the calculation of the parameters of electromagnetic systems in the spherical coordinate system by analytical or semi-analytical methods has become one of the important research topics. In this study, the structure of the multi-winding coil which is in the magnetic core was investigated. First of all, the expressions B, E and A for multi-winding coils were calculated analytically by using the single-winding coil approach. The same geometries were calculated using finite element analysis (FEA) in the ANSYS Maxwell program with circular symmetry assumption. Finally, two concentric coil geometries, one of which is inside the magnetic core with radius r1, are determined. The coefficients of self-inductance (Lii) and mutual inductance (Mij) were investigated in spherical coordinates depending on the γ angle. Simulation studies were carried out by creating a 3D model of coil geometries on the ANSYS Maxwell program. FEA and analytical results were compared and validated.

Anahtar Kelimeler

• Spherical coordinates
• Magnetic fields
• Maxwell’s equations
• Finite element analysis‎ (FEA)
• mutual inductance

Kaynakça

1. Conway J. T., Mutual inductance for an explicitly finite number of turns, Progr. Electromagn. Res. B, vol. 28, pp. 273–287, 2011.
2. Ravaud R., Lemarquand G., Lemarquand V., Babic S. and Akyel C., Mutual inductance and force exerted between thick coils, Progress in Electromagnetics Research, PIER 102, pp. 367–380, 2010
3. Babic S. I., Akyel C., New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air, IEEE Transactions On Magnetics, Vol. 42, No. 6, June 2006 1661
4. Conway J. T., Noncoaxial inductance calculations without the vector potential for axisymmetric coils and planar coils, IEEE Transactions On Magnetics, Vol. 44, No. 4, Aprıl 2008 453
5. Conway J. T., Analytical solutions for the self- and mutual inductances of concentric coplanar disk coils, IEEE Transactions On Magnetics, Vol. 49, No. 3, March 2013 1135
6. YANG T. T., YANG J. J., The effect of cylindrical ferromagnetic shells on the self and mutual inductance of parallel wires, IEEE Transactıons On Electromagnetıc Compatıbılıty, Vol. Emc-17, No. 4, November 1975
7. Babic S. I. and Akyel C., Calculating mutual inductance between circular coils with inclined axes in air, IEEE Transactions On Magnetics., vol. 44, no. 7, pp. 1743–1750, Jul. 2008.
8. Conway J. T., Exact solutions for the mutual inductance of circular coils and elliptic coils, IEEE Transactions On Magnetics, Vol. 48, No. 1, January 2012

9. Kuang, S., Yan, G. Modelling on mutual inductance of wireless power transfer for capsule endoscopy. Biomed Microdevices 22, 54, 2020.
10. Zhou X, Chen B, Luo Y, Zhu R. Analytical calculation of mutual inductance of finite-length coaxial helical filaments and tape coils. Energies., 12(3):566, 2019.
11. Weber H., Baran H., Utermöhlen F. and Schuster C., Macromodeling of mutual inductance for displaced coils based on laplace’s equation, in IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-11, Art no. 9506911, 2021.
12. Zhang X., Quan C. and Li Z., Mutual inductance calculation of circular coils for an arbitrary position with electromagnetic shielding in wireless power transfer systems, IEEE Transactions on Transportation Electrification, vol. 7, no. 3, pp. 1196-1204, 2021.
13. Lipiriski W., Rolicz P., R. Sikora, Application of integral transforms to the analysis of the magnetic field of a spherical coil, IEEE Transactions on Magnetics, vol. Mag-11, no. 5, 1975.
14. Semenov V. G., Synthesıs of spherical methods of determining magnetic field source parameters of internal and external origin, Measurement Techniques, Volume 33, Issue 12, pp 1236–1240 , 1990.
15. Eaton, H., Electric field induced in a spherical volume conductor from arbitrary coils: application to magnetic stimulation and MEG. Medical and Biological Engineering and Computing, 30(4), 433-440, 1992.
16. Matute E.A., On the vector solutions of Maxwell equations in spherical coordinate systems, Rev. Mex. Fis. E 51, 31-36, 2005. arXiv:physics/0512261v1 [physics.class-ph] 29, Dec 2005.
17. Liua C. Y., Andalib T., Ostapchuk D.C.M. and Bidinostia C.P., Analytic models of magnetically enclosed spherical and solenoidal coils, arXiv:1907.03539v1 [physics.acc-ph] 2 Jul 2019.
18. Kok-Meng Lee, Hungsun Son and J. Joni, Concept development and design of a spherical wheel motor (SWM), Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pp. 3652-3657, 2005.
19. B. Dehez, G. Galary, D. Grenier and B. Raucent, Development of a spherical induction motor with two degrees of freedom, IEEE Transactions on Magnetics, vol. 42, no. 8, pp. 2077-2089, 2006.
20. J. F. P. Fernandes and P. J. C. Branco, The shell-like spherical induction motor for low-speed traction: electromagnetic design, analysis, and experimental tests, IEEE Transactions on Industrial Electronics, vol. 63, no. 7, pp. 4325-4335, 2016.
21. C. Zhang et al., Analytical models of electromagnetic field and torques in a novel reaction sphere actuator, 2018 IEEE International Conference on Applied System Invention (ICASI), pp. 271-274, 2018.
22. J. Zhang et al., Torque optimization of a novel reaction sphere actuator based on support vector machines, IEEE International Conference on Applied System Invention (ICASI), 2018, pp. 263-266, 2018.
23. Yıldız H. , Uzal E. , Çalık H., An analytical solution of a multi-winding coil problem with a magnetic core in spherical coordinates, Acta Polytechnica Hungarica, vol.18, no.10, pp.87-112, 2021.
24. Griffiths, D. J., 1998. Introduction to Electrodynamics, Prentice Hall, New Jersey, 3th ed., ISBN 0-13-805326-X.
25. Jackson, J. D., Classical Electrodynamics, John Wiley & Sons, Inc, New York, Chapter 5, LCCCN:62-8774, 1962.
26. Clayton P. R., Inductance Loop and Partial, John Wiley & Sons, ISBN 978-0-470-46188-4, New Jersey A.B.D., Chapter 3, 2010.
27. Smythe W. R., Static and dynamic electricity, Taylor & Francis Publisher, ISBN 0-89116-916-4., New York A.B.D., 1989.
28. Theodoulidis T. P. and Kriezis E. E., Eddy current canonical problems (with applications to nondestructive evaluation), Tech Science Press, 978-0971788015, 2006.