In this paper, the computation of forces and torques mutually applied between a helical toroidal magnet and a magnet shaped like an angular plane sector is illustrated. The evaluation considers the magnetostatic field hypothesis. The main aim of this study is to furnish a tool for performing fast and accurate evaluation of forces and torques based on the method of the magnetic charges with reference to helical toroidal magnetic systems. The particular geometry of the case study concerns the development of unconventional configurations of electrical machines. These configurations should reduce the magnetic flux changing during the machine operation. A small change of the magnetic flux reduces all the losses associated to the flux variation. The illustrated model for the computation of forces and moments also represents a starting point for a reliable analytical numerical evaluation of the external/internal actions applied to parts of other kinds of helical toroidal systems as stellarator and similar ones.
2. Stumberger, B., G. Stumberger, D. Dolinar, A. Hamler, and M. Trlep, "Evaluation of saturationand ˇ cross-magnetization effects in interior permanent-magnet synchronous motor," IEEE Transactions on Industry Applications, Vol. 39, No. 5, 1264-1271, September/October 2003.
3. Stumberger, B., G. Stumberger, and D. Dolinar, "Analysis of cross-saturation effects in a ˇ linear synchronous reluctance motor performed by finite elements method and measurements," Proceedings of 12th International Conference on Power Electronics and Motion Control, EPE-PEMC 2006, IEEE Conferences Publications, 2006.
4. Ruoho, S. and A. Arkkio, "Partial demagnetization of permanent magnets in electrical machines causedby an inclined field," IEEE Transactions on Magnetics, Vol. 44, No. 7, 1773-1778, July 2008.
5. Fratila, R., A. Benabou, A. Tounzi, and J. C. Mipo, "Nonlinear modeling of magnetization loss in permanent magnets," IEEE Transactions on Magnetics, Vol. 48, No. 11, 2957-2960, November 2012.
6. Muscia, R., "Symmetry of magnetostatic fields generated by toroidal helicoidal magnets," IEEE Transactions on Magnetics, Vol. 51, No. 10, 7002614, October 2015.
7. Brouver, L., S. Caspi, D. Robin, and W. Wan, "3D toroidal field multipoles for curved accelerator magnets," Proceedings of PAC 2013, 907-909, Pasadena, CA, USA, 2013.
8. He, Y., Z. Yang, W. Ba, X. Zhang, G. Zhuang, X. Hu, Y. Pan, and Z. Liu, "Calculation of the poloidal magnetic field configuration for the J-TEXT tokamak," IEEE Transactions on Applied Superconductivity, Vol. 20, No. 3, 1840-1843, 2010.
9. Maviglia, F., R. Albanese, M. De Magistris, P. J. Lomas, S. Minucci, F. G. Rimini, A. C. C. Sips, and P. C. De Vries, "Electromagnetic models of plasma breakdown in the JET tokamak," IEEE Transactions on Magnetics, Vol. 50, No. 2, Febraury 2014.
10. Jin, C. G., T. Yu, Y. Zhao, Y. Bo, C. Ye, J. S. Hu, L. J. Zhuge, S. B. Ge, X. M. Wu, H. T. Ji, and J. G. Li, "Helicon plasma discarge in a toroidal magnetic field of the tokamak," IEEE Transactions on Plasma Science, Vol. 39, No. 11, 3103-3107, November 2011.
11. Albanese, R., G. Artaserse, F. Maviglia, F. Piccolo, and F. Sartori, "Identification of vertical instabilities in the JET tokamak," IEEE Transactions on Magnetics, Vol. 44, No. 6, June 2008.
12. Albanese, R., M. De Magistris, R. Fresa, F. Maviglia, and S. Minucci, "Numerical formulations for accurate magnetic field flow tracing in fusion tokamaks," Proceedings of the 9th International Conference on Computation in Electromagnetics (CEM 2014), 2014.
13. Beidler, D. C., E. Harmeyer, F. Herrnegger, J. Kisslinger, Y. Igitkhanov, and H. Wobig, "Stellarator fusion reactors — An overview," Proceedings of Toki Conference ITC12, December 2001.
14. Probert, P., "High-performance interpolation of stellarator magnetic fields," IEEE Transactions on Plasma Science, Vol. 39, No. 4, 1051-1054, April 2011.
15. Neilson, G. H., D. A. Gates, P. J., Heitzenroeder, S. C. Prager, T. Stevenson, P. Titus, M. D. Williams, and M. C. Zarnstorf, "Facilities for Quasi-Axisymmetric Stellarator research," Proceedings of 25th Symposium on Fusion Engineering (SOFE), 2013 IEEE Conferences Publications, 2013.
16. Imagawa, S., A. Sagara, and Y. Kozaki, "Conceptual design of magnets with CIC conductors for LHD-type reactors FFHR2m," Plasma and Fusion Research, Vol. 3, S1050, 1-5, 2008.
17. Rummel, T., K. Riβe, G. Ehrke, K. Rummel, A. John, T. M¨onnich, K. P. Buscer, W. H. Fietz, R. Heller, O. Neubauer, and A. Panin, "The superconducting magnet system of the stellarator wendlstein 7-X," IEEE Transactions on Plasma Science, Vol. 40, No. 3, 769-776, March 2012.
18. Clark, A. W., F. A. Volpe, and D. A. Spong, "Proto-circus tilted-coil tokamak-stellarator hybrid," Proceedings of 25th Symposium on Fusion Engineering (SOFE), 2013 IEEE Conferences Publications, 2013.
19. Dal Maso, A., Screws with curvilinear axis: A CAD analysis of coupling problems, Thesis, Supervisor: R. Muscia, Department of Engineering and Architecture, University of Trieste, Italy, 2014.
20. Chen, M., K. T. Chau, C. H. T. Lee, and C. Liu, "Design and analysis of a new axial-field magnetic variable gear using pole-changing permanent magnets," Progress In Electromagnetics Research, Vol. 153, 23-32, 2015.
21. Boutora, Y., N. Takorabet, and R. Ibtiouen, "Analytical model on real geometries of magnet bars of surface permanent magnet slotless machine," Progress In Electromagnetics Research B, Vol. 66, 31-47, 2016.
22. Sun, X., S. Luo, L. Chen, R. Zhao, and Z. Yang, "Suspension force modelling and electromagnetic characteristics analysis of an interior bearingless permanent magnet synchronous motor," Progress In Electromagnetics Research B, Vol. 69, 31-45, 2016.
23. Brown, Jr., W. F.c, "Electric and magnetic forces: A direct calculation I," Am. J. Phys., Vol. 19, 290-304, 1951.
24. Brown, Jr., W. F., "Electric and magnetic forces: A direct calculation II," Am. J. Phys., Vol. 19, 333-350, 1951.
25. Muscia, R., "Equivalent magnetic charge in helicoidal magnets," J. Appl. Phys., Vol. 104, 103916, 2008.
26. Muscia, R., "Computation of the magnetic field generated by helical toroidal permanent magnets," Electromagnetics, Vol. 32, 8-30, 2012.
27. Furlani, E. P., Permanent Magnet and Electromechanical Devices, Material, Analysis, and Applications, 136, Eq. (3.118), Academic Press, 2001.
28. Furlani, E. P., "A formula for the levitation force between magnetic disks," IEEE Transactions on Magnetics, Vol. 29, No. 6, 4165-4169, November 1993.
29. Furlani, E. P., "Formulas for the force and torque of axial coupling," IEEE Transactions on Magnetics, Vol. 29, No. 5, 2295-2301, September 1993.
30. Furlani, E. P., R. Wang, and H. Kusnadi, "A three-dimensional model for computing the torque of radial couplings," IEEE Transactions on Magnetics, Vol. 31, No. 5, 2522-2526, September 1995.
31. Mathematica 10.3, [Online] Available: http://www.wolfram.com/mathematica.