In this paper, a new numerical method of calculating rectangular busbar impedance is proposed. This method is based on integral equation method and partial inductance theory. In particular, impedances of shielded and unshielded three-phase systems with rectangular phase and neutral busbars, conductive enclosure, and use of the method are described. Results for resistances and reactances for these systems of multiple rectangular conductors have been obtained, and skin and proximity effects have also been taken into consideration. The impact of the enclosure on impedances is also presented. Finally, two applications to three-phase shielded and unshielded systems busbars are described. The validation of the proposed method is carried out through FEM and laboratory measurements, and a reasonable level of accuracy is demonstrated.
1. Salinas, E., "Conductive and ferromagnetic screening of 50 Hz magnetic field from a three-phase system of busbars," Journal of Magnetism and Magnetic Materials, No. 226-230, 1239-1241, 2001. doi:10.1016/S0304-8853(00)01003-9
2. Ducluzaux, A., "Extra losses caused in high current conductors by skin and proximity effects," Schneider Electric, Cahier Technique, Vol. 83, 1983.
3. Du, Y. and J. Burnett, Power-frequency magnetic shielding of heavy-current conductors by rectangular shields, IEE Proc. --- Gener. Transm. Distrib., Vol. 146, No. 5, 223-228, 1999.
4. Koroglu, S., P. Sergeant, and N. Umurkan, "Comparison of analytical, finite element and neural network methods to study magnetic shielding," Simulation Modelling Practice and Theory, Vol. 18, 206-216, 2010. doi:10.1016/j.simpat.2009.10.007
5. Copper Development Association, Copper for Busbars, 2001, available online at: http://www.cda.org.uk/Megab2/elecapps/pub22/index.htm.
6. Sarajcev, P. and R. Goic, "Power loss computation in high-current generator bus ducts of rectangular cross-section," Electric Power Components and Systems, Vol. 38, 1469-1485, 2010. doi:10.1080/15325001003735192
7. Chiampi, M., D. Chiarabaglio, and M. Tartaglia, "A general approach for analyzing power busbar under AC conditions," IEEE Trans. on Magn., Vol. 20, No. 6, 2473-2475, 1993. doi:10.1109/20.280979
8. Clavel, E., J. Roudet, and A. Foggia, "Electrical modeling of transformer connecting bars," IEEE Trans. on Magn., Vol. 38, No. 2, 1378-1382, 2002. doi:10.1109/20.996028
9. Sigg, H. J. and M. J. O. Strutt, "Skin effect and proximity effect in polyphase systems of rectangular conductors calculated on an RC network," IEEE Trans. on Power Apparatus and Systems, Vol. PAS-89, No. 3, 470-477, 1970. doi:10.1109/TPAS.1970.292726
10. Guo, J., A. W. Glisson, and D. Kajfez, "Analysis of resistance and internal reactance in systems of parallel conductors," Int. J. Electron. Commun. AEÜ, Vol. 52, No. 2, 57-64, 1998.
11. Piatek, Z., Impedances of Tubular High Current Busducts, Polish Academy of Sciences, Warsaw, 2008.
12. Piatek, Z., "Self and mutual impedances of a finite length gas insulated transmission line (GIL)," Elec. Pow. Syst. Res., Vol. 77, 191-203, 2007. doi:10.1016/j.epsr.2006.02.017
13. 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.
14. Fazljoo, S. A. and M. R. Besmi, A new method for calculation of impedance in various frequencies, 1st Power Electronic & Drive Systems & Technologies Conference (PEDSTC), 36-40, February 17-18, 2010.
15. Ametani, A., "Approximate method for calculating the impedance of multiconductors with cross section of arbitrary shapes ," Electrical Engineering in Japan, Vol. 111, No. 2, 117-123, 1992. doi:10.1002/eej.4391120213
16. Kazimierczuk, M. K., High-frequency Magnetic Components, J. Wiley & Sons, Chichester, 2009.
17. Paul, C. R., Inductance: Loop and Partial, J. Wiley & Sons, New Jersey, 2010.
18. Paul, C. R., Analysis of Multiconductor Transmission Lines, J. Wiley & Sons, New Jersey, 2010.
19. Silvester, P., "AC resistance and reactance of isolated rectangular conductors," IEEE Trans. on Power Apparatus and Systems, Vol. PAS-86, No. 6, 770-774, June 1967. doi:10.1109/TPAS.1967.291888
20. Goddard, K. F., A. A. Roy, and J. K. Sykulski, Inductance and resistance calculations for isolated conductor, IEE Pro. --- Sci. Meas. Technol., Vol. 152, No. 1, 7-14, January 2005.
21. Goddard, K. F., A. A. Roy, and J. K. Sykulski, "Inductance and resistance calculations for a pair of rectangular conductor," IEE Pro. --- Sci. Meas. Technol., Vol. 152, No. 1, 73-78, January 2005. doi:10.1049/ip-smt:20041058
22. Chen, H. and J. Fang, "Modeling of impedance of rectangular cross-section conductors," IEEE Conference on Electrical Performance of Electronic Packaging, 159-162, 2000.
23. Zhihua, Z. and M. Weiming, "AC impedance of an isolated flat conductor," IEEE Trans. on Electromagnetic Compatibility, Vol. 44, No. 3, 482-486, 2002. doi:10.1109/TEMC.2002.801773
24. Piatek, Z. and B. Baron, "Exact closed form formula for self inductance of conductor of rectangular cross section," Progress In Electromagnetics Research M, Vol. 26, 225-236, 2012.
25. Piatek, Z., et al., "Exact closed form formula for mutual inductance of conductors of rectangular cross section," Przeglad Elektrotechniczny (Electrical Review), R. 89, No. 3a, 61-64, 2013.
26. Piatek, Z., et al., "Self inductance of long conductor of rectangular cross section," Przeglad Elektrotechniczny (Electrical Review), R. 88, No. 8, 323-326, 2012.
27. Piatek, Z., et al., "Mutual inductance of long rectangular conductors," Przeglad Elektrotechniczny (Electrical Review), R. 88, No. 9a, 175-177, 2012.
28. Broydé, F., E. Clavelier, and L. Broydé, A direct current per-unit-length inductance matrix computation using modified partial inductance, Proc. of the CEM 2012 Int. Symp. on Electromagnetic Compatibility, Rouen, April 25-27, 2012.
29. Hoer, C. and C. Love, "Exact inductance equations for rectangular conductors with application to more complicated geometries," J. Res. NBS, Vol. 69C, No. 2, 127-137, 1965.
30. Zhong, G. and C. K. Koh, "Exact form formula for mutual inductance of on-chip interconnects," IEEE Trans. on Circ. and Sys., I: FTA, Vol. 10, 1349-1353, 2003. doi:10.1109/TCSI.2003.817778
31. Antonini, G., A. Orlandi, and C. R. Paul, "Internal impedance of conductor of rectangular cross section," IEEE Trans. on Microwave Theory and Tech., Vol. 47, No. 7, 979-984, 1999. doi:10.1109/22.775429
32. Canova, A. and L. Giaccone, "Numerical and analytical modeling of busbar systems," IEEE Trans. on Power Delivery, Vol. 24, No. 3, 1568-1577, July 2009. doi:10.1109/TPWRD.2009.2014270
33. Weeks, W. T., et al., "Resistive and inductive skin effect in rectangular conductors," IBM J. Res. Develop., Vol. 23, No. 6, 652-660, November 1979. doi:10.1147/rd.236.0652
34. Barr, A. W., "Calculation of frequency dependent impedance for conductor of rectangular cross section," AMP J. of Technology, Vol. 1, 91-100, November 1991.
35. Baron, B., et al., "Impedance of an isolated rectangular conductor," Przeglad Elektrotechniczny (Electrical Review), R. 89, No. 4, 278-280, 2013.
36. Comellini, E., A. Invernizzi, and G. Manzoni, "A computer program for determining electrical resistance and reactance of any transmission line," IEEE Trans. on Power Apparatus and Systems, Vol. PAS-92, 308-314, 1973. doi:10.1109/TPAS.1973.293628
38. Angi, H., M. Weiming, and Z. Zhihua, New numerical methods of computing internal inductance of conductor of rectangular cross-section, Asia-Pacific Symposium on Electromagnetic Compatibility and 19th International Zurich Symposium on Electromagnetic Compatibility, 674-677, 2008.
39. Matsuki, M. and A. Matsushima, "Improved numerical method for computing internal impedance of a rectangular conductor and discussions of its high frequency behavior," Progress In Electromagnetics Research M, Vol. 23, 139-152, 2012. doi:10.2528/PIERM11122105
40. Matsuki, M. and A. Matsushima, "Efficient impedance computation for multiconductor transmission lines of rectangular cross section," Progress In Electromagnetics Research B, Vol. 43, 373-391, 2012.
41. Battauscio, O., M. Chiampi, and D. Chiarabaglio, "Experimental validation of a numerical model of busbar systems," IEE Proceedings --- Generation, Transmission and Distribution, 65-72, 1995. doi:10.1049/ip-gtd:19951489
42. Birtwistle, D. and P. Pearl, "Measurement of impedance, power loss and current distribution in three-phase busbars," J. of Electrical and Electronics Engineering, Australia --- IE Aust. & IREE Aust., Vol. 8, No. 1, 37-46, 1988.
43. Du, J., J. Burnett, and Z. C. Fu, "Experimental and numerical evaluation of busbar trunking impedance," Electric Power Systems Research, No. 55, 113-119, 2000. doi:10.1016/S0378-7796(99)00104-2
44. Battauscio, O., et al., "Numerical and experimental evaluation of magnetic field generated by power busbar systems," IEE Proc. --- Gener. Transm. Distrib., Vol. 143, No. 5, 455-460, 1996. doi:10.1049/ip-gtd:19960557
45. Konrad, A., "Interodifferential finite element formulation of two-dimensional steady-state skin effect problems," IEEE Trans. on Magn., Vol. MAG-18, 284-292, 1982. doi:10.1109/TMAG.1982.1061775
46. Deeley, E. M. and E. E. Okon, "An integral method for computing the inductance and A.C. resistance of parallel conductors," International Journal for Numerical Methods in Engineering, Vol. 12, 625-634, 1978. doi:10.1002/nme.1620120407
47. Kamon, M., M. J. Tsuk, and J. K. White, "FASTHENRY: A multipole-accelerated 3-D inductance extraction program," IEEE Trans. on Microwave Theory and Techniques, Vol. 42, No. 9, 1750-1758, September 1994. doi:10.1109/22.310584
48. Meeker, D., Finite Element Method Magnetics, version 4.2 (April 11, 2012, Mathematica Build), http://www.femm.info .