This paper presents a new integro-differential coupling between partial equivalent electrical circuits (PEEC) and finite difference method (FDM) taking into account the magnetization effect. This coupling is intended for thin plates having simultaneously significant conductive and magnetic properties in presence of exciting coils of complex topologies. These cases exist in eddy current nondestructive testing (ECNDT), eddy current separation, induction or levitation melting devices and more other applications. The choice of FDM, is in relation with rectangular surfaces generated by numerical meshes leading to mathematical integrations of magnetic and electrical quantities with independent variables, unlike more complicated forms of surfaces generated by finite element method (FEM) or others. Fully successful analytical expressions have been realized and implemented in overall coupling process. The PEEC method is mainly used to calculate the magnetic field applied to the nodes of the plate from different inclined polygonal coils. The results of magnetic field and eddy current distributions on thin plates are presented, and parts of them are compared with those realized by Flux 3D software.
Houssem Rafik El-Hana Bouchekara,
"Partial Electrical Equivalent Circuits and Finite Difference Methods Coupling; Application to Eddy Currents Calculation for Conductive and Magnetic Thin Plates," Progress In Electromagnetics Research C,
Vol. 114, 83-96, 2021. doi:10.2528/PIERC21051602
1. Dodd, C. V. and W. E. Deeds, "Analytical solutions to eddy-current probe-coil problems," Journal of Applied Physics, Vol. 39, No. 6, 2829-2838, 1968. doi:
2. Bowler, J. R., "Eddy-current interaction with an ideal crack. 1: The forward problem," Journal of Applied Physics, Vol. 75, No. 12, 8128-8137, 1994. doi:The server didn't respond in time.
3. Chebout, M., M. R. Mekideche, A. Hafaifa, A. Kouzou, and H. Allag, "Impedance measurement and computation for the steam generator tube integrity using the ECNDT technique," Elektrotehniški Vestnik, Vol. 85, No. 5, 235-240, 2018. doi:
4. Maraspin, F. P., P. Bevilacqua, and P. Rem, "Modelling the throw of metals and nonmetals in eddy current separations," International Journal of Mineral Processing, Vol. 73, No. 1, 1-11, 2004. doi:10.1016/S0301-7516(03)00081-4
5. Nagel, J. R., "Fast finite-difference calculation of Eddy currents in thin metal sheets," Applied Computational Electromagnetic Society Journal, Vol. 33, No. 6, 575-584, 2018.
6. Okress, E. C., D. M. Wroughton, G. Comenetz, P. H. Brace, and J. C. R. Kelly, "Electromagnetic levitation of solid and molten metals," Journal of Applied Physics, Vol. 23, No. 5, 545-552, 1952. doi:10.1063/1.1702249
7. Fromm, E. and H. Jehn, "Electromagnetic forces and power absorption in levitation melting," British Journal of Applied Physics, Vol. 16, No. 5, 653-662, 1965. doi:10.1088/0508-3443/16/5/308
8. Dholu, N., J. R. Nagel, D. Cohrs, and R. K. Rajamani, "Eddy current separation of nonferrous metals using a variable-frequency electromagnet," KONA Powder and Particle Journal, Vol. 34, 241-247, 2017. doi:10.14356/kona.2017012
9. Smith, Y. R., J. R. Nagel, and R. K. Rajamani, "Electrodynamic Eddy current separation of end of-life PV," Energy Technology, The Minerals, Metals and Materials Series, 379-386, Springer, Cham., 2017. doi:10.1007/978-3-319-52192-3_37
10. Ray, J. D., J. R. Nagel, D. Cohrs, and R. K. Rajamani, "Forces on particles in time-varying magnetic fields," KONA Powder and Particle Journal, Vol. 35, 251-257, 2018. doi:10.14356/kona.2018016
11. Fireteanu, V., B. Paya, J. Nuns, and T. Tudorache, "Electromagnetic levitation of solid and molten metals," Journal of Applied Physics, Vol. 21, No. 4, 581-590, 2002.
12. Yin, W., A. J. Peyton, and S. J. Dickinson, "Simultaneous measurement of distance and thickness of a thin metal plate with an electromagnetic sensor using a simplified model," IEEE Transactions on Instrumentation and Measurement, Vol. 53, No. 4, 1335-1338, 2004. doi:10.1109/TIM.2004.830585
13. Betta, G., L. Ferrigno, M. Laracca, P. Burrascano, M. Ricci, and G. Silipigni, "An experimental comparison of multi-frequency and chirp excitations for eddy current testing on thin defects," Measurement, Vol. 63, 207-220, 2015. doi:10.1016/j.measurement.2014.12.015
14. Tian, L. L., C. Yin, Y. H. Cheng, and L. Bai, "Successive approximation method for the measurement of thickness using pulsed eddy current," IEEE Instrumentation and Measurement Technology Conference, 848-852, 2015.
15. N. Ida, Y. Le Menach and T. Henneron, "High order surface impedance boundary conditions with the A-Φ formulation," Facta Universitatis (NIS), Vol. 24, 147-155, 2011.
16. Zaidi, H., L. Santandréa, G. Krebs, and Y. Le Bihan, "Modelling of 3D thin regions in magnetostatic NDT using overlapping elements in dual formulations," PIERS Proceedings, 166-170, Marrakesh, Morocco, Mar. 20–23, 2011.
17. Ruehli, A., G. Antonini, and L. Jiang, Circuit Oriented Electromagnetic Modelling Using the PEEC Techniques, Wiley-Blackwell, 2017. doi:10.1002/9781119078388
18. Ruehli, A. E., "Inductance calculation in a complex integrated circuit environment," IBM Journal Research Development, 470-480, 1972. doi:10.1147/rd.165.0470
19. Hoer, C. and C. Love, "Exact inductance equations for rectangular conductors whith applications to more complicated geometries," Journal of Research of the National Bureau of Standards-C. Engineering and Instrumentation, Vol. 69, No. 2, 127-137, 1965. doi:10.6028/jres.069C.016
20. Paul, C. R., Inductance Loop and Partial, John Wiley Sons, Inc., 2010.
21. Antonini, G., A. Orlandi, and C. R. Paul, "Internal impedance of conductors of rectangular cross section," IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 7, 979-985, 1999. doi:10.1109/22.775429
22. Zhong, G. and C. K. Koh, "Exact closed-form formula for partial mutual inductances of rectangular conductors," IEEE Transactions on Magnetics, Vol. 50, No. 1, 1349-1352, 2003.
23. Wollenberg, C. and A. Gurisch, "Analysis of 3-D interconnect structures with PEEC using SPICE," IEEE Trans. on Electromag. Compat., Vol. 41, No. 4, 412-417, 1999. doi:10.1109/15.809841
24. Cao, Y., Z. F. Li, J. F. Mao, and J. F. Mao, "A PEEC with a new capacitance model for circuit simulation of interconnects and packaging structures," IEEE Trans. on Microwave Theory and Tech., Vol. 48, No. 9, 1435-1442, 2000. doi:10.1109/22.868992
25. Gope, D., A. Ruehli, and V. Jandhyala, "Solving low-frequency EM-CKT problems using the PEEC Method," IEEE Trans. on Adv. Packaging, Vol. 30, No. 2, 313-320, 2007. doi:10.1109/TADVP.2007.896000
26. Antonini, G., "PEEC capacitance extraction of 3-D interconnects," IET Science, Measurement and Technology, Vol. 1, No. 4, 201-209, 2007. doi:10.1049/iet-smt:20050047
27. Le-Duc, T., O. Chadebec, J. M. Guichon, and G. Meunier, "Coupling between partial element equivalent circuit method and an integro-differential approach for solving electromagnetics problems," IET Science, Measurement and Technology, Vol. 6, No. 5, 394-397, 2012. doi:10.1049/iet-smt.2011.0121
28. Kalimov, A., F. Kmos, B. Langenbeck, and G. Moritz, "Dynamic processes in laminated magnets: Simulation and comparison with experimental results," IEEE Trans. Appl. Supercond., Vol. 12, No. 1, 98-10, 2002. doi:10.1109/TASC.2002.1018360
29. Salon, S. J., B. Mathewson, and S. Uda, "An intergro-differential approach to eddy currents in thin plates," IEEE Transactions on Magnetics, Vol. 19, No. 6, 2405-2408, 1983. doi:10.1109/TMAG.1983.1062874
30. Tsuboi, H., M. Tanaka, and T. Misaki, "Eddy current and deflection analyses of a thin plate in time-changing magnetic field," IEEE Transactions on Magnetics, Vol. 26, No. 5, 1647-1649, 1990. doi:10.1109/20.104476
31. Demenko, A. and J. Sykulski, "On the equivalence of finite difference and edge element formulations in magnetic field analysis using vector potential," COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 33, No. 1-2, 47-55, 2014.
32. Huang, J., W. Liao, and Z. Li, "A multi-block finite difference method for seismic wave equation in auxiliary coordinate system with irregular fluid-solid interface," Engineering Computations, Vol. 35, No. 1, 334-362, 2018. doi:10.1108/EC-12-2016-0438
33. Chapwanya, M., R. Dozva, and G. Gift Muchatibaya, "A nonstandard finite difference technique for singular Lane-Emden type equations," Engineering Computations, Vol. 36, No. 5, 1566-1578, 2019. doi:10.1108/EC-08-2018-0344
34. Mawlood, M., S. Basri, W. Asrar, A. Omar, A. Mokhta, and M. Ahmad, "Solution of Navier-Stokes equations by fourth-order compact schemes and AUSM flux splitting," International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 16, No. 1, 107-120, 2006. doi:10.1108/09615530610636982
35. Momeni, K., Y. Ji, Y. Wang, S. Paul, S. Neshani, D. E. Yilmaz, Y. K. Shin, D. Zhang, J.-W. Jiang, H. S. Park, S. Sinnott, A. van Duin, V. Crespi, and L.-Q. Chen, "Multiscale computational understanding and growth of 2D materials: A review," NPJ Computational Materials, Vol. 6, No. 22, 2020.
36. Aissaoui, M., H. Allag, and J. P. Yonnet, "Mutual inductance and interaction calculation between conductor or coil of rectangular cross section and parallelepiped permanent magnet," IEEE Transactions on Magnetics, Vol. 50, No. 11, 1-4, 2014. doi:10.1109/TMAG.2014.2324283
37. Aomar, L., H. Allag, M. Feliachi, and J. P. Yonnet, "3-D integral approach for calculating mutual interactions between polygon-shaped massive coils," IEEE Transactions on Magnetics, Vol. 53, No. 11, 1-2, 2017. doi:10.1109/TMAG.2017.2706022