Time domain analysis of thin-wire antennas over lossy ground
using the reflection-coefficient approximation
M. Ferna´ndez Pantoja,1 A. G. Yarovoy,2A. Rubio Bretones,1 and S. Gonza´lez Garcı´a1
Received 5 February 2009; revised 29 July 2009; accepted 2 October 2009; published 23 December 2009.
[1] This paper presents a procedure to extend the methods of moments in time domain for the transient analysis of thin-wire antennas to include those cases where the antennas are located over a lossy half-space. This extended technique is based on the reflection coefficient (RC) approach, which approximates the fields incident on the ground interface as plane waves and calculates the time domain RC using the inverse Fourier transform of Fresnel equations. The implementation presented in this paper uses general expressions for the RC which extend its range of applicability to lossy grounds, and is proven to be accurate and fast for antennas located not too near to the ground. The resulting general purpose procedure, able to treat arbitrarily oriented thin-wire antennas, is appropriate for all kind of half-spaces, including lossy cases, and it has turned out to be as
computationally fast solving the problem of an arbitrary ground as dealing with a perfect electric conductor ground plane. Results show a numerical validation of the method for different half-spaces, paying special attention to the influence of the antenna to ground distance in the accuracy of the results.
Citation: Ferna´ndez Pantoja, M., A. G. Yarovoy, A. Rubio Bretones, and S. Gonza´lez Garcı´a (2009), Time domain analysis of thin-wire antennas over lossy ground using the reflection-coefficient approximation, Radio Sci., 44, RS6009, doi:10.1029/ 2009RS004152.
1. Introduction
[2] Transient analysis of thin-wire radiating structures
in the presence of dissipative half-spaces has been a matter of interest during recent decades, with applications in different fields such as ground penetrating radar (GPR) [Peters et al., 1994], bioelectromagnetics [Iskander, 1991], electromagnetic compatibility [Poljak, 2007], etc. [3] Numerical techniques for the simulation of these
problems can been developed with different approaches. One is to apply a inverse Fourier transform (IFT) to the well known solution of the thin-wire antenna problem in the frequency domain [Rahmat-Samii et al., 1978; Lestari et al., 2004]. Two different approaches constitute the basis of these frequency domain numerical algo-rithms. On the one hand, there are algorithms based on the solution of the Sommerfeld problem for horizontal or
vertical dipoles over lossy half-spaces, which turn out to be accurate but require intensive computational resources [Miller et al., 1972a, 1972b; Sarkar, 1977; Parhami and Mittra, 1980; Burke and Poggio, 1981; Burke et al., 1981; Burke and Miller, 1984; Cui and Chew, 2000a, 2000b]. On the other hand, there are solutions based on approximations such as the reflection-coefficient method [Miller et al., 1972a, 1972b; Sarkar, 1977; Burke and Poggio, 1981], which is computationally faster but, as it assumes that the waves incident on the ground are plane waves, it presents losses of accuracy when the approxi-mation is not valid [Karwoski and Michalski, 1987]. In any case, for wideband or ultrawideband systems the use of IFT is computationally inefficient, and numerical algorithms obtained directly in the time domain are advantageous compared with the aforementioned fre-quency domain techniques [Miller and Landt, 1980]. Therefore, additional efforts were devoted to the devel-opment of new methods of solution in the time domain. In this context, several authors [Rubio Bretones and Tijhuis, 1995, 1997; Tijhuis and Rubio Bretones, 2000; Vossen, 2003] have presented an extension of the Hallen’s time domain electric field integral equation (TD-EFIE) to include lossy half-spaces, based on the transient solution
for
Full Article
1
Departmento de Electromagnetismo, Facultad de Ciencias, University of Granada, Granada, Spain.
2
International Research Center of Telecommunications and Radar, Technical University of Delft, Delft, Netherlands.
Copyright 2009 by the American Geophysical Union. 0048-6604/09/2009RS004152$11.00
of the Sommerfeld problems presented by De Hoop and Frankena [1960] and Frankena [1960]. This approach, as its counterpart in the frequency domain, leads to accurate solutions but with an intensive use of computational resources. To overcome this disadvantage, time domain solutions under the RC approximation have been imple-mented by Poljak [2007], by employing the time domain RC inferred by Barnes and Tesche [1991], and satisfac-tory results have been showed for particular cases of two coupled horizontal wires over dielectric half-spaces [Poljak, 2007; Poljak et al., 2000].
[4] Moreover, parallel studies have recently been
devoted to find improved numerical expressions for the direct time domain calculation of RC [Rothwell and Suk, 2003, 2005]. The main advantages of these expressions are in their range of applicability, being useful for reflections produced over all kind of soils, in contrast to those from Barnes and Tesche [1991], which are restricted to specific conditions over the constitutive parameters of the half-space. Given that those restrictive conditions are not always fulfilled, the use of the more general approach given by Rothwell and Suk [2003, 2005] is advisable for general purpose electromagnetic codes.
[5] An important drawback of using the time domain
techniques developed so far is the poor efficiency in the treatment of strongly conductive soils. In the RC approximation the calculation of the transient response of conductive soils is performed by a convolution oper-ator [Poljak et al., 2000] between the incident electric field and the impulsive response of the soil. In cases where the late-time responses are of interest, this convo-lution is particularly intensive in terms of computational costs, and a bottleneck arises in terms of computational time in the simulations.
[6] In the present paper, we present a new algorithm
for time domain simulation of arbitrarily oriented thin-wire antennas over lossy ground, by applying a RC approximation for the Pocklington’s EFIE [Miller and Landt, 1980; Miller, 1994]. The main contributions of this work are: (1) wider applicability of the algorithm, by using recently proposed RC equations [Rothwell and Suk, 2003, 2005], (2) efficient treatment of all kind of conductive soils, by employing approximations derived from the analysis of impulsive response of the soil which drastically reduces the computational time for lossy grounds, and (3) ability to simulate arbitrarily oriented thin wires, by decomposing the interactions between different parts of the structures due to reflections on the ground, into those corresponding to waves polarized with the electric field parallel or perpendicular to the interface. The results are validated by using IFT to accurate frequency domain solutions.
[7] The paper is organized as follows, in section 2, an
extension of Pocklington EFIE equation to include wires
over lossy ground using the TD-RC method is described. Time domain reflection coefficients (TD-RC) needed for the formulation of the EFIE are presented in section 3, and a numerical approximation to decrease the compu-tational burden of the calculation for the case of con-ductive soils is proposed. Section 4 formulates the numerical procedure of solution of the EFIE, by applying a point-matching method of moments and lagrangian interpolation basis functions both in the time and space domain, and takes into account the numerical decompo-sition of the electric field incident on the ground into its components polarized either parallel or perpendicular to the interface (TE or TM polarization). Finally, section 5 shows results for simulations in different lossy grounds.
2. EFIE for Thin-Wire Antennas Over
Lossy Ground Using RC Method
[8] With the aim of solving the problem of a thin-wire
antenna located over a lossy ground, we first consider in section 2.1 the subproblem of the transient excitation of a single thin-wire segment, embedded in a homoge-neous, lossless dielectric with properties identical to those of the upper medium in which the antenna is placed. Section 2.2 presents the extension of the integral equation to account for the presence of a generally lossy ground located under the arbitrarily oriented thin-wire antenna. The new integral equation includes terms corresponding to the time domain reflection coefficients of the field radiated by a current element located arbitrarily oriented with respect to the interface. The calculation of these time domain RC is described later in section 3. Although the RC method can be inferred as an approximation of the exact solution derived from Sommerfeld’s problem [Sarkar, 1977], or as a multiplication of the Green function of the image source and the reflection coeffi-cients [Miller et al., 1972a, 1972b; Poljak et al., 2000], we have chosen to explain it more easily as a modification of the classical theory of images in which the fields radiated by the image source are scaled by taking into account properly the plane wave reflection coefficients.
2.1. EFIE for Thin-Wire Antennas in a Dielectric Space
[9] The EFIE is inferred taking as a starting point the
theorem of physical equivalent [Balanis, 1989], which states the identity, in terms of the electromagnetic fields outside a region bounded by a perfect electric conductor (PEC) surface S, embedded in a region of characteristics (e, m) excited by an incident field ~Ei(~r, t), between: (1) the original structure and (2) an equivalent problem considering only a set of surface equivalent electric currents ~Jsplaced on S, radiating in a homogeneous inner
region inner (i.e., within S) of constitutive parameters (e, m) equal to those of the outer region in the original problem. This set of equivalent currents radiates an electric field: ~ Esð~r; tÞ ¼ @ @t m 4p Z S ~J sð~r0;t0Þ R dS 0 ! r 1 4pe Z S rsð~r0;t0Þ R dS 0 ð1Þ
where ~r0notes the position of the source, placed on the surface S; ~r is the position vector of the field point; R is the distance between field and source, calculated as the modulus of the vector ~R =~r~r0; t0= tR
vis the retarded
time which assures the causality of the system, accounting for the propagation of electromagnetic fields in the medium of constitutive parameters (e, m) and velocity of propagation v = 1ffiffiffiffi
me
p ; and rs are the electric
charges related to ~Js byrs(~r0, t0) =
R
0 t0
[r0 ~Js(~r0,t)]dt.
[10] As the surface S is located at the contour of the
PEC, the tangential component of the total electric field is null on S, and the scattered field tangential to S, ~Es(~r, t) can be replaced by the incident field ~Ei(~r, t) in (1) in the form:
~Eið~r; tÞ tan¼ m 4p Z S @ @t ~J sð~r0;t0Þ R dS 0 ! tan þ 1 4pe Z S rrsð~r0;t0Þ R dS 0 tan ð2Þ
where ~r is located on the surface S, and ( )tanindicates
the tangential component of the vector inside the
parenthesis. In antenna or scattering problems where the incident field is given, equation (2) constitutes the starting point to determine the electromagnetic behavior of the PEC.
[11] To find the unknown electric currents ~Js(~r0, t0), we
can apply the continuity equation of charge, and with aid of some basic mathematical relationships, equation (2) leads to [Rubio Bretones et al., 1989]:
~Eið~r; tÞ tang¼ 1 4pe Z S 1 v2R @ @t~Js~r 0;t0 ð ÞdS0 tang þ 1 4pe Z S ~ R R3 Z t0 0 " r0 ~Js ~r0;t !# dt ! dS0 ! tang þ 1 4pe Z S ~R vR2 r 0 ~J sð~r0;t0Þ t0dS 0 ! tang ð3Þ which is known as the time domain EFIE for PEC. Brackets []t0 are employed to emphasize that the divergencer0
applies exclusively to the spatial variable in ~Js(~r0, t0).
[12] For a thin-wire structure, i.e., that whose radius a
is negligible compared with its length (Figure 1), two-dimensional surface currents ~Js(~r0, t0) on S can be
approx-imated as one-dimensional total currents ~I (~r0, t0) = 2pa~Js(~r0, t0), placed at the center of the thin-wire structure
and flowing along its axis. Thus, integrals involved in equation (3) are reduced in one order, and singularities associated with the calculation of field points placed on the surface S are removed. Applying this approximation to (3) gives the time domain EFIE for thin-wires PEC:
^s ~Eið~r; tÞ ¼ 1 4pe Z C ^s ^s0 v2R @ @tI ~r 0; t0 ð Þds0 þ 1 4pe Z C ^s ~R R3 Z t0 0 @ @r0I ~r 0;t ð Þdt ! ds0 þ 1 4pe Z C ^s ~R vR2 @ @r0I ~r 0; t0 ð Þds0 ð4Þ
where s0 and s account for positions located on the axis and on the surface of the wire, respectively (see Figure 1); C corresponds to the contour following the axis of the wire; and ^s and ^s0 are the tangential unit vectors on the surface S and at the axis of the wire, respectively.
[13] For those cases including a total of Nwthin wires,
equation (4) holds, by applying the superposition prin-ciple, in the form:
^s ~Eið~r; tÞ ¼ 1 4pe XNw k¼1 " Z Ck ^s ^s0 v2R @ @tIk~r 0;t0 ð Þds0 þ Z Ck ^s ~R R3 Z t0 0 @ @r0Ik~r 0; t ð Þdt ! ds0 # þ 1 4pe XNw k¼1 Z Ck ^s ~R vR2 @ @r0Ik~r 0 ;t0 ð Þds0 " # ð5Þ
[14] For sake of briefness, the rest of the paper is
developed on the basis of equation (4), assuming that the multiple thin-wire case remains valid by only considering (5).
2.2. EFIE Extension to Include Lossy Ground by the RC Method
[15] In cases where the thin wires are placed over a
ground plane, equation (4) is no longer valid, because it fails to take into account the contributions from the reflected electromagnetic field on the surface between the two different half-spaces. In order to consider them, the RC approximation approach holds that the total electric field at any point of the outer space of the antenna can be expressed as a sum of (1) a direct wave, which is determined by the direct scattered electric field (~Ed(~r, t)) given by the right hand side of the equation (4), and (2) a reflected wave, which is calculated by adding
the electric field radiated by each point source forming the image of the thin wire located into the ground plane. Figure 2 shows an example of the direct scattered field and the reflected fields due to the radiation of a thin-wire antenna (called thin-wire 1) above ground. The fields are calculated for a position located on a second antenna (wire 2) also located above ground and parallel to wire 1.
[16] The reflected wave can be effectively calculated
by an integration, along the axis of the image wire, of a convolution operator between the electric field radiated by each point source of the image wire placed at ~r0, and the time domain plane wave reflection coefficient G (~r, ~r0, t) corresponding to the constitutive parameters and type of incidence of the problem at hand. By doing this, equation (4) is modified to:
^s ~Eið~r; tÞ ¼ ^s ~Edð~r; tÞ þ G ~ðr;~r0;tÞ3~Erð~r; tÞ ð6Þ where ~Edð~r; tÞ ¼ 1 4pe Z C ^s0 v2R @ @tI ~r 0; t0 ð Þds0 þ 1 4pe Z C ~R vR2 @ @r0I ~r 0;t0 ð Þds0 þ 1 4pe Z C ~R R3 Z t0 0 @ @r0I ~r 0; t ð Þdt ! ds0 ð7Þ
and the operator3 is defined as a retarded convolution, along the path of the wire, between any reflection coefficientG (~r, ~r0, t) and any electric field ~E(~r, t):
G ~ðr;~r0;tÞ3~Erð~r; tÞ ¼ 1 4pe Z C G ~ðr;~r0;t0Þ " ^s0 v2R @ @tI ~r 0; t0 ð Þ þ ~R vR2 @ @r0I ~r 0; t0 ð Þ # ds0 þ 1 4pe Z C G ~ðr;~r0;t0Þ ~R R3 Z t0 0 @ @r0I ~r 0; t ð Þdt ! " # ds0 ð8Þ where the path C is established according the theory
of images, and accounts for the contour following the axis of the image wire. It bears remarking that, in equation (8), the velocity of propagation v corresponds Figure 2. Example of TE incidence for horizontal
to that from the upper media although the image is located inside the ground.
[17] Furthermore,G (~r,~r0, t) depend on the polarization
of the plane wave incident on the ground plane. Then, it is noted ^n as the unit vector normal to the ground plane in the point of incidence,GTM(~r, ~r0, t) as the RC in case
of the reflection of a vertically polarized wave (magnetic field tangential to the ground plane, as shown in Figure 3), and GTE (~r, ~r0, t) corresponds to the RC for
the case where the electric field is tangential to the ground plane (Figure 2), usually called the horizontal incidence. As shown by Burke and Poggio [1981], the reflected electric field ~Er(~r, t) at the ground plane can be decom-posed into its vertical component, ~Evr(~r, t) = (~Er(~r, t) ^n) ^n,
and horizontal component, ~Eh r
(~r, t) = [~Er(~r, t) (~Er(~r, t) ^
n) ^n]. Then, equation (6) can be written as:
^s ~Eið~r; tÞ ¼^s ~Edð~r; tÞ þ ^s G TMð~r;~r0;tÞ3~Ervð~r; tÞ þ ^s GTEð~r;~r0;tÞ3~Erhð~r; tÞ ¼ ^s ~Edð~r; tÞ þ ^s GTEð~r;~r0;tÞ3~Erð~r; tÞ þ ^s ð½GTMð~r;~r0;tÞ GTEð~r;~r0;tÞ3~Erð~r; tÞ ^n^n ð9Þ
which constitutes, by replacing equations (7) and (8) into (9), the extended EFIE equation that includes the effect of lossy grounds.
3. Computation of TD-RC
[18] Let us consider as a usual case the oblique
incidence of a plane wave from free space onto a
nonmagnetic lossy frequency-independent half-space, of constitutive parameters ( = r0, m0, s), which is
solved in the frequency domain by using the Fresnel RC [Wait, 1962]. Naming q0 as the angle between the
incident wave and normal vector to the interface, which can be inferred from the equation cos(q0) = (~r~r
0
j~r~r0j) ^n, the RC for TM incidence is:
GTMð~r;~r0;wÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r sin2q0 jws0 q cosq0 r jws0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r sin2q 0 jws0 q þ cosq0 r jws0 ð10Þ
and for TE incidence, the RC is:
GTEð~r;~r0;wÞ ¼ cosq0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r sin2q0 jws0 q cosq0þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r sin2q0 jws0 q ð11Þ
[19] Therefore, TD-RCs are defined from the inverse
Fourier transform of the Fresnel RC. In this way, TD-RC for TM polarization is given by [Suk and Rothwell, 2002b]: GTMð~r;~r0;tÞ ¼ 1pffiffiffiffiffiffiffiDM 1þpffiffiffiffiffiffiffiDMdðtÞ þ 2bMpffiffiffiffiffiffiffiDM DM 1 ðK3Mþ K4MÞebMtIðbMtÞuðtÞ 2 ffiffiffiffiffiffiffi DM p ðDM 1Þð1 þpffiffiffiffiffiffiffiDMÞ KM 1 eP M 1tþ KM 2 eP M 2t h i uðtÞ þ2K M 3 PM1 bM ffiffiffiffiffiffiffi DM p DM 1 e PM 1tuðtÞ Z t 0 eðP1Mþb MÞx IðbMxÞdx þ2K M 4 PM2 b M ffiffiffiffiffiffiffi DM p DM 1 e PM 2tuðtÞ Z t 0 eðP2Mþb MÞx IðbMxÞdx ð12Þ
where u(x) corresponds to the unit step function and I(x) = I0(x) + I1(x), being In(x) the modified Bessel
Figure 3. Example of TM incidence for a horizontal wire above ground.
function of the first kind. Additional quantities in equation (12) are: DM ¼ 2 rcos2q0 r sin2q0 BM ¼ s 0 r sin2q0 AM ¼ s 0r bM ¼B M 2 bM¼2D MAM BM 1 DM cM¼D MðAMÞ2 DM 1 P1M¼1 2 b Mþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbMÞ2 4cM q P2M¼1 2 b M ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbMÞ2 4cM q EM¼ BM AM 1pffiffiffiffiffiffiffiDM FM ¼ ðAMÞ2pffiffiffiffiffiffiffiDM K1M ¼E MPM 1 þ FM PM1 PM 2 K2M ¼E MPM 2 þ FM PM2 PM 1 K3M ¼P M 1 þ AM PM 1 P2M K4M ¼P M 2 þ AM PM 2 P1M ð13Þ
For TE-polarized incidence, TD-RC is in the form [Suk and Rothwell, 2002a]:
GTEð~r;~r0;tÞ ¼ 1pffiffiffiffiffiffiDE 1þpffiffiffiffiffiffiDEdðtÞ þ BEe2DEBEt ffiffiffiffiffiffi DE p ðDE 1Þ I0 BEt 2DE þ I1 BEt 2DE uðtÞ ðB EÞ2 ffiffiffiffiffiffi DE p ðDE 1Þ2e BE DE1tuðtÞ Z t 0 e BEð1þDE Þx 2DEðDE 1Þ I 0 BEx 2DE þ I1 BEx 2DE dx 2B E ð1 þpffiffiffiffiffiffiDEÞðDE 1Þe BE DE1tuðtÞ ð14Þ where DE ¼r sin 2q 0 cos2q 0 BE¼ s 0cos2q 0 ð15Þ
[20] Consequently, it can be seen that equations (12)
and (14) have similar structures. They are composed by a nonconductive term, proportional to 1
ffiffiffiffiffiffiffiffiffiffiffiffi
fDM;DE p
g 1þpffiffiffiffiffiffiffiffiffiffiffiffifDM;DEg where the corresponding constants {DM, DE} are given alternatively by equation (13) or (15), plus a conductive term, associated in a more complex way with those constants depending on s. Computation of nonconduc-tive terms is fast, and their contribution to the total field by the convolution operator of equation (9) is computa-tionally cheap due to their dependency to the d(t) function. Nevertheless, computation of the conductive terms are expensive for two reasons: first, an accurate calculation of the integrals in (12) and (14) in general requires numerical integration techniques, and secondly, for those terms, the performance of the convolution operation of (9) increases its computational burden with time, leading to undesirably long computational times for late time responses. The next section presents ways to avoid these drawbacks.
[21] As mentioned above, the computational
imple-mentation of the analytical expressions (13) and (15) by using numerical integration techniques is time con-suming. Furthermore, it should be taken into account that the simulation of thin-wire antennas above ground, where multiple interactions between different parts of the structure have to be considered, requires the calcu-lation of hundreds of RC, making it advisable to use approximate expressions for an acceptably accurate and fast computation of the RC [Rothwell and Suk, 2003, 2005].
[22] For the TM case, equation (12) can be expanded
as an infinite series in the form [Rothwell and Suk, 2005]:
GTMð~r;~r0;tÞ ¼ GdieTMð~r;~r 0
ÞdðtÞ þ RTMðtÞ ð16Þ
withGTMdie as the nonconductive term equal to:
GdieTMð~r;~r0Þ ¼1 ffiffiffiffiffiffiffi DM
p
and RTM(~r, ~r0, t) as the conductive term: RTMð~r;~r0;tÞ ¼ 2bMpffiffiffiffiffiffiffiDM DM 1 uðtÞ X1 n¼1 KM 3 bM PM 1 n þKM 4 bM PM 2 n " # QðnÞðbMtÞ ð18Þ where Q(n)(x) means the (n)th-order derivative of the function Q(x) = exI(x).
[23] The numerical implementation of the equation (18)
requires a truncation of the infinite series, where at least 10 terms are needed for convergence [Ferna´ndez Pantoja et al., 2009]. However, the equation (18) for the compu-tation of RTM(~r, ~r0, t) has the biggest error rate at early
times. The response at these early times is the main con-tribution to the convolution operator of (9), and a higher degree of accuracy is needed for an adequate calculation of the early reflected electric field. With this purpose, an improved approximation of (18) is employed [Rothwell and Suk, 2005]: RappTMð~r;~r0;tÞ ¼ RTMð~r;~r0;tÞ þ R app TMð~r;~r 0;0Þ R TMð~r;~r0;0Þ RappTMð~r;~r0;0Þ bMtþ 1 R½ TMappð~r;~r0;0Þ RTMð~r;~r0;0ÞeB Mt ð19Þ where RappTMð~r;~r0;0Þ ¼ ffiffiffiffiffiffiffi DM p ð1 þpffiffiffiffiffiffiffiDMÞ2ðB M 2AMÞ ð20Þ
where typically three terms are enough to converge [Ferna´ndez Pantoja et al., 2009]. Figure 4 represents the TD TM-RC for the case of normal incidence (q0= 0°)
over lossy ground (er = 72, s = 4) for equations (12),
(18), (19), named as ‘‘exact,’’ ‘‘approximate series,’’ and ‘‘approximate improved,’’ respectively. We can see not only the high error rate at early times of equation (18) but also the higher accuracy of the formulation (19) at these early times, achieved without increasing the computa-tional costs of its implementation.
[24] For the TE polarization, the expanded form of
equation (14) is [Rothwell and Suk, 2003]: GTEð~r;~r0;tÞ ¼ GdieTEð~r;~r
0
ÞdðtÞ þ RTEðtÞ ð21Þ
withGTEdieas the nonconductive term equal to:
GdieTEð~r;~r0Þ ¼1 ffiffiffiffiffiffi DE
p
1þpffiffiffiffiffiffiDE ð22Þ
and RTE(~r, ~r0, t) as the conductive term:
RTEð~r;~r0;tÞ ¼ bEuðtÞ ffiffiffiffiffiffi DE p ðDE 1Þ X 1 n¼1 1 DE 2DE n QðnÞ B Et 2DE ð23Þ
[25] As in the TM case, an improved version of (21)
can be employed [Rothwell and Suk, 2005]: RappTEð~r;~r0;tÞ ¼ RTEð~r;~r0;tÞ R app TEð~r;~r 0;0Þ RTEð~r;~r0;0Þ 1 e2DEBEtþ 1 ð24Þ where RappTEð~r;~r0;0Þ ¼ B E ffiffiffiffiffiffi DE p ð1 þpffiffiffiffiffiffiDEÞ2 ð25Þ
Figure 5 depicts a comparison of the exact RC and the results from equations (23) and (24) for a plane wave incidence of (q0 = 0°) over a weakly lossy ground
(er= 10,s = 0.01), leading to similar conclusions as in
the TM case.
[26] In general, the magnitude of the TD-RCs decrease
with time, as shown in Figures 4 and 5 and their value can be considered negligible for t > tmax, where tmax is
chosen as R{TM,TE}app (~r, ~r0, tmax) 0.1R{TM,TE}app (~r, ~r0, 0).
[27] Therefore, the computational cost of evaluating
equation (6) can be greatly reduced if onlyG(~r,~r0, t) with t tmax is considered. Further results confirm that, for
high conductive soils, there is no significant loss of accuracy in the calculation of the reflected electric field Figure 4. TM TD-RC for normal incidence over lossy
if the approximation G(~r, ~r0, t) 0 for t tmax is
employed in equation (6). For weakly conductive soils, the values of the conductive terms RTMappand RTEappat any
time are negligible compared to their dielectric counter-partsGTMdieandGTEdie, and thus there is no need to consider
them.
4. Computational Implementation of EFIE
for Thin-Wire Antennas Over Lossy Ground
[28] The computational implementation of equation (9)
can be made by applying the method of moments (MOM) [Harrington, 1968]. In this work, the unknown currents I(s0, t0) in equations (7) and (8) are expanded using lagrangian subsectional basis functions both in spatial and temporal dimensions [Miller et al., 1973]. The weighting phase of MOM is performed by applying a point-matching algorithm: along the surface of the wire a discrete set of points ~ru (u = 1. . .NS) are chosen to
define the spatial weight functions (d(~r ~ru)), and a set
of time instants tv (v = 1. . .NT) are considered for the
temporal weight functions (d(t tv)).
[29] The first step in the discretization of (9) is to
establish a rectilinear uniform segmentation of the con-tour of the thin wires of equations (7) and (8). Then, the spatial and temporal dimensions are subdivided into regular intervals: Di is the size of the ith segments of
the wire, and Dt the duration of the time interval into
which the total analysis time is uniformly subdivided. Further, we note the auxiliary variable s00i = s0 si as
the distance of a position s0 located at any segment of the wire i from its center si(Figure 6), and the variable
t00j= t0 tj is the time distance referred to a chosen jth
time tj. With this notation, electric currents I(s0, t0) can
be referred as: Iðs0;t0Þ ¼X NS i¼1 XNT j¼1 Iijðs00i;t 00 jÞU ðs 00 iÞV ðt 00 jÞ ð26Þ
where U(x) and V(x) correspond to rectangular pulse functions of widthsDiandDt, respectively.
[30] Hence, applying the point-matching delta
func-tions to equation (6) and a subsequent substitution of the electric currents in (7) and (8) results in:
^su ~Eið~ru;tvÞ ¼^su ~Edð~rv;tvÞ þ ^su GTEð~ru;~ri;tvÞ3~Erð~ru;tvÞ þ ^su n GTMð~ru;~ri;tvÞ ½ GTEð~ru;~ri;tvÞ3~Erð~ru;tvÞ ^no^n ð27Þ in which the electric field ~Edis given by:
~Edð~r u;tvÞ ¼ 1 4pe XNS i¼1 Z Di ^si v2R iu @Iij s00i;t 00 j @t00j ds 00 i þ 1 4pe XNS i¼1 Z Di ~R iu R3 iu Z t00 j 0 @Iij s00i;t @s00i dt ! ds00i þ 1 4pe XNS i¼1 Z Di ~R iu vR2 iu @Iij s00i;tj00 @s00i 2 4 3 5 t00 j ds00i ð28Þ Figure 5. TE TD-RC for incidence (0 = 60°) over
lossy ground ("r= 10, s = 0.01).
and terms including the RCs are: GAð~ru;~ri;tvÞ3~Erð~ru;tvÞ ¼ 1 4pe XNS i¼1 Z D;i GAð~ru;~ri;tvÞ ^si v2R iu @Iij s00i;t 00 j @t00j ds 00 i þ 1 4pe XNS i¼1 Z D;i GAð~ru;~ri;tvÞ ~Riu R3 iu Z t00 j 0 @Iij s00i;t @s00i dt ! ds00i þ 1 4pe XNS i¼1 Z D;i GAð~ru;~ri;tvÞ ~Riu vR2 iu @Iij s00i;tj00 @s00 i 2 4 3 5 t00 j ds00i ð29Þ where A is either TE or TM, ~Riu corresponds to the
vector between source and field points, and fulfills the relation ~Riu= ~ru ~ri s00i^sias it can be seen in Figure 6.
^suand ^sistand for the tangential vectors, respectively, to
the contour of the wire at the field point ~ruand to the axis
of the wire at the source point ~ri. Notation refers to
image wires, subdivided in NSsegments of lengthD.
[31] The next step in the implementation of the MOM
to solve equation (27) is to specify the particular depen-dence on s00i and t00j of the currents in equations (28) and
(29), i.e., to choose appropriate basis functions to expand I(s00i, t00j). In this work, following Miller et al. [1973] and
Rubio Bretones et al. [1989], we have applied two-dimensional second-order lagrangian functions, which have been recognized to provide both accurate and numerically stable solutions.
5. Results
[32] The method proposed in this paper is validated by
solving several canonical cases and comparing the results found with the new technique with those produced by algorithms based on the solution of the integral equa-tion in the frequency domain, where Green’s funcequa-tions account for the presence of a ground plane [Sommerfeld, 1964]. As pointed out in section 1, this latter method shows great accuracy but its excessive computational time, mainly due to the numerical evaluation of the Sommerfeld integrals [Lager and Lytle, 1975], greatly limits its use.
[33] Two kinds of soils have been chosen to perform
the study: a dielectric soil with constitutive parameters
resembling those of dry earth (er,mr,s) = (2.7, 1, 0), and
a strongly conductive soil matching the typical parame-ters of seawater (er,mr,s) = (72, 1, 4 S/m). These soils
are considered as limit cases of permittivity and conduc-tivity, being mostly the constitutive parameters of soils in nature in the midrange of the chosen examples at radio frequencies [Sternberg and Levitskaya, 2001].
[34] Moreover, a key point in the use of the RC
approximation is the loss of accuracy in cases where the antenna is in the vicinity of the ground. It has been estimated [Sarkar, 1977] that accurate results for RC approximation in the frequency domain are given for heights h fulfilling h > 0:25l er ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ s jwere0 q ð30Þ
Taking into account that the usual feedings are made with wideband or utltrawideband pulses, precise results are achieved if equation (30) holds for the spectra of the feeding pulse. To check the accuracy of the new technique in terms of the distance of the thin wire to the ground, we present graphs at different heights for each example.
[35] The first example validates the method in case of
only TM incidence by considering a thin-wire antenna of a total length of 0.5 m and a diameter of 2 mm, identical to that proposed by Lestari et al. [2004], located parallel to the interface over the soils described above. The antenna is fed at its central point by a normalized derivative gaussian pulse in the form:
vðtÞ ¼ e0:5gpffiffiffi2ðt t maxÞeg
2ðtt
maxÞ2 ð31Þ with parameters g = 1.25 109 s1, and tmax = 4g.
Figures 7 – 12 show comparative results for different heights. As can be observed, a good resemblance is reached between exact (MOM-FD) and approximate (MOM-TD) solutions in cases where the equation (30) holds. Truly, the spectra of the feeding pulse of (31) is centered at approximately 300 MHz and, by replacing this value and the parameters of soils in (30), accurate simulations should be achieved with antennas located at minimum heights of 9.25 cm and 0.3 cm for dry earth and seawater, respectively. This is confirmed by inspection of Figures 9 and 12. Another fact derived from the same reasoning is that, as predicted by equation (30), the higher the permittivity and conductivity of the ground, the higher the accuracy of the solutions for antennas closer to the ground (see also Figures 9 and 12).
[36] An example composed of two thin-wire antennas
Figure 2. The feeding of the transmitter is placed in the center of the first antenna, and the receiver terminals are located at the center of the second thin wire. In this case, adequate simulations are achieved only by considering both TE and TM incidences. Figures 13 – 18 depict the electric current induced at the center of the receiver for different soils and different distances between transmitter and receiver, the lengths, heights above ground and diameters of the wires being fixed to 0.5, 0.15 and 0.002 meters, respectively. The height above ground in this case has been chosen not to degrade the accuracy of the results.
[37] Our first impression looking at the graphs for the
two thin-wire case is that RC-TD method produces satisfactory results. Further, deeper examination of Figures 13 – 18 leads to some conclusions about sources of errors when the RC approximation is employed. A comparative analysis shows that: (1) a better resemblance is achieved for the nonconductive earth and (2) greater accuracy is reached for shorter distances between trans-mitter and receiver. Different reasons have to be consid-ered to explain this behavior. On the one hand, the results for conductive grounds (Figures 16 – 18) deteriorated because equations (12) and (14) are not precise for incidence angles near the Brewster angle [Suk and Rothwell, 2002b], which appear more frequently in
Figure 10. Current at the feeding point of the thin-wire antenna of L = 0.5 m at a height of h = 0.25 m over seawater.
Figure 8. Current at the feeding point of the thin-wire antenna of L = 0.5 m at a height of h = 0.15 m over dry earth.
Figure 7. Current at the feeding point of the thin-wire antenna of L = 0.5 m at a height of h = 0.25 m over dry earth.
Figure 9. Current at the feeding point of the thin-wire antenna of L = 0.5 m at a height of h = 0.05 m over dry earth.
simulations including conductive grounds and greater distances between antennas. On the other hand, the closer the distance between wires the greater the direct wave compared to the reflected wave. As a result, greater differences appear for farer wires, as noted by comparing Figures 16 and 18, or alternatively in a weaker form in Figures 13 and 15. Reasons for the discrepancies of the reflected field are inherent to the RC approximation, which assumes that radiated electromagnetic fields are composed exclusively of plane waves, which is true for fields located in the electromagnetic far-field zone, but it is not strictly certain for interactions between sources placed at the near- or intermediate-field zones.
[38] It worths to remark that late time oscillations have
not appeared in the above described examples. The two-dimensional lagrangian basis functions employed in this paper [Miller et al., 1973] lead to diagonally dominant matrix equations and, consequently, to stable schemes. The use of the RC approach, at heights accomplishing equation 30, does not origin any kind of instabilities because it does not modify the diagonal properties of the numerical equation corresponding to identical thin-wire antennas located at free space.
[39] A final aspect to be considered in the analysis of
the results is the reduction in the computational time.
Figure 12. Current at the feeding point of the thin-wire antenna of L = 0.5 m at a height of h = 0.05 m over seawater.
Figure 13. Current at the center of the receiver antenna. Distance between wires is of 0.25 m, and they are both located at a height of 0.15 cm above dry earth.
Figure 11. Current at the feeding point of the thin-wire antenna of L = 0.5 m at a height of h = 0.15 m over seawater.
Figure 14. Current at the center of the receiver antenna. Distance between wires is of 0.15 m, and they are both located at a height of 0.15 cm above dry earth.
Following the guidelines for the design of MOM-FD and MOM-TD given by Burke and Poggio [1981] and Rubio Bretones et al. [1989], a total of 147 segments have been used to model each thin-wire antenna. Table 1 shows the reduction in computational time of the RC approxima-tion compared to MOM-FD, and gives numerical details of each simulation of section 5. Number of frequency and time intervals have been chosen for an adequate representation of the temporal signals, this choice being a key point in the effective reduction of computational time. As pointed out by Miller [1994], MOM-TD is
advantageous when small frequency intervals are needed for the analysis of ultrawideband spectrum. This situa-tion appears in resonant narrowband structures with unknown resonance frequency, where an exhaustive search in frequency range has to be performed by using small frequency intervals. An illustrative example of this highly resonant case is presented in Figures 15 and 18, showing plot graphs of two closely spaced thin wires. On the other hand, in cases where the number of frequency points of analysis can be diminished, computational Figure 15. Current at the center of the receiver antenna.
Distance between wires is of 0.05 m, and they are both located at a height of 0.15 cm above dry earth.
Figure 16. Current at the center of the receiver antenna. Distance between wires is of 0.25 m, and they are both located at a height of 0.15 cm above seawater.
Figure 17. Current at the center of the receiver antenna. Distance between wires is of 0.15 m, and they are both located at a height of 0.15 cm above seawater.
Figure 18. Current at the center of the receiver antenna. Distance between wires is of 0.05 m, and they are both located at a height of 0.15 cm above seawater.
times are in the same order, the being advantageous even for MOM-FD, for example corresponding to Figure 13.
6. Conclusions
[40] The proposed RC-TD method constitutes an
alternative to simulate the transient electromagnetic behavior of thin-wire structures located above homoge-neous lossy ground, directly in the time domain. This approach is faster than methods based on solutions of the Sommerfeld equation for wideband analysis, and this paper has been proven to provide satisfactory results for several simple cases involving one or two wires. The examples shown identify the main factors that influence the accuracy of the method, which are: height above ground, distance between different points of the wire structures, both related to the validity of the plane wave approximation for the radiated electromagnetic fields incident on the ground. Further studies of more complex structures or/and practical applications can be performed in future works.
[41] Acknowledgments. This work has been supported by the EU FP7/2007 – 2013, under GA 205294 (HIRF-SE project), from the Spanish National projects TEC2007-66698-C04-02, CSD200800068, and DEX-5300002008105, and from the Junta de Andalucia project TIC1541.
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