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Abstract

Multi-pole permanent magnetic encoders are used for wheel speed sensing in automotive systems. This paper discusses rings and discs magnetized along axial direction. The vector field is calculated analytically in 3D as sums over all poles. For the case of discs with vanishing inner and infinite outer diameter the summation is done in closed form with a new summation formula. The results are discussed and several plots of field patterns are given in normalized form: At very small air-gap the field shows an overshoot. At moderate and large air-gaps it is a sine-function with exponentially decaying amplitude. The amplitude versus air gap, reading radius, thickness of the magnetic layer, and number of poles is studied and excellent agreement with measurements is found. The effect of a steel-back on the field is explained. It is also shown how to maximize the torque transmitted in axially magnetized couplings.

1. Nachtigal, D. and G. Bergmann, "Multi-pole magnetic encoders for active speed-measurement systems," SAE-paper, Paper No. 1999-01-3402, Proc. of the 17th Annual Brake Colloquium & Engineering Display (SAE Proc.), Vol. 349, 53-58, Oct. 1999.

2. Saito, H., M. Kume, S. Kawamura, and O. Shimizu, "Wheel rotation sensor for use in a navigation/location system," Sumitomo Electric Technical Review, Vol. 28, 241-247, Jan. 1989.

3. Miyashita, K., T. Takahashi, and M. Yamanaka, "Features of a magnetic rotary encoder," IEEE Trans. Magn., Vol. 23, No. 5, 2182-2184, Sept. 1987.
doi:10.1109/TMAG.1987.1065634

4. Kikuchi, Y., F. Nakamura, H. Wakiwaka, and H. Yamada, "Index phase output characteristics of magnetic rotary encoder using a magneto-resistive element," IEEE Trans. Magn., Vol. 33, No. 5, 3370-3372, Sept. 1997.
doi:10.1109/20.617947

5. Kikuchi, Y., F. Nakamura, H. Wakiwaka, and H. Yamada, "Consideration of magnetization and detection on magnetic rotary encoder using finite element method," IEEE Trans. Magn., Vol. 33, No. 2, 2159-2162, Mar. 1997.
doi:10.1109/20.582759

6. Campbell, P., "Miniature magnetic encoder," Machine Design, 105-108, Mar. 8, 1990.

7. Campbell, P., "Magnetic encoding technology," Motor-Con Proceedings, 211-219, Apr. 1987.

8. Kafader, U., Für anspruchsvolle antriebstechnik, Vol. 10, 55-57 MegaLink, Oct. 2001.

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

10. Bancel, F., "Magnetic nodes," J. Phys. D: Appl. Phys., Vol. 32, 2155-2161, 1999.
doi:10.1088/0022-3727/32/17/304

11. Liu, W. Z., C. Y. Xu, and Z. Y. Ren, "Research of the surface magnetic field of multi-pole magnetic drum of magnetic encoder," Int'l Conf. Sensors and Control Techniques (ICSC2000), Proceedings of SPIE, Vol. 4077, 288-291, D.-S. Jiang and A.-B. Wang (eds.), 2000.

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

13. Furlani, E. P., "A three-dimensional field solution for axially-polarized multipole discs," J. Magn. Magn. Mat., Vol. 135, 205-214, 1994.
doi:10.1016/0304-8853(94)90347-6

14. Ravaud, R. and G. Lemarquand, "Magnetic field created by a uniformly magnetized tile permanent magnet," Progress In Electromagnetics Research B, Vol. 24, 17-32, 2010.
doi:10.2528/PIERB10062209

15. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "Magnetic field produced by a tile permanent magnet whose polarization is both uniform and tangential," Progress In Electromagnetics Research B, Vol. 13, 1-20, 2009.
doi:10.2528/PIERB08121901

16. Ravaud, R. and G. Lemarquand, "Analytical expression of the magnetic field created by tile permanent magnets tangentially magnetized and radial currents in massive disks," Progress In Electromagnetics Research B, Vol. 13, 309-328, 2009.
doi:10.2528/PIERB09012704

17. Forrester, A. T. and J. Busnardo-Neto, "Magnetic fields for surface containment of plasmas," J. Appl. Phys., Vol. 47, No. 9, 3935-3941, Sept. 1976.
doi:10.1063/1.323267

18. Viana, R. L. and I. L. Caldas, "Comments on the magnetic field generated by an infinite current grid," Eur. J. Phys., Vol. 12, 293-296, Dec. 1991.
doi:10.1088/0143-0807/12/6/009

19. Da Silva, E. C., I. L. Caldas, and R. L. Viana, "Ergodic magnetic limiter for the TCABR," Brazilian J. Phys., Vol. 32, No. 1, 39-45, Jan. 2002.

20. Leung, K. N., N. Hershkowitz, and K. R. MacKenzie, "Plasma confinement by localized cusps," Phys. Fluids, Vol. 19, 1045-1053, 1976.
doi:10.1063/1.861575

21. Nihei, H., "Analytic expressions of magnetic multipole field generated by a row of permanent magnets," Jap. J. Appl. Phys., Vol. 29, No. 9, 1831-1832, Sept. 1990.
doi:10.1143/JJAP.29.1831

22. Ozeretskovskiy, V., "Calculation of two-dimensional nonperiodic multipole magnetic systems," Sov. J. Commun. Techn. and Electr., Vol. 36, No. 8, 81-92, Aug. 1991.

23. Grinberg, E., On determination of properties of some potential fields, Vol. 12, 147-154 Applied Magnetohydrodynamics, Reports of the Physics Inst. Riga, 1961.

24. Avilov, V. V., Electric and magnetic fields for the riga plate, Internal Report FZR Forschungszentrum Rossendorf, Dresden, Germany, 1998. Published in a report by E. Kneisel, ``Numerische und experimentelle untersuchungen zur grenzschichtbee-influssung in schwach leitfähigen flüssigkeiten,'' Nov. 24, 2003, http://www.hzdr.de/FWS/FWSH/Mutschke/kleinerbeleg.pdf.

25. De Visschere, P., "An exact two-dimensional model for a periodic circular array of head-to-head permanent magnets," J. Phys. D: Appl. Phys., Vol. 38, 355-362, 2005.
doi:10.1088/0022-3727/38/3/001

26. Wang, R., M. Kamper, K. Westhuizen, and J. Gieras, "Optimal design of a coreless stator axial flux permanent-magnet generator," IEEE Trans. Magn., Vol. 41, No. 1, 55-64, Jan. 2005.
doi:10.1109/TMAG.2004.840183

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

28. Gridnev, A. I., N. A. Kelin, N. I. Klevets, et al. "Synthesis and optimization of rods made of high-coercivity permanent magnets and systems for magnetization," Izv. AN SSSR. Ser. Energetika i Transport, No. 6, 1987.

29. Jackson, J. D., Classical Electrodynamics, 2nd German Ed., 219, 5.79, Walter de Gruyter, 1982, ISBN 3-11-009579-3.

30. 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, Vol. 88, 307-319, 2008.
doi:10.2528/PIER08112708

31. Smythe, W. R., "Current images in plane face," Static and Dynamic Electricity, 3rd edition, Chapter 7.23, Taylor & Francis, 1989, ISBN 0-89116-917-2.

32. Hansen, E. R., A Table of Series and Products, 271, 41.2.16, Prentice Hall, 1975.

33. Hansen, E. R., A Table of Series and Products, 271, 41.2.14, Prentice Hall, 1975.

34. Ausserlechner, U., Vorrichtung und verfahren zum erzeugen eines logischen ausgangssignals, German Patent Application DE 10214524, Apr. 2002.

35. Becker/Sauter Theorie der Elektrizität, 21st Ed., Vol. 1, 112, 5.4.22, B. G. Teubner, 1973.

36. Arfken, G., Mathematical Methods for Physicists, 3rd Ed., 618, Academic Press, 1985.

37. Furlani, E. P., "Formulas for the force and torque of axial couplings," IEEE Trans. Magn., Vol. 29, No. 5, 2295-2301, Sept. 1993.
doi:10.1109/20.231636

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

39. Furlani, E. P., "Analytical analysis of magnetically coupled multipole cylinders," J. Phys. D: Appl. Phys., Vol. 33, 28-33, 2000.
doi:10.1088/0022-3727/33/1/305

40. Ravaud, R. and G. Lemarquand, "Magnetic couplings with cylindrical and plane air gaps: Influence of the magnet polarization direction," Progress In Electromagnetics Research B, Vol. 16, 333-349, 2009.
doi:10.2528/PIERB09051903

41. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "Torque in PM couplings: Comparison of uniform and radial magnetization," J. Appl. Phys., Vol. 105, 053904, 2009, DOI: 10.1063/1.3074108.
doi:10.1063/1.3074108

42. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "Permanent magnet couplings: Field and torque three-dimensional expressions based on the coulombian model," IEEE Trans. Magn., Vol. 45, No. 4, 1950-1964, 2009.
doi:10.1109/TMAG.2008.2010623

43. Huang, D. R., G.-J. Chiou, Y.-D. Yao, and S.-J. Wang, "Effect of magnetization profiles on the torque of magnetic coupling," J. Appl. Phys., Vol. 76, No. 10, 6862-6864, Nov. 15, 1994.
doi:10.1063/1.358094

44. Yao, Y. D., D. R. Huang, C. C. Hsieh, D. Y. Chiang, S. J. Wang, and T. F. Ying, "The radial magnetic coupling studies of perpendicular magnetic gears," IEEE Trans. Magn., Vol. 32, No. 5, 5061-5063, Sept. 1996.
doi:10.1109/20.539490

45. Yao, Y. D., G. J. Chiou, D. R. Huang, and S. J. Wang, "Theoretical computations for the torque of magnetic coupling," IEEE Trans. Magn., Vol. 31, No. 3, 1881-1884, May 1995.
doi:10.1109/20.376405

46. Tsamakis, D., M. Ioannides, and G. Nicolaides, "Torque transfer through plastic bonded Nd2Fe14B magnetic gear system," J. Alloys Compounds, Vol. 241, 175-179, 1996.
doi:10.1016/0925-8388(96)02353-5

47. Hansen, E. R., A Table of Series and Products, Prentice Hall, 1975.

48. Ryshik, I. M. and I. S. Gradstein, Tables of Series, Products, and Integrals, Vol. 34, 1.392, VEB Deutscher Verlag der Wissenschaften, 1957.

Dr. Udo Ausserlechner