108 IEEE TRANSACTIONS ONELECTROMAGNETIC COMPATIBILITY, VOL. 52, NO. 1, FEBRUARY2010 Transient Response of Straight Thin Wires Located at Different Heights Above a Ground Plane Using Antenna Theory and Transmission Line Approach Dragan Poljak, Member, IEEE, Sinisa Antonijevic, Khalil El Khamlichi Drissi, and Kamal Kerroum Abstract—Transient electromagnetic field coupling to straight thin wires parallel to each other and located at different heights above a perfectly conducting or dielectric ground plane is ana- lyzed using wire antenna theory and a transmission line method. The time-domain antenna theory formulation is based on a set of the space-time Hallen integral equations. The transmission line approximation is based on the corresponding time-domain Telegrapher’s equations. The space-time integral equations arising from the wire antenna theory are handled by the time- domain Galerkin–Bubnov scheme of the indirect boundary ele- ment method. The time-domain Telegrapher’s equations are solved using the finite-difference time-domain method. Time-domain nu- merical results obtained with both approaches are compared to the results computed via NEC 2 code combined with an inverse Fourier transform procedure. Some illustrative comparisons of re- sults obtained by means of antenna theory and transmission line approach are presented throughout the paper. Index Terms—Dielectric half-space, straight thin wires, trans- mission line approximation, wire antenna theory. I. INTRODUCTION T IME-DOMAIN study of electromagnetic field coupling to arbitrary configurations of multiple wires finds numer- ous applications in many areas of electromagnetic compatibility (EMC) or antennas and propagation such as; antenna arrays, short-pulse radar, lightning protection systems, wideband radio communications, electromagnetic interference (EMI) coupling to printed circuit boards (PCB), power lines, etc. Generally, electromagnetic field coupling to multiple overhead wires can be analyzed by using wire antenna theory (full wave model) or a transmission line model [1]. The antenna theory approach is the exact one as the transmission line (TL) approximation fails to accurately account for resonances, the presence of a lossy ground, and the effects at the wire ends. The principal advantage of the TL approximation is simplicity and relatively low computational cost, and it is, under certain cir- cumstances, considered as a satisfactory compromise between a quasi-static approximation and the full-wave approach [1]. Though sufficient for longer lines with electrically small cross Manuscript received November 28, 2008; revised April 27, 2009 and September 20, 2009. First published January 26, 2010; current version published February 18, 2010. D. Poljak and S. Antonijevic are with the Department of Electronics, University of Split, Split HR-21000, Croatia (e-mail: Dragan.Poljak@fesb.hr; santon@fesb.hr). K. El Khamlichi Drissi and K. Kerroum are with the Laboratoire des Sci- ences et Mat´ eriaux pour l ’Electronique et, d’ Automatique (LASMEA), Blaise Pascal University, Clermont-Ferrand 63177, France (e-mail: drissi@lasmea. univ-bpclermont.fr; kamal.kerroum@lasmea.univ-bpclermont.fr). Digital Object Identifier 10.1109/TEMC.2009.2038486 sections, the classical TL approach is not valid for straight thin wires of finite length and for high-frequency excitations. Con- sequently, the analysis of multiple overhead finite-length wires requires the antenna theory model. On the other hand, the main restriction of the wire antenna model applied to longer lines is related to a rather high computational cost. Under certain cir- cumstances, it is possible to improve a TL model to overcome some limitations of the method. The classical TL approach has been extended to the combined electromagnetic field to TL cou- pling equations valid for overhead wires above perfectly con- ducting (PEC) ground in [2] and [3], while a relationship be- tween TL equations and integral equations arising from the wire antenna theory in the frequency domain including the effect of a lossy ground has been presented in [4]. In particular, in [4], the standard set of Telegrapher’s equations is deduced from the gen- eralized Telegrapher’s equations derived from antenna theory. An efficient TL approach valid for multiple-wire configurations has been presented in [5]. As it is extremely difficult to handle the case of a finitely conducting half-space directly in the time domain, in this pa- per, the electromagnetic field coupling to straight thin wires above a dielectric or perfect ground at different heights has been analyzed by using both wire antenna theory and the TL method. Basically, this paper is a direct extension of the work related to the case of a single wire presented in [6]. The for- mulation arising from the time-domain antenna theory is based on the space-time Hallen integral equations. The time-domain TL model is based on the corresponding Telegrapher’s equa- tions. The set of Hallen integral equations is numerically han- dled via the time-domain Galerkin–Bubnov scheme of the in- direct boundary element method (GB-IBEM) [7], [8], while the time-domain Telegrapher’s equations are treated using the finite-difference time-domain (FDTD) method [5]. Computational examples are related to two-wire and three- wire TLs, respectively, excited by a transient incident electric field. The numerical results calculated via direct time-domain codes based on the antenna and TL approach, respectively, are compared to the results obtained via NEC2 [9] and inverse Fourier transform. II. FORMULATION A. Geometry of Interest An arbitrary number of multiple thin wires located above an infinite ground plane (lower medium is assumed to be either an 0018-9375/$26.00 © 2010 IEEE