Analytical Solution of a Multi-Winding Coil Problem With an Air Core in Spherical Coordinates

Yıl 2024, Cilt: 27 Sayı: 1 , 255 - 261 , 29.02.2024

Hüseyin Yıldız , Erol Uzal Hüseyin Çalık

https://doi.org/10.2339/politeknik.1052918

https://izlik.org/JA56RM35UM

Öz

In this study, electromagnetic fields and mutual inductance of two interacting coils were computed analytically, and the results were compared with numerical solutions and measurements. For the analytical solution, the fields were expressed in spherical coordinates in each coil’s frame. The total field due to two coils was calculated using superposition, resulting in infinite series involving associated Legendre functions. The numerical computations were carried out by finite element analysis capabilities of ANSYS Maxwell. At the end, the self-inductances and the mutual inductance coefficients between the coils (which depend on the angle between the coils) were computed. The results (analytical, finite element method, and the measurements) were compared, obtaining good agreement for various relative positions of the coils. 

Anahtar Kelimeler

Magnetic field , Maxwell’s equations , mutual inductance , spherical coordinates

Hava Çekirdekli Çok Sargılı Bobbin Porbleminin Küresel Koordinatlarda Analitik Çözümü

https://izlik.org/JA56RM35UM

Öz

Bu çalışmada, bir biri ile etkileşen iki bobinin elektromanyetik alanı ve karşılıklı endüktans katsayısı analitik olarak hesaplanmış, sonuçlar sayısal çözümler ve ölçümlerle karşılaştırılmıştır. Analitik çözüm her bir bobinin etkisi küresel koordinatlarda genişletilerek hesaplandı. Her iki bobinden kaynaklanan toplam alanlar, süperpozisyon ilkesi gereği uyarlanmış legendre fonksiyonlarını içeren seri çözümler olarak elde edildi. Sayısal hesaplamalar ANSYS Maxwell programında sonlu elemanlar metodu kullanılarak elde edildi. Bobinlerin öz endüktans ve karşılıklı endüktans katsayıları (bobinler arasındaki açıya bağlı olarak) elde edildi. Sonuç olarak, elde edilen sonuçlar (analitik, sonlu elemanlar yöntemi ve ölçümler) kayaslanarak, bobinlerin çeşitli göreli konumları için analitik formüllerin geçerli oldukları gösterildi.

Anahtar Kelimeler

Manyetik alan , karşılıklı endüktans , küresel koordinatlar

Kaynakça

• [1] Ravaud, R., Lemarquand, G., Lemarquand, V., Babic, S., & Akyel, C., “Mutual inductance and force exerted between thick coils”, Progress In Electromagnetics Research, 102: 367-380, (2010).
• [2] Conway, J. T., “Mutual inductance for an explicitly finite number of turns”, Progress In Electromagnetics Research B, 28: 273-287, (2011)
• [3] Babic S. I. and 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, 42(6): 1661- 1669., (2006)
• [4] Conway, J. T., “Exact solutions for the mutual inductance of circular coils and elliptic coils”, IEEE transactions on magnetics, 48(1): 81-94, (2012)
• [5] Lipinski, W., Rolicz, P., & Sikora, R., “Application of integral transforms to the analysis of the magnetic field of a spherical coil”, IEEE Transactions on Magnetics, 11(5): 1552-1554, (1975).
• [6] Semenov, V. G., “Synthesis of spherical methods of determining magnetic field source parameters of internal and external origin”, Measurement Techniques, 33(12): 1236-1240, (1990).
• [7] 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).
• [8] Matute, E. A., “On the vector solutions of Maxwell equations in spherical coordinate systems”. arXiv preprint physics, 0512261,(2005).
• [9] Liu, C. Y., Andalib, T., Ostapchuk, D. C. M., & Bidinosti, C. P., “Analytic models of magnetically enclosed spherical and solenoidal coils”, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 949: 162837, (2020).
• [10] Lee, K. M., Son, H., & Joni, J., “Concept development and design of a spherical wheel motor (SWM)”, In Proceedings of the 2005 IEEE International Conference on Robotics and Automation: 3652- 3657, (2005).
• [11] Dehez, B., Galary, G., Grenier, D., & Raucent, B.,”Development of a spherical induction motor with two degrees of freedom”, IEEE Transactions on Magnetics, 42(8): 2077-2089, (2006)
• [12] Fernandes, J. F., & Branco, P. C., “The shell-like spherical induction motor for low-speed traction: electromagnetic design, analysis, and experimental tests”. IEEE transactions on industrial electronics, 63(7): 4325-4335, (2016).
• [13] Zhang, C., Yuan, L., Zhang, J., Chen, J., Chen, C. Y., Chen, S., & Yang, G., “Analytical models of electromagnetic field and torques in a novel reaction sphere actuator”, In 2018 IEEE International Conference on Applied System Invention (ICASI), 271-274, (2018a).
• [14] Zhang, J., Yuan, L., Liao, Y., Zhang, C., Chen, C. Y., Chen, S., & Yang, G., “Torque optimization of a novel reaction sphere actuator based on support vector machines”. In 2018 IEEE International Conference on Applied System Invention (ICASI), 263-266, (2018b).
• [15] Zhou, H., Fang, Y., & Ma, J. Structural Design and FEA of a New Type of Multi-DOF Spherical Induction Motor without Output Shaft. In 2019 22nd International Conference on Electrical Machines and Systems (ICEMS), (1-7). IEEE., (2019a, August).
• [16] Akkaya Oy, S. , Arslan, S. & Gürdal, O., “Finite Element Analysis of the Inductance and Magnetic Field in the Permanent Magnet Spherical Motor”, Politeknik Dergisi , 23 (4) , 1387-1394 (2020)
• [17] Kuang, S., & Yan, G., “Modelling on mutual inductance of wireless power transfer for capsule endoscopy”, Biomedical Microdevices, 22(3), 1-11, (2020)
• [18] Jackson, J. D., “Classical electrodynamics”, John Wiley & Sons, ISBN-13: 978-0471309321, Inc, New York, (1962).
• [19] Griffiths, D. J., “Introduction to Electrodynamics”, Prentice Hall, ISBN 0-13-805326-X, New Jersey, (1998).
• [20] Smythe, W. R., “Static and dynamic electricity”, Taylor & Francis Publisher, ISBN-13: 978-0891169161, New York, (1989).
• [21] Hacıoglu, A. , Yılmazer, H. & Ustundag, C. B., “3D Printing for Tissue Engineering Applications”, Politeknik Dergisi, 21 (1) , 221-227, (2018).
• [22] Seçkin, M. , Seçkin, A. Ç. & Yaman Turan, N., “Investigation of Discharge Characteristics of Hinges Produced with 3D Printing for Prosthetic Fingers” . Politeknik Dergisi, 24 (2) , 575-583, (2021).