volume | PIER Journals

Abstract

This paper presents a novel contribution for the analysis of skin effect phenomena in inhomogeneous tubular conductors. For homogeneous tubular geometries the skin effect diffusion equation has an analytical solution described by a combination of Bessel functions, but, when the conductivity and magnetic permeability of the tubular conductor arbitrarily depend on the radial coordinate an analytical solution cannot be found. However, this work shows that closed form solutions for the electromagnetic field and conductor internal impedance do exist, provided that a specific type of radial variation of medium parameters is considered --- tubular structures like these are coined here Euler-Cauchy Structures (ECS). Analytic and computation results concerning general and particular ECS are presented, validated, and discussed. Besides its intrinsic theoretical importance, the simple closed form solutions that we have found can be of interest as benchmark tools for testing the accuracy and performance of EM field software solvers.

1. Dwight, H., "Skin effect in tubular and flat conductors," AIEE Trans., Vol. 37, Part. II, 1379-1403, 1918.

2. Cockcroft, J., "Skin effect in rectangular conductors at high frequencies," Proc. Roy. Soc., Vol. 122, 533-542, 1929.
doi:10.1098/rspa.1929.0038

3. Arnold, A., "The alternating current resistance of tubular conductors," J. IEE, Vol. 78, 580-593, 1936.

4. Silvester, P., "The accurate calculation of skin effect in conductors of complicated shape," IEEE Trans. Power App. Syst., Vol. 87, No. 3, 735-742, 1968.
doi:10.1109/TPAS.1968.292187

5. Tegopoulos, J. and E. Kriezis, "Eddy current distribution in cylindrical shells of infinite length due to axial currents," IEEE Trans. Power App. Syst., Vol. 90, No. 3, 1278-1286, 1971.
doi:10.1109/TPAS.1971.292929

6. Waldow, P. and I. Wolff, "The skin-effect at high frequencies," IEEE Trans. Microw. Theory Tech., Vol. 33, No. 10, 1076-1081, 1985.
doi:10.1109/TMTT.1985.1133172

7. Costache, G., "Finite element method applied to skin-effect problems in strip transmission lines," IEEE Trans. Microw. Theory Tech., Vol. 35, No. 11, 1009-1013, 1987.
doi:10.1109/TMTT.1987.1133799

8. Tsuk, M. and J. Kong, "A hybrid method for the calculation of the resistance and inductance of transmission lines with arbitrary cross sections," IEEE Trans. Microw. Theory Tech., Vol. 39, No. 8, 1338-1347, 1991.
doi:10.1109/22.85409

9. Silveira, F. and J. Lima, "Skin effect from extended irreversible thermodynamics perspective," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 2-3, 151-160, 2010.
doi:10.1163/156939310790735787

10. Barmada, S., L. Rienzo, N. Ida, and S. Yuferev, "The use of surface impedance conditions in time domain problems: Numerical and experimental validation," Appl. Comput. Electromag. Soc. Journal, Vol. 19, No. 2, 76-83, 2004.

11. Barmada, S., L. Rienzo, N. Ida, and S. Yuferev, "Time domain surface impedance concept for low frequency electromagnetic problems. Part II: Application to transient skin and proximity effect problems in cylindrical conductors," IEE Proc. Sci. Meas. Technol., Vol. 154, No. 5, 207-216, 2005.
doi:10.1049/ip-smt:20049036

12. Rong, A. and A. Cangelari, "Note on the definition and calculation of the per-unit-length internal impedance of a uniform conducting wire," IEEE Trans. Electrom. Comp., Vol. 49, No. 3, 677-681, 2007.
doi:10.1109/TEMC.2007.903043

13. Morsey, J., V. Okhmatovski, and A. Cangelaris, "Finite-thickness conductor models for full-wave analysis of interconnects with a fast integral equation method," Proc. IEEE 11th Topical Meeting on Electrical Performance of Electronic Packaging, Monterey, USA, Oct. 2002.

14. Zutter, D., H. Rogier, L. Knockaert, and J. Sercu, "Surface current modelling of the skin effect for on-chip interconnections," IEEE Trans. Adv. Packaging, Vol. 30, No. 2, 342-349, 2007.
doi:10.1109/TADVP.2007.895984

15. Morgan, V., R. Findlay, and S. Derrah, "New formula to calculate the skin e®ect in isolated tubular conductors at low frequencies," IEE Proc. Sci. Meas. Technol., Vol. 147, No. 4, 169-171, 2000.
doi:10.1049/ip-smt:20000420

16. Mingli, W. and F. Yu, "Numerical calculations of internal impedance of solid and tubular cylindrical conductors under large parameters," IEE Proc. Generation Transm. Distribution, Vol. 151, No. 1, 67-72, 2004.
doi:10.1049/ip-gtd:20030981

17. Coufal, O., "Current density in a long solitary tubular conductor," J. Physics A: Math. Theor., Vol. 41, No. 14, 145401 (14pp), 2008.

18. Rienzo, L., S. Yuferev, and N. Ida, "Calculation of energy-related quantities of conductors using surface impedance concept," IEEE Trans. Magnetics, Vol. 44, No. 6, 1322-1325, 2008.
doi:10.1109/TMAG.2008.915850

19. Vujevic, S., V. Boras, and P. Sarajcev, "A novel algorithm for internal impedance computation of solid and tubular cylindrical conductors," Int. Rev. Electric. Eng., Vol. 4, No. 6, 1418-1425, 2009.

20. Knight, D., "Practical continuous functions and formulae for the internal impedance of cylindrical conductors,", http//www.g3ynh.info/ zdocs/comps/Zint.pdf, March 2010.

21. Lovric, D., V. Boras, and S. Vujevic, "Accuracy of approximate formulas for internal impedance of tubular cylindrical conductors for large parameters," Progress In Electromagnetics Research M, Vol. 16, 171-184, 2011.

22. Wylie, C., Advanced Engineering Mathematics, McGraw-Hill, 1975.

23. Faria, J., Electromagnetic Foundations of Electrical Engineering, Wiley, Chichester, UK, 2008.
doi:10.1002/9780470697498

24. Faria, J., "A matrix approach for the evaluation of the internal impedance of multilayered cylindrical structures," Progress In Electromagnetics Research B, Vol. 28, 351-367, 2011.

Prof. Jose Antonio Marinho Brandao Faria