volume | PIER Journals

Abstract

This paper uses a three-dimensional analytical approach based on the Coulombian model for studying the magnetic field produced by cylindrical Halbach structures. Such structures, commonly used in magnetic couplings or in electrical machines, are composed of tile permanent magnets with rotating magnetizations. Such assemblies of tile permanent magnets allow one to easily optimize the radial field shape in the air gap of electrical machines. In addition, Halbach structures can be used in magnetic couplings for improving the torque transmitted between the two rotors. Analytical studies dealing with the optimization of such structures generally use a twodimensional analytical approach for calculating either the magnetic field produced by tile permanent magnets or the forces exerted between them. These two-dimensional expressions are useful because they have a very low computational cost. However, their accuracy depends greatly on the structure dimensions. We propose in this paper to use a three-dimensional analytical model based on the Coulombian model for determining the exact shape of the magnetic field produced by a Halbach structure. Such an approach also allows one to determine the demagnetizing magnetic field inside the tile permanent magnets. This element of information is important for the design of tile permanent magnets. In addition, we show that some effects cannot be predicted with the linearized analytical model. This implies that a linearized dimensional optimization is not accurate. This study has been carried out without any simplifying assumptions. Therefore, the calculations of the three magnetic field components are exact for all points in space, whatever the magnet dimensions. We can say that such a three-dimensional analytical approach is a good alternative to a finite element one because it has a lower computational cost and is more accurate.

1. Halbach, K., "Strong rare earth cobalt quadrupoles," IEEE Trans. Magn., Vol. 26, No. 3, 3882-3884, 1979.

2. Halbach, K., "Design of permanent multiple magnets with oriented rec material," Nucl. Inst. Meth., Vol. 169, 1-10, 1980.
doi:10.1016/0029-554X(80)90094-4

3. Marinescu, M. and N. Marinescu, "Compensation of anisotropy effects in flux-confining permanent-magnet structures," IEEE Trans. Magn., Vol. 25, No. 5, 3899-3901, 1989.
doi:10.1109/20.42470

4. Marinescu, M. and N. Marinescu, "Anisotropy effects in permanent multiple magnets," IEEE Trans. Magn., Vol. 20, No. 5, 3882-3884, 1984.
doi:10.1109/TMAG.1984.1063430

5. Marinescu, M. and N. Marinescu, "New concept of permanentmagnet excitation for electrical machines," IEEE Trans. Magn., Vol. 28, 1390-1393, 1992.
doi:10.1109/20.123952

6. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "Analytical calculation of the magnetic field created by permanent-magnet rings," IEEE Trans. Magn., Vol. 44, No. 8, 1982-1989, 2008.
doi:10.1109/TMAG.2008.923096

7. Babic, S. and C. Akyel, "Improvement of the analytical calculation of the magnetic field produced by permanent magnet rings," Progress In Electromagnetics Research C, Vol. 5, 71-82, 2008.

8. Furlani, E. P., Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications, Academic Press, 2001.

9. Selvaggi, J. P., S. Salon, O. M. Kwon, and M. Chari, "Computation of the three-dimensional magnetic field from solid permanent-magnet bipolar cylinders by employing toroidal harmonics," IEEE Trans. Magn., Vol. 43, No. 10, 3833-3839, 2007.
doi:10.1109/TMAG.2007.902995

10. Azzerboni, B. and G. Saraceno, "Three-dimensional calculation of the magnetic field created by current-carrying massive disks," IEEE Trans. Magn., Vol. 34, No. 5, 2601-2604, 1998.
doi:10.1109/20.717601

11. Rakotoarison, H. L., J. P. Yonnet, and B. Delinchant, "Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization," IEEE Trans. Magn., Vol. 43, No. 4, 1261-1264, 2007.
doi:10.1109/TMAG.2007.892316

12. Durand, E., Magnetostatique, Masson Editeur, Paris, France, 1968.

13. Babic, S. and C. Akyel, "Magnetic force calculation between thin coaxial circular coils in air," IEEE Trans. Magn., Vol. 44, No. 4, 445-452, 2008.
doi:10.1109/TMAG.2007.915292

14. Babic, S., C. Akyel, S. Salon, and S. Kincic, "New expressions for calculating the magnetic field created by radial current in massive disks," IEEE Trans. Magn., Vol. 38, No. 2, 497-500, 2002.
doi:10.1109/20.996131

15. Babic, S., S. Salon, and C. Akyel, "The mutual inductance of two thin coaxial disk coils in air," IEEE Trans. Magn., Vol. 40, No. 2, 822-825, 2004.
doi:10.1109/TMAG.2004.824810

16. Furlani, E. P., S. Reznik, and A. Kroll, "A three-dimensonal field solution for radially polarized cylinders," IEEE Trans. Magn., Vol. 31, No. 1, 844-851, 1995.
doi:10.1109/20.364587

17. Furlani, E. P., "Field analysis and optimization of ndfeb axial field permanent magnet motors," IEEE Trans. Magn., Vol. 33, No. 5, 3883-3885, 1997.
doi:10.1109/20.619603

18. Furlani, E. P. and M. Knewston, "A three-dimensional field solution for permanent-magnet axial-field motors," IEEE Trans. Magn., Vol. 33, No. 3, 2322-2325, 1997.
doi:10.1109/20.573849

19. Furlani, E. P., "A two-dimensional analysis for the coupling of magnetic gears," IEEE Trans. Magn., Vol. 33, No. 3, 2317-2321, 1997.
doi:10.1109/20.573848

20. Mayergoyz, D. and E. P. Furlani, "The computation of magnetic fields of permanent magnet cylinders used in the electrophotographic process," J. Appl. Phys., Vol. 73, No. 10, 5440-5442, 1993.
doi:10.1063/1.353709

21. Azzerboni, B. and E. Cardelli, "Magnetic field evaluation for disk conductors," IEEE Trans. Magn., Vol. 29, No. 6, 2419-2421, 1993.
doi:10.1109/20.280997

22. Azzerboni, B., E. Cardelli, M. Raugi, A. Tellini, and G. Tina, "Magnetic field evaluation for thick annular conductors," IEEE Trans. Magn., Vol. 29, No. 3, 2090-2094, 1993.
doi:10.1109/20.211324

23. Yonnet, J. P., Rare-earth Iron Permanent Magnets, Ch. Magnetomechanical Devices, Oxford Science Publications, 1996.

24. Blache, C. and G. Lemarquand, "New structures for linear displacement sensor with hight magnetic field gradient," IEEE Trans. Magn., Vol. 28, No. 5, 2196-2198, 1992.
doi:10.1109/20.179441

25. Zhu, Z. and D. Howe, "Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors," IEEE Trans. Magn., Vol. 28, No. 2, 1371-1374, 1992.
doi:10.1109/20.123947

26. Blache, C. and G. Lemarquand, "High magnetic field gradients in flux confining permanent magnet structures," Journal of Magnetism and Magnetic Materials, Vol. 104, 1111-1112, 1992.
doi:10.1016/0304-8853(92)90510-U

27. Wang, J., G. W. Jewell, and D. Howe, "Design optimisation and comparison of permanent magnet machines topologies," IEE. Proc. Elect. Power Appl., Vol. 148, 456-464, 2001.
doi:10.1049/ip-epa:20010512

28. Lemarquand, V., J. F. Charpentier, and G. Lemarquand, "Nonsinusoidal torque of permanent-magnet couplings," IEEE Trans. Magn., Vol. 35, No. 5, 4200-4205, 1999.
doi:10.1109/20.799068

29. Abele, M., J. Jensen, and H. Rusinek, "Generation of uniform high fields with magnetized wedges," IEEE Trans. Magn., Vol. 33, No. 5, 3874-3876, 1997.
doi:10.1109/20.619600

30. Lemarquand, G. and V. Lemarquand, "Annular magnet position sensor," IEEE. Trans. Magn., Vol. 26, No. 5, 2041-2043, 1990.
doi:10.1109/20.104612

31. Aydin, M., Z. Zhu, T. Lipo, and D. Howe, "Minimization of cogging torque in axial-flux permanent-magnet machines: design concepts," IEEE Trans. Magn., Vol. 43, No. 9, 3614-3622, 2007.
doi:10.1109/TMAG.2007.902818

32. Yong, L., J. B. Zou, and Y. P. Lu, "Optimum design of magnet shape in permanent-magnet synchronous motors," IEEE Trans. Magn., Vol. 39, No. 11, 3523-4205, 2003.
doi:10.1109/TMAG.2003.819462

33. Bancel, F. and G. Lemarquand, "Three-dimensional analytical optimization of permanent magnets alternated structure," IEEE Trans. Magn., Vol. 34, No. 1, 242-247, 1998.
doi:10.1109/20.650248

34. Elies, P. and G. Lemarquand, "Analytical study of radial stability of permanent magnet synchronous couplings," IEEE Trans. Magn., Vol. 35, No. 4, 2133-2136, 1999.
doi:10.1109/20.774183

35. Charpentier, J. F., V. Lemarquand, and G. Lemarquand, "A study of permanent-magnet couplings with progressive magnetization using analytical exact formulation," IEEE Trans. Magn., Vol. 35, No. 5, 4206-4217, 1999.
doi:10.1109/20.799069

36. Berkouk, M., V. Lemarquand, and G. Lemarquand, "Analytical calculation of ironless loudspeaker motors," IEEE Trans. Magn., Vol. 37, No. 2, 1011-1014, 2001.
doi:10.1109/20.917185

37. Lemarquand, G., "Ironless loudspeakers," IEEE Trans. Magn., Vol. 43, No. 8, 3371-3374, 2007.
doi:10.1109/TMAG.2007.897739

38. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "The three exact components of the magnetic field created by a radially magnetized tile permanent magnet," Progress In Electromagnetics Research, PIER 88, 307-319, 2008.

39. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "Discussion about the analytical calculation of the magnetic field created by permanent magnets," Progress In Electromagnetics Research B, Vol. 11, 281-297, 2009.
doi:10.2528/PIERB08112102

40. http://www.univ-lemans.fr/∼glemar, , .

Dr. Romain Ravaud

Prof. Guy Lemarquand