A 2-D analytic based eddy-current transient model for a conducting plate is derived that is capable of accounting for continuous changes in the input conditions. Only the source field on the surface of the conducting plate needs to be known. In addition, a 2-D steady-state analytic based eddy-current model that is capable of accounting for frequency and velocity changes in two directions is derived. Both analytic based models have been validated using finite element code. The transient and steady-state models are integrated into an electromechanical system where the magnetic source is a Halbach rotor. The accuracy of both calculation methods is compared. The stiffness and damping coefficients are derived using the steady-state model.
Jonathan Z. Bird,
"Modeling the Dynamic Electromechanical Suspension Behavior of an Electrodynamic Eddy Current Maglev Device," Progress In Electromagnetics Research B,
Vol. 49, 1-30, 2013. doi:10.2528/PIERB12121115
1. Bohn, G. H. and G. Steinmetz, "The electromagnetic levitation and guidance technology of the `transrapid' test facility Emsland," IEEE Transactions on Magnetics, Vol. 20, No. 5, 1666-1671, Sep. 1984. doi:10.1109/TMAG.1984.1063246
2. Davey, K. R., "Designing with null flux coils," IEEE Transactions on Magnetics, Vol. 33, No. 5, 4327-4334, Sep. 1997. doi:10.1109/20.620442
3. Karoly, K., M. K. Volus, and K. Robert, "Design considerations for the hunting and braking performance of Maglev vehicle utilizing permanent magnet EDS levitation system," MAGLEV 2006, 363-370, Dresden, Germany, 2006.
4. Bird, J. and T. A. Lipo, "Characteristics of an electrodynamic wheel using a 2-D steady-state model," IEEE Transactions on Magnetics, Vol. 43, 3395-3405, Aug. 2007. doi:10.1109/TMAG.2007.900572
5. Bird, J. and T. A. Lipo, "Calculating the forces created by an electrodynamic wheel using a 2D steady-state finite element model," IEEE Transactions on Magnetics, Vol. 44, No. 3, 365-372, Mar. 2008. doi:10.1109/TMAG.2007.913038
6. Halbach, K., "Design of permanent multipole magnets with oriented rare earth cobalt material," Nucl. Instr. and Meth., Vol. 187, 1-10, 1980. doi:10.1016/0029-554X(80)90094-4
7. Shute, H. A., J. C. Mallinson, D. T. Wilton, and D. J. Mapps, "One-sided °uxes in planar, cylindrical, and spherical magnetized structures," IEEE Transactions on Magnetics, Vol. 36, No. 2, 440-451, 2000. doi:10.1109/20.825805
8. Ravaud, R. and G. Lemarquand, "Discussion about the magnetic field produced by cylindrical halbach structures," Progress In Electromagnetics Research B, Vol. 13, 275-308, 2009. doi:10.2528/PIERB09012004
9. Rote, D. M. and Y. Cai, "Review of dynamic stability of repulsive-force maglev suspension systems," IEEE Transactions on Magnetics, Vol. 38, No. 2, 1383-1390, Mar. 2002. doi:10.1109/20.996030
10. Davis, L. C. and D. F. Wilkie, "Analysis of motion of magnetic levitation systems: Implications for high-speed vehicles," Jour. of Appl. Phy., Vol. 42, No. 12, 4779-4793, Nov. 1971. doi:10.1063/1.1659855
11. Baiko, A. V., K. E. Voevodskii, and V. M. Kochetkov, "Vertical unstable stability of electrodynamic suspension of high speed ground transport," Cryogenics, Vol. 20, No. 5, 271-276, May 1980. doi:10.1016/0011-2275(80)90052-1
12. Yoshida, K. and M. Takakura, "Magnetic damping and stiffness coefficients in superconducting maglev system with sheet guideways," Electr. Eng. Japan, Vol. 99, No. 12, 797-804, 1979.
13. Urankar, L., "Intrinsic damping in basic magnetic levitation systems with a continuous sheet track," Siemens Forschungs and Entwicklungsberichte, Vol. 5, No. 2, 110-119, 1996.
14. Yamada, T., M. Iwamoto, and T. Ito, "Magnetic damping force in inductive magnetic levitation system for high-speed trains," Electr. Eng. Japan, Vol. 94, No. 1, 49-54, 1974. doi:10.1002/eej.4390940112
15. Iwamoto, M., T. Yamada, and E. Ohno, "Magnetic damping force in electrodynamically suspended trains," IEEE Transactions on Magnetics, Vol. 10, No. 3, 458-461, 1974. doi:10.1109/TMAG.1974.1058446
16. Fujiwara, S., "Damping characteristics of the repulsive magnetic levitation vehicle," Japanese Railway Technial Res. Inst., Quart. Rep., Vol. 21, No. 1, 49-52, 1980.
18. Ooi, B. T., "Electromechanical stiffness and damping coefficients in the repulsive magnetic levitation system," IEEE Trans. Power App. Syst., Vol. 95, No. 3, 936-943, May-Jun. 1976.
19. Takano, I. and H. Ogiwara, "Magnetic damping characteristics of magnetically suspended ultrahigh-speed vehicles," Electr. Eng. Japan, Vol. 98, No. 5, 14-23, Sep. 1978. doi:10.1002/eej.4390980503
20. Kratki, N. and K. Oberretl, "Ausgleichsvorgiinge und schwingungen beim elektrodynamischen magnetkissen-system," Archiv fur Elektrotechnik, Vol. 57, 59-64, 1975. doi:10.1007/BF01407654
21. He, J. L. and H. T. Coffey, "Magnetic daming forces in figure-eight-shaped null-flux coil suspension systems," IEEE Transactions on Magnetics, Vol. 33, No. 5, 4230-4232, Sep. 1997. doi:10.1109/20.619719
22. Zhu, S., Y. Cai, D. M. Rote, and S. S. Chen, "Magnetic damping for maglev," Shock and Vibration, Vol. 5, 119-128, 1998.
23. Smythe, W. R., Static & Dynamic Electricity, 5th Ed., 1989.
24. Bird, J. and T. A. Lipo, "Modeling the 3-D rotational and translational motion of a Halbach rotor above a split-sheet guideway," IEEE Transactions on Magnetics, Vol. 45, No. 9, 3233-3242, Sep. 2009. doi:10.1109/TMAG.2009.2021160
25. Goldstein, H., C. Poole, and J. Safko, Classical Mechanics, 2002.
26. Paudel, N. and J. Z. Bird, "General 2D steady-state force and power equations for a traveling time-varying magnetic source above a conductive plate," IEEE Transactions on Magnetics, Vol. 48, No. 1, 95-100, Jan. 2012. doi:10.1109/TMAG.2011.2161638
27. Sankaran, R., K. P. P. Pillai, and D. A. Muraleedharan, "Transient performance of linear induction machines following reconnection of supply," Proc. IEE, Vol. 126, No. 10, 979-983, 1979.
28. Paudel, N., J. Z. Bird, S. Paul, and D. Bobba, "Modeling the dynamic suspension behavior of an eddy current device," Energy Conversion Congress and Exposition (ECCE), 1692-1699, 2011.
29. Yamamura, S., Theory of Linear Induction Motors, University of Tokyo Press, 1979.
30. Rodger, D. and J. F. Eastham, "Dynamic behavior of linear induction machines in the heave mode," IEE Trans. Vehicular Technology, Vol. 31, No. 2, 1982.
31. Paudel, N., S. Paul, and J. Z. Bird, "General 2-D transient eddy current force equations for a magnetic source moving above a conductive plate," Progress In Electromagnetics Research B, Vol. 43, 255-277, 2012.
32. Rodger, D., P. J. Leonard, and . Karaguler, "An optimal formulation for 3D moving conductor eddy current problems with smooth rotors," IEEE Transactions on Magnetics, Vol. 26, 2359-2363, Sep. 1990. doi:10.1109/20.104731
33. Paudel, N., "Dynamic suspension modeling of an eddy-current device: An application to MAGLEV,", Ph.D. Thesis, Electrical and Computer Engineering, University of North Carolina at Charlotte, Charlotte, NC, 2012.
34. Polyanin, A. D. and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall, 2007.
35. Freeman, E. M. and C. Papageorgiou, "Spatial Fourier transforms: A new view of end effects in linear induction motors," Proc. IEE, Vol. 125, No. 8, 747-753, Aug. 1978.
37. Xia, Z. P., Z. Q. Zhu, and D. Howe, "Analytical magnetic field analysis of Halbach magnetized permanent-magnet machines," IEEE Transactions on Magnetics, Vol. 40, No. 4, 1864-1872, Jul. 2004. doi:10.1109/TMAG.2004.828933
38. Atallah, K., D. Howe, P. H. Mellor, and D. A. Stone, "Rotor loss in permanent-magnet brushless AC machines," IEEE Trans. Ind. Appl., Vol. 36, No. 6, 1612-1618, Nov.-Dec. 2000. doi:10.1109/28.887213
39. Zhu, S., Y. Cai, D. M. Rote, and S. S. Chen, "Magnetic damping for maglev,", ANL/ET/CP-82419, CONF-9411194-2, Energy Technology Division, Argonne National Laboratory, 1995.
40. Gillespire, T. D., Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Inc., 1992.
41. Moon, F. C., "Superconducting Levitation: Applications to Bearings and Magnetic Transportation," WILEY-VCH Verlag GmbH & Co. KGaA, 2004.
42. Brown, J. W. and R. V. Churchill, Complex Variables and Applications, 6th Ed., McGraw-Hill Inc., New York, 1996.
43. Earnshaw, S., "On the nature of the molecular forces which regulate the constitution of the luminferous ether," Transactions Camb. Phil. Soc., Vol. 7, 97-112, 1842.
44. Di Puccio, F., A. Musolino, R. Rizzo, and E. Tripodi, "A self-controlled maglev system," Progress In Electromagnetics Research M, Vol. 26, 187-203, 2012.
45. Di Puccio, F., R. Bassani, E. Ciulli, A. Musolino, and R. Rizzo, "Permanent magnet bearings: Analysis of plane and axisymmetric V-shaed element design," Progress In Electromagnetics Research M, Vol. 26, 205-223, 2012.
46. Post, R. F. and D. D. Ryutov, "Ambient-temperature passive magnetic bearings: Theory and design equations," 6th International Symposium on Magnetic Bearings, Cambridge, Massachusetts, 1998.
47. Higashi, K., S. Ohashi, H. Ohsaki, and E. Masada, "Damping mechanism of the superconducting electrodynamic suspension system," Nonlinear Electromagnetic Systems: Proceedings of the International IOS Press, 1996.