The electromagnetic field of a current conductor in the presence of lossy half-space

Main Article Content

Marijana Gavrilović Božović
Mina Vasković Jovanović
Milan Čabarkapa
Nikola Mijailović
Milan Vesković
Jasna Radulović


In this paper, the electromagnetic field problem of a line current conductor in the presence of homogenous and isotropic lossy half-space is analyzed. The main difficulty in this analysis is to take correctly into account the influence of ground finite conduct since the use of integral transformations in Maxwell's equations leads to Sommerfeld integral, hard to evaluate even numerically. Charge Simulation Method, based on the equivalence theorem of different electromagnetic systems, is proposed here for the numerical approach. Two independent, equivalent systems created of appropriately chosen, shaped, and arranged fictitious sources are suggested for the determination of EM field components, in the air and in the lossy half-space. Point-matching method based on the boundary conditions between media is applied for the determination of the fictitious source intensities. The proposed procedure has the advantage that it takes into account the influence of the finite ground conductivity, without calculation of Sommerfeld integral and it ensures excel-lent accuracy with a small number of fictitious sources. Also, the features of the proposed method are simple implementation on a standard PC, fast and accurate simulation procedure, and easy generalization for some other cases of arbitrarily oriented wires of finite lengths above a lossy half space.

Article Details

How to Cite
M. Gavrilović Božović, M. Vasković Jovanović, M. Čabarkapa, N. Mijailović, M. Vesković, and J. Radulović, “The electromagnetic field of a current conductor in the presence of lossy half-space”, ET, Jul. 2024.
Original Scientific Papers


K. A. Michalski and J. R. Mosig, “On the complete radiation pattern of a vertical Hertzian dipole above a low-loss ground”, IEEE Journal of Microwaves, Vol.1(3), pp. 747-762., (2021)

K. Ioannidi, Ch . Christakis, S. Sautbekov, P. Frangos, and S. Atanov, “The radiation problem from a vertical short dipole antenna above flat and lossy ground: Novel formulation in the spectral domain with closed – form analytical solution in the high frequency regime”, International Journal of Antennas and Propagation, Vol. 2014, pp. 1-9,, (2014)

A. Chrysostomou, S. Bourgiotis, S. Sautbekov, K. Ioannidi, and P. Frangos, “radiation of a vertical dipole antenna over flat and lossy ground: accurate electromagnetic field calculation using the spectral domain approach along with redefined integral representations and corresponding novel analytical solution”, Elektronika ir elektrotechnika, Vol. 22(2), pp. 54-61,, (2016)

J. Mosig, K. Michalski, “Sommerfeld integrals and their relation to the development of planar microwave devices”, IEEE Journal of Microwaves, Vol. 1(1), pp. 470-480,, (2021)

A. Sommerfeld, “Über die Ausbreitung der Wellen in der Drachtlosen Telegraphie“, Annalen der Physik, Vol. 333(4), pp. 665-736,, (1909)

J. R. Carson, “Wave propagation in overhead wires with ground return”, Proceedings of the 32nd Annual North American Power Symposium, Waterloo (Canada), (reprinted of the 1926 original edition), pp. 539–554, (2000)

R. G. Olsen, S. L. Backus, and R. D. Steams, “Development and validation of software for predicting ELF magnetic fields near power lines”, IEEE Transactions on Power Delivery, Vol. 10(3), pp.1525–1534,, (1995)

A. Geri, A. Locatelli, G. M. Veca, “Magnetic fields generated by power lines”, IEEE Transactions on Magnetics, Vol. 31(3) pp. 1508–1511,, (1995)

M. Abdel-Salam, H. Abdallah, M. Th. El-Mohandes, and H. El-Kishky, “Calculation of magnetic fields from electric power transmission lines”, Electric Power Systems Research, Vol. 49(2), pp. 99–105,, (1999)

G. Filippopoulos and D. Tsanakas, “Analytical calculation of the magnetic field produced by electric power lines” IEEE Transactions on Power Delivery, Vol. 20(2), pp. 1474–1482,, (2005)

R. Djekidel, A. Bessedik, A. Hadjadj, “Electric field modeling and analysis of ehv power line using improved calculation method”, Facta universitatis Series: Electronics and Energetics, Vol. 31(3), pp. 425-445,, (2018)

R. Wang , J. Tian, F. Wu, Z. Zhang, and H. Liu , “PSO/GA combined with charge simulation method for the electric field under transmission lines in 3D calculation model”, Electronics, Vol. 8(10), p. 1140,, (2019)

J. Radulović, V. Ranković, M. Bojić, and J.Skerlić, “Environmental impacts of the electromagnetic field levels near overhead transmission lines”, Environmental Engineering and Management Journal, Vol.13(3), pp. 627-633,, (2014)

D. Dib and M. Mordjaoui, “Study of the influence high-voltage power lines on environment and human health (Case study: The electromagnetic pollution in Tebessa City, Algeria)”, Journal of Electrical and Electronic Engineering, Vol. 2(1), pp. 1-8,, (2014)

S. Boumous, N.Guettaf, A.Hamel, I. Lariche, and H. Nouri, “Effect on human body of the magnetic field induced by the high voltage transmission line”, European Journal of Electrical Engineering, Vol. 23(2), pp.149-155,, (2022)

J. G. Gurney, S. Davis, S. M. Schwartz, B. A. Mueller, W. T. Kaune, and R. G. Stevens, “Childhood cancer occurrence in relation to power line configurations: A study of potential selection”, Epidemiology, Vol. 6 (1), pp. 31-35,, (1995)

J. Radulović and V. Ranković, “Feedforward neural network and adaptive network-based fuzzy inference system in study of power lines”, Expert Systems with Applications, Vol. 37(1), pp. 165-170,, (2010)

J. Radulović and V. Ranković, “Application of RBF neural network and ANFIS for electromagnetic field prediction around the power lines”, Proceedings of the ninth International Conference on Telecommunications in Modern Satellite, Cable and abroadcasting Services “TELSIKS 2009”, Niš (Serbia), pp. 457-460,, (2009)

E. Turajlić, A. Alihodžić, and A. Mujezinović, “Artificial neural network models for estimation of electric field intensity and magnetic flux density in the proximity of overhead transmission line”, Radiation Protection Dosimetry, Vol. 199(2), pp. 107-115,, (2022)

M. Rančić, R. Jankoski, S. Silvestrov, and S. Aleksić, “Analysis of horizontal thin-wire conductor buried in lossy ground: New model for Sommerfeld type integral”, Springer Proceedings in Mathematics & Statistics “Engineering Mathematics I”, Vol 178, pp. 33-49,, (2016)

G. A. Skianis, “A class of Fourier integrals based on the electric potential of an elongated dipole”, SpringerPlus, Vol. 3(1), p. 729,, (2014)

J.R. Wait, “EM fields of a phased line current over a conducting half-space”, IEEE Transactions on Electromagnetic Compatibility, Vol. 38(4), pp. 608–611,, (1996)

O. M. Abo-Seida, “Estimation of the electromagnetic field created at the earth’s surface by an overhead line current”, Applied Mathematics and Computation, Vol. 149(2), pp. 559–565,, (2004)

P. L. Tokarsky, “Dipole field over a lossy half-space: An approximate analysis”, Telecommunications and Radio Engineering, Vol. 55(5), p. 6,, (2001)

S. Sautbekov, M. Sautbekova, K. Baisalova, and M. Pshikov, “Calculation of Sommerfeld integrals in dipole radiation problems”, Mathematics, Vol. 12(2), p. 298,, (2024)

K. A. Michalski and H.-I. Lin, “On the Sommerfeld half-space problem: appraisal of approximate solutions with extensions to plasmonics”, Journal of Electromagnetic Waves and Applications, Vol. 32(4), pp. 483–503,, (2018)

S. Bourgiotis, P. Frangos, S. Sautbekov, and M. Pshikov, “The evaluation of an asymptotic solution to the Sommerfeld radiation problem using an efficient method for the calculation of Sommerfeld integrals in the spectral domain”, Electronics, Vol. 10(11), p. 1339,, (2021)

M. Rančić and S. Aleksić, “Horizontal dipole antenna very close to lossy half-space surface”, Przegląd Elektrotechniczny (Electrical Review), Vol. 88, pp. 82-85, (2012)

M. P. Rančić and P. D. Rančić, “Vertical linear antennas in the presence of a lossy half-space: An improved approximate model”, AEU - International Journal of Electronics and Communications, Vol. 60(5), pp. 376–386,, (2006)

D. Poljak, B. Jajac, and R. Šimundic, “Current induced along horizontal wire above an imperfectly conducting half-space”, Engineering Analysis with Boundary Elements, Vol. 23(10), pp. 835–840,, 1999

R. M. Shubair and Y. L. Chow, “A simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half-spaces”, IEEE Transactions on Antennas and Propagation, Vol. 41(6), pp. 806–812,, (1993)

B. A. Arand, M. Hakkak, K. Forooraghi, and J. R. Mohassel, “Analysis of vertical wire antenna above lossy ground using discrete complex image method”, AEU - International Journal of Electronics and Communications, Vol. 57(5), pp. 333–337,, (2003)

B. A. Arand and M. Hakkak, “Horizontal wire antenna above lossy half-space”, Proceedings of the 2003 IEEE International Symposium on Antennas and Propagation: URSI North American Radio Science Meeting, Columbus (USA), Vol. 4, pp. 970–973,, (2003)

H. F. Arnoldus, “Numerical evaluation of Sommerfeld-type integrals for reflection and transmission of dipole radiation”, Computer Physics Communications, Vol. 257, p. 107510,, (2020)

A. Khalatpour, R. Sarraf Shirazi, and G. Moradi, “Analysis of vertical wire antennas above lossy half-space using matrix pencil method”, AEU - International Journal of Electronics and Communications, Vol. 64(8), pp. 784–789,, (2010)

M. P. Rančić and P. D. Rančić, “Field pattern of the vertical dipole antenna above a lossy half-space,” Serbian Journal of Electrical Engineering, Vol. 2(2), pp. 125–136,, (2005)

D. Veličković and J. Radulović, “Line conductor over semi-conducting half space”, Proceedings of the Fourth International Conference of Applied Electrostatics “PES ’96”, Niš, 1996, pp. 53-58, (1996)

J. Radulović, “Jedan novi pristup za proračun elektromagnetnog polja lineičnog provodnika iznad poluprovodnog poluprostora”, PhD Thesis, University of Kragujevac (Serbia), (2001)

M. Štafl, “Electrodynamics of electrical machines”, Academia, Prague (Czechoslovakia), (1967)

D. Rabah, C. Abdelghani, and H. Abdelchafik, “Efficiency of some optimisation approaches with the charge simulation method for calculating the electric field under extra high voltage power lines”, IET Generation, Transmission & Distribution, Vol. 11(17), pp. 4167–4174,, (2017)

H. Das, “Implementation of basic charge configurations to charge simulation method for electric field calculations”, International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, Vol. 3(5), pp. 9607-9611, (2014)

M. Dhamodaran and R. Dhanasekaran, “A comparative study of computational electromagnetics”, Proceedings of ICMTSET 2013 & IRCEBM 2013, Dubai (UAE), pp. 128-132, (2013)

G. B. Arfken, H. J. Weber, F. E. Harris, “Mathematical methods for physicists - A comprehensive guide”, Academic Press, Cambridge (USA),, (2012)