Time domain analysis of thin-wire antennas over lossy ground using the reflection-coefficient approximation



[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.

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 algorithms. 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 approximation 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 frequency domain techniques [Miller and Landt, 1980]. Therefore, additional efforts were devoted to the development 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 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 implemented by Poljak [2007], by employing the time domain RC inferred by Barnes and Tesche [1991], and satisfactory 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 operator [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 convolution 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 computational burden of the calculation for the case of conductive 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 decomposition 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 homogeneous, 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 coefficients [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 (ɛ, μ) excited by an incident field equation imagei(equation image, t), between: (1) the original structure and (2) an equivalent problem considering only a set of surface equivalent electric currents equation images placed on S, radiating in a homogeneous inner region inner (i.e., within S) of constitutive parameters (ɛ, μ) equal to those of the outer region in the original problem. This set of equivalent currents radiates an electric field:

equation image

where equation image′ notes the position of the source, placed on the surface S; equation image is the position vector of the field point; R is the distance between field and source, calculated as the modulus of the vector equation image = equation imageequation image′; t′ = tequation image is the retarded time which assures the causality of the system, accounting for the propagation of electromagnetic fields in the medium of constitutive parameters (ɛ, μ) and velocity of propagation v = equation image; and ρs are the electric charges related to equation images by ρs(equation image′, t′) = ∫0t[∇′ · equation images(equation image′, τ)].

[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, equation images(equation image, t) can be replaced by the incident field equation imagei(equation image, t) in (1) in the form:

equation image

where equation image is located on the surface S, and ( )tan indicates 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 equation images(equation image′, t′), 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]:

equation image

which is known as the time domain EFIE for PEC. Brackets []t are employed to emphasize that the divergence ∇′ · applies exclusively to the spatial variable in equation images(equation image′, t′).

[12] For a thin-wire structure, i.e., that whose radius a is negligible compared with its length (Figure 1), two-dimensional surface currents equation images(equation image′, t′) on S can be approximated as one-dimensional total currents equation image(equation image′, t′) = 2πaequation images(equation image′, t′), 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:

equation image

where s′ 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 equation image and equation image′ are the tangential unit vectors on the surface S and at the axis of the wire, respectively.

Figure 1.

Geometry of a thin-wire structure.

[13] For those cases including a total of Nw thin wires, equation (4) holds, by applying the superposition principle, in the form:

equation image

[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 (equation imaged(equation image, 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 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.

Figure 2.

Example of TE incidence for horizontal wires above ground.

[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 equation image′, and the time domain plane wave reflection coefficient Γ (equation image, equation image′, t) corresponding to the constitutive parameters and type of incidence of the problem at hand. By doing this, equation (4) is modified to:

equation image


equation image

and the operator ⊛ is defined as a retarded convolution, along the path of the wire, between any reflection coefficient Γ (equation image, equation image′, t) and any electric field equation image(equation image, t):

equation image

where the path Cϒ 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 to that from the upper media although the image is located inside the ground.

[17] Furthermore, Γ (equation image, equation image′, t) depend on the polarization of the plane wave incident on the ground plane. Then, it is noted equation image as the unit vector normal to the ground plane in the point of incidence, ΓTM (equation image, equation image′, 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 ΓTE (equation image, equation image′, 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 equation imager(equation image, t) at the ground plane can be decomposed into its vertical component, equation imagevr(equation image, t) = (equation imager(equation image, t) · equation image) equation image, and horizontal component, equation imagehr(equation image, t) = [equation imager(equation image, t) − (equation imager(equation image, t) · equation image) equation image]. Then, equation (6) can be written as:

equation image

which constitutes, by replacing equations (7) and (8) into (9), the extended EFIE equation that includes the effect of lossy grounds.

Figure 3.

Example of TM incidence for a horizontal wire above ground.

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 (ε = εrε0, μ0, σ), which is solved in the frequency domain by using the Fresnel RC [Wait, 1962]. Naming θ0 as the angle between the incident wave and normal vector to the interface, which can be inferred from the equation cos(θ0) = (equation image) · equation image, the RC for TM incidence is:

equation image

and for TE incidence, the RC is:

equation image

[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]:

equation image

where u(x) corresponds to the unit step function and I(x) = I0(x) + I1(x), being In(x) the modified Bessel function of the first kind. Additional quantities in equation (12) are:

equation image

For TE-polarized incidence, TD-RC is in the form [Suk and Rothwell, 2002a]:

equation image


equation image

[20] Consequently, it can be seen that equations (12) and (14) have similar structures. They are composed by a nonconductive term, proportional to equation image 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 σ. Computation of nonconductive terms is fast, and their contribution to the total field by the convolution operator of equation (9) is computationally cheap due to their dependency to the δ(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 implementation of the analytical expressions (13) and (15) by using numerical integration techniques is time consuming. 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 calculation 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]:

equation image

with ΓTMdie as the nonconductive term equal to:

equation image

and RTM(equation image, equation image′, t) as the conductive term:

equation image

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 [Fernández Pantoja et al., 2009]. However, the equation (18) for the computation of RTM(equation image, equation image′, t) has the biggest error rate at early times. The response at these early times is the main contribution 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]:

equation image


equation image

where typically three terms are enough to converge [Fernández Pantoja et al., 2009]. Figure 4 represents the TD TM-RC for the case of normal incidence (θ0 = 0°) over lossy ground (ɛr = 72, σ = 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 computational costs of its implementation.

Figure 4.

TM TD-RC for normal incidence over lossy ground (ɛr = 72, σ = 4).

[24] For the TE polarization, the expanded form of equation (14) is [Rothwell and Suk, 2003]:

equation image

with ΓTEdie as the nonconductive term equal to:

equation image

and RTE(equation image, equation image′, t) as the conductive term:

equation image

[25] As in the TM case, an improved version of (21) can be employed [Rothwell and Suk, 2005]:

equation image


equation image

Figure 5 depicts a comparison of the exact RC and the results from equations (23) and (24) for a plane wave incidence of (θ0 = 0°) over a weakly lossy ground (ɛr = 10, σ = 0.01), leading to similar conclusions as in the TM case.

Figure 5.

TE TD-RC for incidence (θ0 = 60°) over lossy ground (ɛr = 10, σ = 0.01).

[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(equation image, equation image′, tmax) ≤ 0.1R{TM,TE}app(equation image, equation image′, 0).

[27] Therefore, the computational cost of evaluating equation (6) can be greatly reduced if only Γ(equation image, equation image′, t) with ttmax 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 if the approximation Γ(equation image, equation image′, t) ≈ 0 for ttmax is employed in equation (6). For weakly conductive soils, the values of the conductive terms RTMapp and RTEapp at any time are negligible compared to their dielectric counterparts ΓTMdie and ΓTEdie, 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(s′, t′) 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 equation imageu (u = 1…NS) are chosen to define the spatial weight functions (δ(equation imageequation imageu)), and a set of time instants tv (v = 1…NT) are considered for the temporal weight functions (δ(ttv)).

[29] The first step in the discretization of (9) is to establish a rectilinear uniform segmentation of the contour of the thin wires of equations (7) and (8). Then, the spatial and temporal dimensions are subdivided into regular intervals: Δi is the size of the ith segments of the wire, and Δt the duration of the time interval into which the total analysis time is uniformly subdivided. Further, we note the auxiliary variable si = s′ − si as the distance of a position s′ located at any segment of the wire i from its center si (Figure 6), and the variable tj = t′ − tj is the time distance referred to a chosen jth time tj. With this notation, electric currents I(s′, t′) can be referred as:

equation image

where U(x) and V(x) correspond to rectangular pulse functions of widths Δi and Δt, respectively.

Figure 6.

Local coordinates on a thin-wire segment.

[30] Hence, applying the point-matching delta functions to equation (6) and a subsequent substitution of the electric currents in (7) and (8) results in:

equation image

in which the electric field equation imaged is given by:

equation image

and terms including the RCs are:

equation image

where A is either TE or TM, equation imageiu corresponds to the vector between source and field points, and fulfills the relation equation imageiu = equation imageuequation imageisiequation imagei as it can be seen in Figure 6. equation imageu and equation imagei stand for the tangential vectors, respectively, to the contour of the wire at the field point equation imageu and to the axis of the wire at the source point equation imagei. Notation ϒ refers to image wires, subdivided in NS segments of length Δϒ.

[31] The next step in the implementation of the MOM to solve equation (27) is to specify the particular dependence on si and tj of the currents in equations (28) and (29), i.e., to choose appropriate basis functions to expand I(si, tj). 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 equation in the frequency domain, where Green's functions 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 (ɛr, μr, σ) = (2.7, 1, 0), and a strongly conductive soil matching the typical parameters of seawater (ɛr, μr, σ) = (72, 1, 4 S/m). These soils are considered as limit cases of permittivity and conductivity, 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

equation image

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:

equation image

with parameters g = 1.25 · 109s−1, and tmax = equation image. 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).

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 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 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.

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 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 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.

[36] An example composed of two thin-wire antennas acting as a separate transmitter and receiver is depicted in 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.

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 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.

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.

[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 transmitter and receiver. Different reasons have to be considered 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 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. 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 approximation 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 situation 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 times are in the same order, the being advantageous even for MOM-FD, for example corresponding to Figure 13.

Table 1. Numerical Details and Reduction of Execution Time for the Examples of Section 5
ExampleNtΔt (s)NfΔf (MHz)Percent Reduction Time
Dry Earth
Figures 7–920481.14 × 10−113001015.61
Figure 1330721.14 × 10−1130010−44.89
Figure 1430721.14 × 10−116002.516.43
Figure 1530721.14 × 10−1120000.579.23
Figures 10–1220481.14 × 10−113001015.61
Figure 1630721.14 × 10−116002.56.03
Figure 1730721.14 × 10−116002.56.03
Figure 1830721.14 × 10−1120000.576.64

6. Conclusions

[40] The proposed RC-TD method constitutes an alternative to simulate the transient electromagnetic behavior of thin-wire structures located above homogeneous 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] 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.