In this paper, a new technique is developed to evaluate efficiently the Sommerfeld integrals arising from the problem of a current element radiating over a lossy half-space. The annihilation of the asymptote and the branch-point singular behavior of the spectral Green's function is used in this technique. The contributions of the subtracted asymptotic and singularity terms are calculated analytically. The annihilation results in a remaining integral that is very smooth and can be calculated adaptively by using Gaussian quadratures and extrapolation methods to accelerate the convergence of the oscillating integrand. The accuracy and efficiency of the new technique has been confirmed by comparison with literature, and the commercial software NEC. The application of the proposed technique provides a robust and rapid procedure to calculate spatial Green's functions which can be used in ground-wave propagation, and lightning return stroke channel modeling.
Zaker Hossein Firouzeh,
"Efficient Evaluation of Green's Functions for Lossy Half-Space Problems," Progress In Electromagnetics Research,
Vol. 109, 139-157, 2010. doi:10.2528/PIER10062310
1. Arand, B. A., M. Hakkak, K. Forooraghi, and J. R. Mohassel, "Analysis of aperture antennas above lossy half-space," Progress In Electromagnetics Research, Vol. 44, 39-55, 2004. doi:10.2528/PIER03022203
2. Wait, J. R., "The ancient and modern history of EM ground-wave propagation," IEEE Antenna & Propag. Magazine, Vol. 40, No. 5, 7-24, Oct. 1998. doi:10.1109/74.735961
3. Daniels, D. J., Ground Penetrating Radar, 2 Ed., IEE Radar, Sonar, And Navigation, Series 15, 2004. doi:10.1049/PBRA015E
4. Poljak, D. and V. Doric, "Wire antenna model for transient analysis of simple grounding systems, Part II: The horizontal grounding electrode," Progress In Electromagnetics Research, Vol. 64, 167-189, 2006. doi:10.2528/PIER06062102
5. Shoory, A., R. Moini, S. H. H. Sadeghi, and V. A. Rakov, "Analysis of lightning radiated electromagnetic fields in the vicinity of lossy ground," IEEE Trans. Electromagn. Compat., Vol. 47, No. 1, 131-145, 2005. doi:10.1109/TEMC.2004.842104
6. Sommerfeld, A., "Uber die Ausbreitung der Wellen in der drahtlosen Telegraphie," Ann. Phys., Vol. 28, 665-736, 1909. doi:10.1002/andp.19093330402
7. Sommerfeld, A., Partial Differential Equations, 246-267, Academic Press, New York, 1949.
8. Banos, A., Dipole Radiation in the Presence of a Conducting Halfspace, Pergamon Press, New York, 1966.
9. Rahmat-Samii, Y., R. Mittra, and P. Parhami, "Evaluation of Sommerfeld integrals for lossy half-space problems," Electromagnetics, Vol. 1, No. 1, 1-28, 1981. doi:10.1080/02726348108915122
10. Burke, G. J., E. K. Miller, J. N. Brittingham, D. L. Lager, R. J. Lytle, and J. T. Okada, "Computer modeling of antennas near the ground," Electromagnetics, Vol. 1, No. 1, 29-49, 1981. doi:10.1080/02726348108915123
11. Mohsen, A., "On the evaluation of Sommerfeld integrals," IEE Proc. H, Microwaves, Opt. & Antennas, Vol. 129, No. 4, 177-182, 1982. doi:10.1049/ip-h-1.1982.0036
12. Michalski, K. A. and C. M. Butler, "Evaluation of Sommerfeld integrals arising in the ground stake antenna problem," IEE Proc., Vol. 134, No. 1, Pt. H, Feb. 1987.
13. Siegel, M. and R. W. P. King, "Electromagnetic fields in a dissipative half-space, a numerical approach," J. Appl. Phys., Vol. 41, No. 6, 2415-2423, 1970. doi:10.1063/1.1659237
14. Tsang, L., R. Brown, J. A. Kong, and G. Simmons, "Numerical evaluation of electromagnetic fields due to dipole antennas in the presence of stratified media," Geophys. Res., Vol. 79, 2077-2080, 1974. doi:10.1029/JB079i014p02077
15. Kuo, W. C. and K. K. Mei, "Numerical approximations of the Sommerfeld integral for fast convergence," Radio Sci., Vol. 13, No. 3, 407-415, 1978. doi:10.1029/RS013i003p00407
16. Lin, C. C. and K. K. Mei, "Radiation and scattering from partially buried vertical wires," Electromagnetics, Vol. 2, No. 4, 309-334, 1982. doi:10.1080/02726348208915170
17. Katehi, P. B. and N. G. Alexopoulos, "Real axis integration of Sommerfeld integrals with applications to printed circuit antennas," J. Math. Phys., Vol. 24, No. 3, 527-533, 1983. doi:10.1063/1.525722
18. Johnson, W. A. and D. G. Dudley, "Real axis integration of Sommerfeld integrals: Source and observation points in air," Radio Sci., Vol. 18, 175-186, 1983. doi:10.1029/RS018i002p00175
19. Michalski, K. A., "On the efficient evaluation of integrals arising in the Sommerfeld half space problem," IEE Proc. H, Microw. Antenna & Propap., Vol. 132, 312-318, 1985. doi:10.1049/ip-h-2.1985.0056
20. Miller, E. K., A. J. Poggio, G. J. Burke, and E. S. Selden, "Analysis of wire antennas in the presence of a conducting halfspace. Part I. The vertical antenna in free space," Canadian J. Phys., Vol. 50, 879-888, 1972. doi:10.1139/p72-125
21. Sarkar, T. K., "Analysis of arbitrarily oriented thin wire antennas over a plane imperfect ground," Arch. Elek. Ubertragungstech. Electronics and Communications, Vol. 31, 449-457, 1977.
22. Fuller, J. A. and J. R. Wait, "A pulsed dipole in the earth," Transient Electromagnetic Fields, L. B. Felsen (ed.), 237-269, Springer, 1976.
23. Kong, J. A., L.-C. Shen, and L. Tsang, "Field of an antenna submerged in a dissipative dielectric medium," IEEE Trans. Antennas Propag., Vol. 25, 887-889, 1977. doi:10.1109/TAP.1977.1141703
24. Franssens, G. R., "Calculation of synthetic seismograms in layered media by means of a modified propagator matrix method," Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen (ed.), 357-373, Martinus Nijhoff,1984.
25. Yokoyama, A., "Dipole radiation affected by the plane earth," Phys. Soc. Jpn., Vol. 27, 224-229, 1969.
26. Chang, D. C. and J. R. Wait, "Appraisal of near-field solutions for a Hertzian dipole over a conducting half-space," Canadian J. Phys., Vol. 48, 737-743, 1970. doi:10.1139/p70-092
27. Kuo, W. C., "Numerical treatment of Sommerfeld's integrals and its application to antenna and scattering problems,", Ph.D. Thesis, Electrical Engineering Department, University of California, Berkeley, 1973.
28. Chang, D. C. and R. J. Fisher, "A unified theory on radiation of a vertical electric dipole above a dissipative earth," Earth Sci., Vol. 9, 1129-1138, 1974.
29. Linsay, Jr., J. E., "A direct approach to the numerical evaluation of electromagnetic fields due to sources buried beneath the earth's surface," IEEE-AP International Sympiosium, Boulder, Colorado, 1973.
30. Wait, J. R., "Image theory of a quasistatic magnetic dipole over a dissipative half-space," Electron. Lett., Vol. 5, No. 13, 281-282, 1969. doi:10.1049/el:19690214
31. Weaver, J. T., "Image theory for an arbitrary quasistatic field in the presence of a conducting half-space," Radio Sci., Vol. 6, 647-653, 1971. doi:10.1029/RS006i006p00647
32. Bannister, P. R., "Summary of image theory expansions for the quasistatic fields of antennas at or above the earth's surface," Proc. IEEE, Vol. 67, 1001-1008, 1979. doi:10.1109/PROC.1979.11381
33. Mittra, R., P. Parhami, and Y. Rahmat-Samii, "Solving the current element problem over lossy half-space without Sommerfeld's integrals," IEEE Trans. Antennas Propag., Vol. 27, 778-782, 1979. doi:10.1109/TAP.1979.1142177
34. Mahmoud, S. F., "Image theory for electric dipoles above a conducting anisotropic earth," IEEE Trans. Antenna Propag., Vol. 32, 679-683, 1984. doi:10.1109/TAP.1984.1143399
35. Lindell, I. and E. Alanen, "Exact image theory for the Sommerfeld half-space problem --- Part I: Vertical magnetic dipole," IEEE Trans. Antennas Propag., Vol. 32, 126-133, 1984. doi:10.1109/TAP.1984.1143278
36. Lindell, I. and E. Alanen, "Exact image theory for the Sommerfeld half-space problem --- Part II: Vertical electric dipole," IEEE Trans. Antennas Propag., Vol. 32, 841-847, 1984. doi:10.1109/TAP.1984.1143431
37. Lindell, I. and E. Alanen, "Exact image theory for the Sommerfeld half-space problem --- Part III: General formulation," IEEE Trans. Antennas Propag., Vol. 32, 1027-1032, 1984. doi:10.1109/TAP.1984.1143204
38. Lindell, I., E. Alanen, and H. Von Bagh, "Exact image theory for the calculation of fields transmitted through a planar interface of two media," IEEE Trans. Antennas Propag., Vol. 34, No. 2, 129-137, 1986. doi:10.1109/TAP.1986.1143788
39. Xu, X.-B. and Y. Huang, "An efficient analysis of vertical dipole antennas above a lossy half-space," Progress In Electromagnetics Research, Vol. 74, 353-377, 2007. doi:10.2528/PIER07052202
40. Collin, R. E., Field Theory of Guided Waves, McGraw-Hill, New York, 1960.
41. Kong, J. A., Electromagnetic Wave Theory, Wiley, New York, 1981.
42. Chew, W. C., Waves and Fields in Inhomogeneous Media, Van Nostrand, New York, 1990.
43. Felsen, L. B. and N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall, 1973.
44. Brekhovskikh, L. M., Waves in Layered Media, Academic Press, 1980.
45. Parhami, P., Y. Rahmat-Samii, and R. Mittra, "An efficient approach for evaluating Sommerfeld integrals encountered in the problem of a current element radiating over lossy ground," IEEE Trans. Antennas Propag., Vol. 28, 100-104, 1980. doi:10.1109/TAP.1980.1142273
46. Ling, R. T., J. D. Scholler, and P. Ya. Ufimtsev, "The propagation and excitation of surface waves in an absorbing layer," Progress In Electromagnetics Research, Vol. 19, 49-91, 1998. doi:10.2528/PIER97071800
47. Tsang, L. and J. A. Kong, "Electromagnetic fields due to a horizontal electric dipole antenna laid on the surface of a two-layer medium," IEEE Trans. Antennas Propag., Vol. 22, 709-711, 1974. doi:10.1109/TAP.1974.1140868
48. Dos Santos, A. F., "Electromagnetic-wave propagation along a horizontal wire above ground," Proc. IEE, Vol. 119, 1103-1109, 1972.
49. Tsang, L., C.-J. Ong, and B. Wu, "Electromagnetic fields of Hertzian dipoles in thin-layered media," IEEE Antennas Wireless Propag. Lett., Vol. 5, 537-540, Dec. 2006. doi:10.1109/LAWP.2006.889553
50. Mosig, J. R. and T. K. Sarkar, "Comparison of quasistatic and exact electromagnetic fields from a horizontal electric dipole above a lossy dielectric backed by an imperfect ground plane," IEEE Trans. Microw. Theory Tech., Vol. 34, 379-387, Apr. 1986. doi:10.1109/TMTT.1986.1133357
51. Michalski, K. A., "Extrapolation methods for Sommerfeld integral tails," IEEE Trans. Antennas Propag., Vol. 46, No. 10, 1405-1418, Oct. 1998. doi:10.1109/8.725271
52. Mosig, J. R. and F. E. Gardiol, "Analytical and numerical techniques in the Green's functions treatment of microstrip antennas and scatterers," Proc. Inst. Elect. Eng. Microwave Antennas and Propagation, Vol. 130, 175-182, Mar. 1983.
53. Mosig, J. R. and A. A. Melcón, "Green's functions in lossy layered media: Integration along the imaginary axis and asymptotic behavior," IEEE Trans. Antenna Propag., Vol. 51, 3200-3208, Dec. 2003.
54. Demuynck, F. J., G. A. E. Vandenbosch, and A. R. Van de Capelle, "The expansion wave concept --- Part I: Efficient calculation of the spatial Green's functions in a stratified dielectric medium," IEEE Trans. Antenna Propag., Vol. 46, 397-406, 1998. doi:10.1109/8.662659
55. Simsek, E., Q. H. Liu, and B. Wei, "Singularity subtraction for evaluation of Green's functions for multilayer media," IEEE Trans. Microw. Theory Tech., Vol. 54, 216-225, Jan. 2006. doi:10.1109/TMTT.2005.860304
56. Chow, Y. L., J. J. Yang, D. G. Fang, and G. E. Howard, "A closed-form spatial Green's function for the thick microstrip substrate," IEEE Trans. Microw. Theory Tech., Vol. 39, No. 3, 588-592, Mar. 1991. doi:10.1109/22.75309
57. Yang, J. J., Y. L. Chow, and D. G. Fang, "Discrete complex images of a three-dimensional dipole above and within a lossy ground," IEE Proc., Vol. 138, No. 4, Pt. H, 319-326, Aug. 1991.
58. Tang, J. L. and W. Hong, "The electromagnetic field produc by a horizontal electric dipole over a dielectric coated perfect conductor," Progress In Electromagnetics Research, Vol. 36, 139-152, 2005.
59. Zhang, H.-Q., W.-Y. Pan, K. Li, and K.-X. Shen, "Electromagnetic field for a horizontal electric dipole buried inside a dielectric layer coated high lossy half space," Progress In Electromagnetics Research, Vol. 50, 163-186, 2005. doi:10.2528/PIER04052301
60. Mei, J. P. and K. Li, "Electromagnetic field from a horizontal electric dipole on the surface of a high lossy medium coated with a uniaxial layer," Progress In Electromagnetics Research, Vol. 73, 71-91, 2007. doi:10.2528/PIER07032604
61. King, R. W. P., "On the radiation efficiency and the electromagnetic field of a vertical electric dipole in the air above a dielectric or conducting half-space," Progress In Electromagnetics Research, Vol. 4, 1-43, 1991.
62. Abramowitz, M. and I. Stegun, Handbook of Mathematical Functions, 488, Dover, New York, 1970.
63. Burke, G. J. and A. J. Poggio, Numerical Electromagnetic Code (NEC)-method of Moments, Vol. 116, Naval Ocean Syst. Control, San Diego, CA, 1980.
64. Burke, G. J., Numerical Electromagnetic Code (NEC4)-method of Moments, Part 2: Program Description-theory, Lawrence Livermore National Laboratory, UCRL-MA-109338, 1992.