IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 3, AUGUST 2007 689 Generalized Form of Telegrapher’s Equations Electromagnetic Field Coupling to Finite-Len Lines Above a Lossy Ground Dragan Poljak, Member, IEEE, Farhad Rachidi, Senior Member, IEEE, and Sergey V. Tkachenko, Senior Member, IEEE Abstract—In this paper, a generalized form of the Telegrapher’s equations for electromagnetic field coupling to finite-length trans- mission lines above a lossy ground is derived. The approach is fully based on the thin-wire antenna theory. The effect of a lossy half- space is taken into account by means of the reflection coefficient approximation. The conductor losses can also be taken into account via surface impedance per unit length. The resulting equations are handled numerically via the Galerkin–Bubnov indirect boundary element method. Numerical results are presented for induced cur- rentalong the line, and compared with transmission line (TL) approximation, for the case of lossless conductor. It is shown that the TL approximation can result in a significant underestimation of the induced currents. Index Terms—Antenna theory,electromagnetic coupling, ground plane, radiation effects, transmission line modeling. I. I NTRODUCTION T HE ANALYSIS of electromagnetic field coupling to lines can be performed by using the thin-wire antenna or trans- mission line (TL) model [1]. The first approach, based on an- tenna theory, is the more rigorous one as the TL theory does notprovide a complete solution for the excitation of victim equipment by an incident field if the wavelength of the elec- tromagnetic field coupling to a transmission line is comparable to or less than the transverse electrical dimensions of the line. The TL approximation is usually considered as a compromise between a quasi-static approximation and the receiving antenna model [1]. This approach, though sufficient if long lines with electrically-small cross sections are considered, fails if one deals with the lines of finite length and high-frequency excitations. The TL model, therefore, fails to predict resonances and is not able to recognize the presence of a lossy ground. Also, the ef- fects at the line ends cannot be taken into account utilizing this approach [1]– [6]. Hence, when the lines of finite length are of interest, the receiving antenna theory has to be used. Thus,the wave-like behavior of the induced responses at higher frequencies requires a more general approach, which is Manuscript received November 3, 2005; revised June 30, 2006. D. Poljak is with the Department of Electronics, University of Split, HR- 21000 Split, Croatia, and also with the Wessex Insitute of Technology, Ashurst, Southampton S040 7AA, U.K. (e-mail: dpoljak@fesb.hr). F. Rachidiis with the Swiss Federal Institute of Technology, CH-1015 Laussane, Switzerland (e-mail: Farhad.Rachidi@epfl.ch). S. V. Tkachenko is with the Institute for Fundamental Electrical Engineer- ing and Electromagnetic Compatibility, Otto-von-Guericke University, D-39106 Magdeburg, Germany (e-mail: sergey.tkachenko@etechnik.uni-magdeburg.de). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2007.902179 Fig. 1. Finite-length line above a lossy ground. based on integral equations arising from wire antenna the the other hand, the restrictions of the wire antenna model to overhead lines are often related to the long computation required for calculations pertaining to the long lines. An extension of the standard TL approach to the combin electromagnetic field to transmission line coupling equatio valid foroverhead wires has been presented in [2] and [3]. The approach presented in [2] and [3] is based on the perf conducting (PEC) ground approximation. The improvemen this important contribution can be done in two directions: 1) to include the effect of a lossy ground in Tkatchenko et al.’s [3] corrected form of Telegrapher’s equations; 2) to provide a clear relationship between TL equations a antenna theory. These represent the objectives of the present study in w a generalized type of the Telegrapher’s equations, including the presence of a lossy ground and conductor loss, is deriv The presented derivations provide a clear correlation betw the rigorous antenna theory approach to the analysis of fin length lines [7]–[9] based on the standard Pocklington inte equation formulation and the classic Telegrapher’s equatio proach arising from the transmission line theory. The influ of a lossy ground is taken into account via the Fresnel refle coefficient appearing within the Green function, while con tor loss is considered via the surface impedance per unit le In particular, the standard set of Telegrapher’s equations is deduced from the generalized Telegrapher’s equations. II. M ATHEMATICAL M ODEL OF S INGLE -WIRETRANSMISSION LINEABOVELOSSYGROUND A transmission line of finite-length L and radius a located at height h above an imperfect ground and illuminated by incident electromagnetic field (see Fig. 1) is considered. 0018-9375/$25.00 © 2007 IEEE