Electromagnetic Field Coupling to Overhead Wire Configurations: Antenna Model versus Transmission Line Approach

Abstract

The paper deals with two different approaches for the analysis of electromagnetic field coupling to finite length overhead wire: the wire antenna theory (AT) and the transmission line (TL) method. The analysis is carried out in the frequency and time domain, respectively. Within the frequency domain analysis the wire antenna formulation deals with the corresponding set of Pocklington integrodifferential equation, while the transmission line model uses the telegrapher's equations. The set of Pocklington equations is solved via the Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM), while the telegrapher’s equations are treated using the chain matrix method and the modal equation to derive per-unit-length parameters. For the case of the time domain analysis AT model uses the space-time Hallen integral equation set, while TL approach deals with the time domain version of the telegrapher’s equations. Hallen equations are handled via time domain version of GB-IBEM, while time domain telegrapher’s equations are solved by using Finite Difference Time Domain (FDTD) method. Many illustrative computational examples for the frequency and time domain response, respectively, for several configurations of overhead wires, obtained via different approaches, are given in this paper.

1. Introduction

The electromagnetic field coupling to overhead wires is of great practical interest for many EMC applications [1–11], such as transient excitation of antennas, power, or communications cables. The electromagnetic field coupling to finite length overhead wires can be determined by means of the transmission line model or the thin wire antenna theory in either frequency or time domain [1]. In particular, the transient response of a wire configuration of interest can be computed directly, by solving the related time domain equations or by the indirect approach, that is, by solving their frequency domain counterpart. When the indirect approach is used the frequency spectrum has to be calculated, and then the transient response is computed by means of the Inverse Fourier Transform (IFT).

Many practical engineering problems dealing with electromagnetic field coupling to thin wires can be analyzed by using the Transmission Line (TL) models [1–6]. These models include the analysis of incident electromagnetic field exciting the line and the propagation of induced currents and voltages along the line.

The TL models yield valid results if the line length is significantly larger than the separation between the wires and also larger than the actual height above ground [8].

On the other hand the TL approximation cannot provide a complete solution for the excitation of a given wire configuration by an incident field if the wavelength of the electromagnetic field exciting a wire structure is comparable to or less than the transverse electrical dimensions of the structure. Namely, the TL model fails to predict resonances and accounts for the presence of a lossy ground only approximately [1]. One of the serious problems with TL approach occurs due to the fact that current grows to infinity at resonant points as there are no losses and radiation resistance to limit its flow [8]. The full wave approach, based on the wire antenna theory and related integral equations, is more rigorous and should be used whenever the above-ground transmission lines of the finite length are considered. However, a serious drawback of AT approach is rather long computational time required for the calculations pertaining to long lines.

This paper deals with the analysis of electromagnetic field coupling to overhead wires in either frequency or time domain by using both antenna model and transmission line approach, respectively.

A number of illustrative computational examples regarding electromagnetic coupling to overhead wires are given in the paper.

The aboveground wires are subjected to electromagnetic fields arriving from a distant source and inducing current to flow along the wires. The key to understanding the behaviour of induced fields is the knowledge of current distribution induced along the wires. These currents generate scattered fields propagating away from the equipment.

The paper is organized as follows: Section 2 deals with the frequency domain analysis followed by related numerical solution methods for overhead wires. Section 2 ends up with many illustrative examples related to the aboveground lines and PLC (power line communications) systems.

Section 3 outlines the time domain analysis and related method of solutions of governing equations. Some computational examples pertaining to the multiconductor aboveground lines are given. Finally, the conclusion summarizes what has been discussed throughout this work.

2. Frequency Domain Models and Methods

This section deals with the wire antenna theory and transmission line (TL) approximation, respectively, for the analysis of electromagnetic field coupling to overhead lines of finite length in the frequency domain. The formulation arising from the wire antenna theory is based on the set of coupled Pocklington integrodifferential equation for half-space problems. The effect of a two-media configuration is taken into account by means of the reflection coefficient approximation [12]. The resulting integro-differential expressions are numerically handled via the frequency domain Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM) [8].

Transmission line model in the frequency domain is based on the corresponding telegrapher's equations which are handled by using the chain matrix method [10].

2.1. Antenna Theory Approach: Set of Coupled Pocklington Equations

Modeling of arbitrarily shaped wires located at different heights above a lossy ground is an important task in both antenna and electromagnetic compatibility (EMC) studies [1].

This section firstly deals with an assessment of the current induced along multiple wire configurations above a lossy ground. Once the currents along the wire array have been obtained, the radiated field components could be determined.

The set of Pocklington equations for a configuration of overhead wires can be obtained as an extension of the Pocklington integro-differential equation for a single wire of arbitrary shape. The Pocklington equation for a single wire above a lossy ground can be derived by enforcing the continuity conditions for the tangential components of the electric field along the perfectly conducting (PEC) wire surface. First, a single wire of arbitrary shape, insulated in free space, as shown in Figure 1 is considered.

Figure 1 

Single wire of arbitrary shape in free space.

For the PEC wire the total field composed from the excitation field 𝐸^exc and scattered field 𝐸^sct vanishes [1, 8]: 𝑒_𝑥⃗⋅𝐸^exc+𝐸^sct=0onthewiresurface.(1) Starting from Maxwell’s equations and Lorentz gauge the scattered electric field can be expressed in terms of the vector potential 𝐴⃗: 𝐸^sct=−𝑗𝜔𝐴⃗+1𝑗𝜔𝜇𝜀∇∇𝐴⃗.(2) The vector potential is defined by the particular integral over a given path C (considered conductive wire structure): 𝐴⃗(𝑠)=𝜇4𝜋_𝐶𝐼𝑠^𝑔₀𝑠,𝑠^,𝑠^∗𝑠^⃗𝑑𝑠^,(3) where 𝐼(𝑠^) is the induced current along the line and 𝑔₀(𝑠,𝑠^) denotes the lossless medium Green function:𝑔₀𝑠,𝑠^=𝑒^{−𝑗𝑘𝑅}𝑅,(4) and 𝑅 is the distance from the source point to the observation point, respectively, while the propagation constant of the homogeneous medium is given by 𝑘²=𝜔²𝜇₀𝜀₀.(5) Inserting (3) into (2) gives the relation for the scattered electric field: 𝐸^sct=1𝑗4𝜋𝜔𝜀₀_𝐶𝐼𝑠^⋅𝑠^⃗⋅𝑘²+∇∇𝑔₀𝑠,𝑠^𝑑𝑠′.(6) Combining (6) and (1) results in the Pocklington integral equation for the unknown current distribution along the wire of arbitrary shape insulated in free space: 𝐸^exc_tan(𝑠)=−1𝑗4𝜋𝜔𝜀₀_𝐶𝐼𝑠^⋅𝑠⃗⋅𝑠^⃗⋅𝑘²+∇∇𝑔₀𝑠,𝑠^𝑑𝑠^,(7) where 𝐸^exc_tan denotes the tangential component of the electric field illuminating the wire.

Now the case of curved wire located above an imperfectly conducting ground can be analyzed by extending integro-differential equation (7) using the reflection coefficient approach [12]. The geometry of an arbitrary wire and its image, respectively, is shown in Figure 2.

Figure 2 

The wire of arbitrary shape and its image.

The excitation function 𝐸^exc is now composed from incident and reflected field, respectively,𝐸^exc=𝐸^inc+𝐸^ref.(8) Performing certain mathematical manipulations the Pocklington integro-differential equation for a curved wire above a lossy ground becomes [12]𝐸^exc_𝑠(𝑠)=𝑗4𝜋𝜔𝜀₀^𝐿0𝑘²𝑒_𝑠⃗𝑒_𝑠′−𝜕²+𝑅_TM𝜕𝑠𝜕𝑠^𝑔⁰𝑠,𝑠^𝑘²𝑒_𝑠⃗𝑒_𝑠∗−𝜕²+𝑅_TE𝜕𝑠𝜕𝑠^∗𝑔^𝑖𝑠,𝑠^∗−𝑅_TM𝑒_𝑠⃗𝑒_𝑝⃗⋅𝑘²𝑒_𝑝⃗𝑒_𝑠∗−𝜕²𝜕𝑝𝜕𝑠^∗𝑔^𝑖𝑠,𝑠^∗𝐼𝑠^𝑑𝑠^,(9) where 𝑒_𝑝⃗ is the unit vector normal to the incident plane, while 𝑔_𝑖(𝑠,𝑠^∗) arises from the image theory and is given by 𝑔_𝑖𝑠,𝑠^∗=𝑒^{−𝑗𝑘𝑅∗}𝑅^∗,(10) and 𝑅^∗ is the distance from the image source point to the observation point, respectively.

An extension to the case of multiple curved wires is straightforward, that is, it follows [12]𝐸^exc_𝑠𝑚=(𝑠)𝑗4𝜋𝜔𝜀_0𝑁^𝑤_𝑛=1^𝐿𝑛₀𝑘𝑒^𝑠𝑚𝑒_𝑠^′𝑛−𝜕²𝜕𝑠_𝑚𝜕𝑠^𝑛𝑔^0𝑛+𝑅_TM𝑠_𝑚,𝑠^𝑛𝑘²𝑒_𝑠𝑚𝑒_𝑠^∗𝑛−𝜕²𝜕𝑠_𝑚𝜕𝑠^∗𝑛𝑔^𝑖𝑛𝑠_𝑚+𝑅_TE,𝑠^∗𝑛−𝑅_TM𝑒_𝑠𝑚𝑒_𝑝⃗⋅𝑘²𝑒_𝑝⃗𝑒_𝑠∗−𝜕²𝜕𝑝𝜕𝑠^∗𝑔^𝑖𝑠_𝑚,𝑠^∗𝑛𝐼𝑠^𝑛𝑑𝑠^,(11) where 𝑁_𝑤 is the total number of wires and 𝐼_𝑛(𝑠^𝑛) is the unknown current distribution induced on the nth wire. Furthermore, 𝑔_0𝑚𝑛(𝑥,𝑥^) and 𝑔_𝑖𝑚𝑛(𝑠,𝑠^)are the Green functions of the form 𝑔_0𝑚𝑛𝑠_𝑚,𝑠^𝑛=𝑒^{−𝑗𝑘𝑅1𝑚𝑛}𝑅_1𝑚𝑛,𝑔_𝑖𝑚𝑛𝑠_𝑚,𝑠^𝑛=𝑒^{−𝑗𝑘𝑅2𝑚𝑛}𝑅_2𝑚𝑛,(12) where 𝑅_1𝑚𝑛 and 𝑅_2𝑚𝑛 are distances from the source point and from the corresponding image, respectively, to the observation point of interest.

The influence of a lossy half-space is taken into account via the Fresnel plane wave reflection coefficient (RC) for TM and TE polarization, respectively [12], 𝑅^_TM=𝑛cos𝜃′−𝑛−sin²𝜃′𝑅𝑛cos𝜃′+𝑛−sin²𝜃′,(13)_TE=cos𝜃′−𝑛−sin²𝜃′cos𝜃′+𝑛−sin²𝜃′,(14) where θ′ is the angle of incidence and 𝑛 is given by, 𝑛=𝜀^eff𝜀₀,𝜀_eff=𝜀_𝑟𝜀₀−𝑗𝜎𝜔,(15) and 𝜀_eff is the complex permittivity of the ground.

For the special case of single horizontal straight wire above a lossy half-space, Figure 3 integro-differential equation (9) simplifies into 𝐸^exc_𝑥=𝑗𝜔𝜇4𝜋^𝐿0𝐼𝑥^𝑔𝑥,𝑥^𝑑𝑥^−1𝑗4𝜋𝜔𝜀𝜕𝜕𝑥^𝐿0𝜕𝐼𝑥^𝜕𝑥^𝑔𝑥,𝑥^𝑑𝑥^,(16) where 𝐼(𝑥^) is the induced current along the horizontal wire and 𝑔(𝑥,𝑥^) denotes the Green’s function given by 𝑔𝑥,𝑥^=𝑔₀𝑥,𝑥^−𝑅_TM𝑔_𝑖𝑥,𝑥^.(17)

Figure 3 

Horizontal wire above a lossy ground.

Furthermore, if an array of multiple horizontal wires is considered (Figure 4), system of (11) becomes 𝐸^exc_𝑥𝑚=−1𝑗4𝜋𝜔𝜀_0𝑁^𝑤_𝑛=1^𝐿𝑛₀𝜕𝜕𝑥²²+𝑘²¹𝑔^𝑚𝑛𝑥,𝑥^𝐼_𝑛𝑥^𝑑𝑥^𝑚=1,2,…,𝑀,,(18) where 𝐼_𝑛(𝑥^) is the unknown current distribution induced along nth wire, 𝐸^exc_𝑥𝑚 is the known excitation field tangential to the jth wire surface, and 𝑔_𝑚𝑛 is the corresponding Green function:𝑔_𝑚𝑛𝑥,𝑥^=𝑔_0𝑚𝑛𝑥,𝑥^−𝑚𝑅=^_TM1𝑔,2_𝑖𝑚𝑛,…𝑥,,𝑥𝑀^.(19) It is worth noting that a trade-off between the rigorous Sommerfeld integral approach and approximate RC approach is presented in [12]. Although reflection coefficient approximation causes certain error (up to 10%) it takes a significantly less computational effort then a rigorous Sommerfeld approach [8].

Figure 4 

Horizontal wires above a lossy half-space at different heights.

The total electric field irradiated by configuration of multiple wires of arbitrary shape is given by [13, 14] 𝐸⃗=^𝑁𝑤_𝑛=1𝐸^0𝑛+𝑅_TM𝐸_𝑖𝑛+𝑅_TE−𝑅_TM𝐸_𝑖𝑛⋅𝑒_𝑝⃗𝑒^𝑝⃗,(20) where 𝐸_0𝑛=1𝑗4𝜋𝜔𝜀₀⎡⎢⎢⎢⎢⎣𝑘²¹^𝐿𝑛₀+_𝐿0𝑒_𝑠𝑛𝐼𝑠^𝑛𝑔_0𝑛𝑟,⃗𝑟^⃗𝑑𝑠^𝑛′𝜕𝐼𝑠^𝑛𝜕𝑠_𝑛′∇𝑔^0𝑛𝑟,⃗𝑟^⃗𝑑𝑠^𝑛𝐸_𝑖𝑛′⎤⎥⎥⎥⎥⎦,=1𝑗4𝜋𝜔𝜀₀⎡⎢⎢⎢⎢⎣𝑘²¹^𝐿𝑛₀𝑒_𝑠𝑛∗𝐼𝑠^𝑛𝑔_𝑖𝑛𝑟,⃗𝑟^∗⃗𝑑𝑤^−^𝐿𝑛₀𝜕𝐼𝑠^∗𝜕𝑠^∗𝑛∇𝑔_𝑖𝑛𝑟,⃗𝑟^∗⃗𝑑𝑠^⎤⎥⎥⎥⎥⎦.(21) Note that index 0 and i are related to the source and image wire, respectively.

For the special case of single horizontal straight wire above a lossy half-space (Figure 3), it follows [15]𝐸_𝑥=1𝑗4𝜋𝜔𝜀₀−^𝐿0𝜕𝐼𝑥^𝜕𝑥^𝜕𝑔𝑥,𝑥^𝜕𝑥^𝑑𝑥^+𝑘²^𝐿0𝑔𝑥,𝑥^𝐼𝑥^𝑑𝑥^𝐸,(22)_𝑦=1𝑗4𝜋𝜔𝜀₀^𝐿0𝜕𝐼𝑥^𝜕𝑥^𝜕𝑔𝑥^,𝑦𝜕𝑦𝑑𝑥^𝐸,(23)_𝑧=1𝑗4𝜋𝜔𝜀₀^𝐿0𝜕𝐼𝑥^𝜕𝑥^𝜕𝑔𝑥^,𝑧𝜕𝑧𝑑𝑥^.(24) For the case of multiple horizontal wires the expressions for electric field are given by [15]𝐸_𝑥=1𝑗4𝜋𝜔𝜀_0𝑁^𝑤_𝑛=1−^𝐿𝑛₀𝜕𝐼_𝑛𝑥^𝜕𝑥^𝜕𝑔_𝑛𝑚𝑥,𝑥^𝜕𝑥^𝑑𝑥^+𝑘²^𝐿𝑛₀𝑔_𝑛𝑚𝑥,𝑥^𝐼_𝑛𝑥^𝑑𝑥^,(25)𝐸_𝑦=1𝑗4𝜋𝜔𝜀_0𝑁^𝑤_𝑛=1^𝐿𝑛₀𝜕𝐼_𝑛𝑥^𝜕𝑥^𝜕𝑔_𝑛𝑚𝑥,𝑥^𝜕𝑦𝑑𝑥^,(26)𝐸_𝑧=1𝑗4𝜋𝜔𝜀_0𝑁^𝑤_𝑛=1^𝐿𝑛₀𝜕𝐼_𝑛𝑥^𝜕𝑥′𝜕𝑔^𝑛𝑚𝑥,𝑥^𝜕𝑧𝑑𝑥^.(27) The radiated magnetic field of the curved wire system can be written as follows [13, 14]:𝐻=^𝑁𝑤_𝑛=1𝐻^𝑆𝑛+𝑅_TE𝐻_𝐼𝑛+𝑅_TM−𝑅_TE𝐻_𝐼𝑛⋅𝑒_𝑝⃗𝑒^𝑝⃗,(28) where𝐻_𝑆𝑛=−14𝜋^𝐿𝑛₀𝐻_𝐼𝑛𝐼𝑠^𝑛𝑒_𝑠^′×∇𝑔_0𝑛𝑟,⃗𝑟^⃗𝑑𝑠^,=−14𝜋^𝐿𝑛₀𝐼𝑠^𝑛𝑒_𝑠^∗×∇𝑔_𝑖𝑛𝑟,⃗𝑟^∗⃗𝑑𝑠′.(29) The reduction to the case of a single straight wire or straight wire array is straightforward, as in the case of electric field given by (22)–(27).

2.2. Numerical Solution

The set of Pocklington integro-differential equations (11) has been solved by using the Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM). An outline of the method is given here, for the sake of completeness while the method has been presented in detail elsewhere, for example, in [8].

Performing the Galerkin-Bubnov scheme of (GB-IBEM) in the frequency domain the set of coupled integro-differential equations (11) is transformed into the following matrix equation [13] ^𝑀_𝑁𝑛=1^𝑛_𝑖=1[𝑍]^𝑒𝑗𝑖{𝐼}^𝑒𝑖={𝑉}^𝑒𝑗,(30) where the mutual impedance matrix is given by [13]: [𝑍]^𝑒𝑖𝑗=−¹⁻¹{𝐷}_𝑗𝐷^^𝑇𝑖𝑔_0𝑛𝑚𝑠_𝑛,𝑠^𝑚𝑑s^𝑚+𝑘₂₁𝑑𝜉′𝑑𝜉′𝑑𝑑𝑠𝜉^𝑛𝑑𝜉𝑒_𝑠𝑛𝑒_𝑠𝑚¹⁻¹{𝑓}_𝑗𝑓^^𝑇𝑖𝑔_0𝑛𝑚𝑠_𝑛,𝑠^𝑚𝑑𝑠^𝑚−𝑅_TM𝑑𝜉′𝑑𝜉′𝑑𝑑𝑠𝜉^𝑛𝑑𝜉¹⁻¹{𝐷}_𝑗𝐷^^𝑇𝑖𝑔_𝑖𝑛𝑚𝑠_𝑛,𝑠^∗𝑚𝑑𝑠^𝑚+𝑅_TM𝑑𝜉′𝑑𝜉′𝑑𝑑𝑠𝜉^𝑛𝑑𝜉𝑘²¹𝑒_𝑠𝑛𝑒_s^∗𝑚×¹⁻¹{𝑓}_𝑗𝑓^^𝑇𝑖𝑔_𝑖𝑛𝑚𝑠_𝑛,𝑠^∗𝑚𝑑𝑠^𝑚+𝑗𝑑𝜉′𝑑𝜉′𝑑𝑑𝑠𝜉^𝑛𝑑𝜉4𝜋𝜔𝜀₀¹⁻¹𝑍^𝑇{𝑓}_𝑗{𝑓′}^𝑇𝑗𝑑𝑠_𝑛𝑑𝜉𝑑𝜉,(31) while the voltage vector is given by [13] {𝑉}^𝑛𝑗=−𝑗4𝜋𝜔𝜀₀¹⁻¹𝐸^exc_𝑠𝑛𝑠_𝑛𝑓_𝑗𝑛𝑠_𝑛𝑑𝑠^𝑛𝑑𝜉𝑑𝜉^𝑛.(32) Once the current distribution is obtained, the radiated field can be obtained applying the similar BEM formalism [13]. Thus, the total field is given by 𝐸⃗=^𝑁_𝑘=1𝐸^𝑒𝑆𝑘+𝑅_TM𝐸^𝑒𝐼𝑘+𝑅_TE−𝑅_TM𝐸^𝑒𝐼𝑘⋅𝑒_𝑝⃗𝑒^𝑝⃗,(33) where the field components due to a wire segment radiation are given by𝐸^𝑒𝑆𝑘=1𝑗4𝜋𝜔𝜀_0𝑛𝑖=1𝑘²¹⁻¹𝑒_𝑘𝑠′𝐼^𝑒𝑖𝑘𝑓_𝑖(𝜉)𝑔_0𝑘𝑟,⃗𝑟^⃗𝑑𝑠^𝑘′+𝑑𝜉𝑑𝜉₁₋₁𝐼^𝑒𝑖𝑘𝜕𝑓_𝑖(𝜉)𝜕𝜉∇𝑔^0𝑘𝑟,⃗𝑟^⃗𝑑𝑠^𝑘′𝐸𝑑𝜉𝑑𝜉_𝑒𝐼,=1𝑗4𝜋𝜔𝜀_0𝑛𝑖=1𝑘²¹⁻¹𝑒_𝑘𝑠∗𝐼^𝑒𝑖𝑘𝑓_𝑖(𝜉)𝑔_𝑖𝑘𝑟,⃗𝑟^∗⃗𝑑𝑠^𝑘′−𝑑𝜉𝑑𝜉₁₋₁𝐼^𝑒𝑖𝑘𝜕𝑓_𝑖(𝜉)𝜕𝜉^∇𝑔_𝑖𝑘𝑟,⃗𝑟^∗⃗𝑑𝑠^𝑘′𝑑𝜉𝑑𝜉.(34) The total magnetic field is given by [13]𝐻=^𝑁_𝑘=1𝐻^𝑒𝑆𝑘+𝑅_TE𝐻^𝑒𝐼𝑘+𝑅_TM−𝑅_TE𝐻^𝑒𝐼𝑘⋅𝑒_𝑝⃗𝑒^𝑝⃗,(35) while the magnetic field components are given by [13]𝐻^𝑒𝑆𝑘=−14𝜋^𝑛_𝑖=1¹⁻¹𝐼_𝑖𝑘𝑓_𝑖(𝜉)𝑒_𝑠𝑘×∇𝑔_0𝑘𝑟,⃗𝑟^⃗𝑑𝑠^𝑘′𝐻𝑑𝜉𝑑𝜉,_𝑒𝐼𝑘=−14𝜋^𝑛_𝑖=1¹⁻¹𝐼^𝑒𝑖𝑘𝑓_𝑖(𝜉)𝑒_𝑘𝑠∗×∇𝑔_𝑖𝑘𝑟,⃗𝑟^∗⃗𝑑𝑠^𝑘′𝑑𝜉𝑑𝜉.(36)

The reduction to the case of a single straight wire or straight wire array is straightforward and can be found elsewhere, for example, in [8].

2.3. Transmission Line Approximation: Telegrapher’s Equations in the Frequency Domain

Voltages and currents along the multiconductor transmission line shown in Figure 4 induced by an external field excitation can be obtained using the field-to-transmission line matrix equations in the frequency domain [10]: 𝑑𝑑𝑥𝑉(𝑥)+𝑍⋅𝐼(𝑥)=−𝑗𝜔𝜇⁰^ℎ0𝐻^exc𝑦𝑑(𝑥,𝑧)𝑑𝑧,𝑑𝑥𝐼(𝑥)+𝑌⋅𝑉(𝑥)=−𝑗𝜔𝜇⁰^ℎ0𝐸^exc𝑧(𝑥,𝑧)𝑑𝑧,(37) where the longitudinal impedance matrix is given by𝑍=𝑗𝜔[𝐿]+𝑍^𝑤+𝑍^𝑔,(38) and the transversal admittance matrix can be written as 𝑌=𝑗𝜔[𝐶]+[𝐺],(39) where [𝐿] is the per-unit-length longitudinal inductance matrix for a perfect soil [𝐶] and [𝐺] are the per-unit length transverse capacitance and conductance matrix of the multiconductor line, respectively. Furthermore, ⌊𝑍_𝑤⌋ is the per-unit length internal impedance matrix of the conductors, and ⌊𝑍_𝑔⌋ is the per-unit length ground impedance matrix. Finally, [𝐻^exc_𝑦(𝑥,𝑧)] and [𝐸^{𝑧𝑒𝑥𝑐}(𝑥,𝑧)] are sources vectors expressed in terms of the incident magnetic and electric field, respectively [1, 8].

2.4. Computational Examples

The first computational example is related to the analysis of an overhead wire (Figure 3) of length 𝐿=20m, radius 𝑎=0.005m located at height ℎ=1 m above PEC ground and illuminated by the plane wave. The amplitude of the electric field is 𝐸₀=1V/m and it is parallel to x-axis. Figure 5 shows the frequency response at the center of the line. The results computed via GB-IBEM and TL are compared to the results obtained via NEC using RC and Sommerfeld integral approach, respectively, to account for the presence of a lossy half-space. The agreement between the results obtained via the different approaches is found to be satisfactory.

Figure 5 

Current induced at the center of the line above a PEC ground versus frequency.

Figure 6 shows the frequency response for the same line located above an imperfectly conducting half-space for various values of ground conductivity 𝜎=1 mS/m. The results calculated via different approaches agree satisfactorily again.

Figure 6 

Current induced at the center of the line above a lossy ground versus frequency (𝜎=0.001S/m,𝜀𝑟=10).

Next computational example is related to a simple Power Line Communications (PLCs) system. PLC technology aims to provide users with necessary communication means by using the already existing and widely distributed power line network and electrical installations in houses and buildings. However, one of the principal drawbacks of this technology is related to electromagnetic interference (EMI) problems, as overhead power lines at the PLC frequency range (1 MHz to 30 MHz) act as transmitting or receiving antennas, respectively [13].

Figure 7 shows the geometry of a simple PLC system consisting of two conductors placed in parallel above each other at the distance 𝑑. The conductors are suspended between two poles of equal height, thus heaving the shape of the catenary.

Figure 7 

Simple PLC circuit.

The geometry of a catenary is fully defined by such parameters as the distance between the points of suspension, 𝐿, the sag of the conductor, 𝑠, and the height of the suspension point, ℎ, as shown in Figure 7. The imperfectly conducting ground is characterized with electrical permeability ε_r and conductivity σ.

The conductors are modeled as thin wire antennas excited by the voltage generator 𝑉_𝑔 at one end and terminated by the load impedance 𝑍_𝐿 at the other end.

The influence of the load impedance is taken into account by modifying continuity condition for the tangential components of the electric field at the wire surface: 𝐸^inc_𝑠+𝐸^sct_𝑠=𝑍^𝐿𝐼(𝑠),(40) where 𝑍^𝐿 is the corresponding conductor per length impedance of the conductor.

The modified Pocklington equation for the wire containing the load impedance is now given by 𝐸^inc(𝑠)=−1𝑗4𝜋𝜔𝜀₀^𝐿0𝑘²¹𝑒_𝑠⃗𝑒_𝑠′−𝜕²+𝑅_𝑇𝑀𝜕𝑠𝜕𝑠^𝑔⁰𝑠,𝑠^𝑘²¹𝑒_𝑠⃗𝑒_𝑠∗−𝜕²×𝐼𝑠_𝜕𝑠𝜕𝑠^∗𝑔^𝑖𝑠,𝑠^∗𝑑𝑠^+𝑍^𝐿𝐼(𝑠).(41) Set of integral equations (41) is numerically solved using via GB-IBEM.

The actual example is related to the simple PLC circuit shown in Figure 7. The distance between poles is 𝐿=200m, with the radii of wires 𝑎=6.35mm. The wires are suspended on the poles at heights ℎ₁=10m and ℎ₂=11m. The maximum sag of the conductor is assumed to be 𝑠=2m. Ground parameters are 𝜀_𝑟=13 and =0.005S/mσ. The power of the applied voltage generator is 2.5 μW (minimum power required for the PLC system operation) and operating frequency is chosen to be 14 MHz. The value of the terminating load 𝑍_𝐿 is 500 Ω. Figure 8 shows the current distribution along the simple PLC system for different values of sag.

Figure 8 

The current distribution along a simple PLC system.

Radiated electric and magnetic fields at the distance of 30 m from the wires and 10 m above ground are shown at the Figures 9 and 10, respectively.

Figure 9 

Radiated electric field.

Figure 10 

Radiated magnetic field.

Analysis of the radiated field distributions shows that the conductor sag does not influence the far-field region significantly while the near-field distribution is mainly determined by the conductor geometry. Finally, the power of the applied voltage generator is changed to 1 mW (average power used at the actual PLC systems) and operating frequency is varied between 1 and 30 MHz. The values of the terminating load 𝑍_𝐿 are chosen to be 50 Ω, 500 Ω, 5000 Ω, thus simulating different conditions within the power grid. The maximum values of the radiated electric field at the distance of 30 m for different arrangements are shown in Table 1.

Table 1 

Maximum values of the radiated electric field at the 30 m distance.

According to the available international standards [16, 17], radiated electric fields should not excide level of 30 μV/m at the distance of 30 m. Obviously, the radiated field levels are at best case more than 10 times higher than the proposed limit. The spatial distributions of the radiated electric field have been calculated for the number of frequencies in the frequency range from 1 to 30 MHz. Maximum levels of the calculated electric fields values are shown to excide the limits defined by the standard for the disturbances caused by information technology equipment.

3. Time Domain Models and Methods

This section deals with direct time domain analysis of transient electromagnetic field coupling to straight overhead wires using the wire antenna theory and the transmission line method, respectively. 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 scheme of GB-IBEM. The time domain Telegrapher's equations are solved using the Finite Difference Time Domain (FDTD) method. Time domain numerical results obtained with both approaches are compared to the results computed via NEC 2 code combined with Inverse Fourier Transform procedure. Some illustrative comparisons of results obtained by means of antenna theory and transmission line approach are presented in this section.

It is worth mentioning that, for the sake of simplicity, only straight wires are analyzed in this paper.

3.1. The Antenna Theory Model

Generally, a direct time-domain analysis of thin wire in the presence of a lossy half-space can be carried out via the appropriate space-time integral equations of either Pocklington or Hallen type [1, 8]. When applied to the solution of the Hallen integral equation the Galerkin-Bubnov Indirect Boundary Element Method (GB-IBEM) [8] results in relatively complex procedures compared to various procedures for the solution of Pocklington equations, but, at the same time, it is proven to be highly efficient, accurate, and unconditionally stable [8, 18, 19]. On the other hand, the implementation of GB-IBEM to the solution of the Pocklington-type equation is relatively simple, but suffers from serious numerical instabilities. The origin of these instabilities is the discretization of space-time differential operator [8]. The GB-IBEM solution of the Pocklington equation in free space for certain values of time domain integration parameters has been presented elsewhere, for example, in [19], while the Hallen integral equation solution by means of GB-IBEM has been obtained for thin wire structures in the presence of a dielectric half-space, for example, in [11]. In both cases, the influence of imperfect ground has been taken into account via the corresponding reflection coefficient. The numerical solution was mostly limited to scenarios in which the finite conductivity of the ground could be ignored. This approximation involves cases where the wires are sufficiently far from the two-media interface or where the ground conductivity is appreciably low or very high, that is, where the approximation of pure dielectric medium or perfect ground is applied. Through these approximations the time-dependent part of the reflection coefficient function vanishes, and the resulting matrix equation simplifies significantly.

However, for the cases where these approximations are not valid, modifications to the original methods are required in order to include the ground conductivity [8]. Namely, the related reflection coefficient is space-time dependent, and the resulting convolution integrals have to be included in the matrix system and numerically computed. This leads to a significant increase in the overall computational cost of the method and consequently requires several modifications.

This section deals with the transient analysis of multiple horizontal wires above a lossy ground using the Hallen integral equation approach.

The set of space-time Hallen's integral equations can be derived as an extension of the single wire case. First, a single wire insulated in free space is considered.

Thin wire antenna or scatterer of length 𝐿 and radius 𝑎, oriented along the x-axis, is considered. The wire is assumed to be perfectly conducting and excited by a plane wave electric field. For the sake of simplicity, the analysis is restricted to the case of a normally incident electric field.

The tangential component of the total field vanishes on the PEC wire surface, that is,𝐸^inc_𝑥+𝐸^sct_𝑥=0,(42) where 𝐸^inc_𝑥 is the incident and 𝐸^sct_𝑥 scattered field on the metallic wire surface. Starting from Maxwell equations and obeying the Lorentz gauge one obtains a time domain version of (2):𝜕²𝐴⃗𝜕𝑡²−1𝜇𝜀∇∇𝐴⃗||||_tan=𝜕𝐸^inc𝜕𝑡||||_tan,(43) where 𝐴⃗ is the space-time-dependent vector potential.

According to the thin wire approximation, only the axial component of the vector potential exists, that is, (43) becomes 𝜕²𝐴_𝑥𝜕𝑥²−1𝑐²𝜕²𝐴_𝑥𝜕𝑡²=−1𝑐²𝜕𝐸^inc_𝑥𝜕𝑡,(44) where c denotes the velocity of light.

The corresponding solution of (44) can be expressed in terms of a sum of the general solution of the homogeneous equation and the particular solution of the inhomogeneous equation:𝐴_𝑥(𝑥,𝑡)=𝐴^ℎ𝑥(𝑥,𝑡)+𝐴^𝑝𝑥(𝑥,𝑡).(45) The solution of the homogeneous wave equation is given as a superposition of incident and reflected wave [8]:𝐴^ℎ𝑥(𝑥,𝑡)=𝐹₁𝑡−𝑥𝑐+𝐹²𝑡+𝑥𝑐,(46) while the particular solution is given by the integral [8]:𝐴^𝑝𝑥(𝑥,𝑡)=12𝑍₀^L₀𝐸^inc_𝑥𝑥^,𝑡−||𝑥−𝑥^||𝑐𝑑𝑥^,(47) where L denotes the total antenna length.

On the other hand, the magnetic vector potential on the PEC wire surface is given by the particular integral:𝐴_𝑥(𝑥,𝑡)=𝜇4𝜋_𝑆𝐼𝑥^,𝑡−𝑅/𝑐𝑅𝑑𝑥^.(48)

Combining (45)–(48) yields the space-time Hallen equation:^𝐿0𝐼𝑥^,𝑡−𝑅/𝑐4𝜋𝑅𝑑𝑥^=𝐹₀𝑡−𝑥+1𝑐+𝐹^𝐿𝑡−𝐿−𝑐𝑥2𝑍₀^𝐿0𝐸^inc_𝑥𝑥^,𝑡−||𝑥−𝑥^||𝑐𝑑𝑥^,(49) where 𝐼(𝑥^) is the equivalent axial current to be determined, 𝐸^inc_𝑥 is the known tangential incident field, 𝑅=[(𝑥−𝑥^)²+𝑎²]^1/2 is the distance from the source point (the equivalent current in the antenna axis) to the observation point, and 𝑍₀ is the wave impedance of a free space.

The unknown functions 𝐹₀(𝑡) and 𝐹_𝐿(𝑡) account for the multiple reflections of the current at the free ends of the wire.

A direct time formulation for a straight thin wire above a dissipative half-space can be obtained as the extension of the free-space Hallen equation (49).

The free space Hallen equation (49) is first transferred into the Laplace frequency domain:^𝐿0𝐼𝑥^,𝑠𝑒^{−𝑠𝑅/𝑐}4𝜋𝑅𝑑𝑥^=𝐹₀(𝑠)𝑒^{−𝑠𝑥/𝑐}+𝐹_𝐿(𝑠)𝑒^{−𝑠((𝐿−𝑥)/𝑐)}+12𝑍₀^𝐿0𝐸^inc_𝑥𝑥^,𝑠𝑒^{−𝑠|𝑥−𝑥′|/𝑐}𝑑𝑥^,(50) where 𝑠=𝑗𝜔 is the Laplace variable.

According to the theory of images the free space integral equation (50) is extended by an additional term multiplying the Green function of the image source by space-frequency-dependent reflection coefficient 𝑅_TM(𝜃^,𝑠) for TM polarization. The integral equation in the frequency domain is given by^𝐿0𝐼(𝑥,𝑠)𝑒^{𝑠𝑅/𝑐}4𝜋𝑅𝑑𝑥′−^𝐿0𝑅_TM(𝜃,𝑠)𝐼(𝑥,𝑠)𝑒^{𝑠𝑅∗/𝑐}4𝜋𝑅^∗=𝐹₀𝑑𝑥′(𝑠)𝑒^{−𝑠𝑥/𝑐}+𝐹_𝐿(𝑠)𝑒^{−𝑠((𝐿−𝑥)/𝑐)}+12𝑍₀^𝐿0𝐸^exc_𝑥𝑥^,𝑠𝑒^{−𝑠(|𝑥−𝑥′|/𝑐)}𝑑𝑥′,(51) where 𝑅^∗=√(𝑥−𝑥′)²+4ℎ² and 𝑅_TM(𝜃^,𝑠) is determined by the expression [1]𝑅_TM𝜃^,𝑠=𝜀^𝑟(1+𝜎/𝜀𝑠)cos𝜃^−𝜀_𝑟(1+𝜎/𝜀𝑠)−sin²𝜃^𝜀_𝑟(1+𝜎/𝜀𝑠)cos𝜃^+𝜀_𝑟(1+𝜎/𝜀𝑠)−sin²𝜃^,(52) where 𝜎 and 𝜀 are the lossy medium conductivity and permittivity, respectively, and𝜃^=arct𝑔(|𝑥−𝑥^|/2ℎ).

The reflection coefficient (RC) approach is a satisfactory approximation in half-space calculations, as long as the field is calculated far away from the source, and the imperfect ground, respectively, to ensure 𝜃^<𝜋/2 [8].

Performing the convolution, the time domain counterpart of (52) is obtained in the form^𝐿0𝐼𝑥^,𝑡−𝑅/𝑐−4𝜋𝑅𝑑𝑥′_𝑡−∞^𝐿0𝑟(𝜃,𝜏)𝐼𝑥^,𝑡−𝑅^∗/𝑐−𝜏4𝜋𝑅^∗=1𝑑𝑥′𝑑𝜏2𝑍₀^𝐿0𝐸^exc_𝑥𝑥^,𝑡−||𝑥−𝑥^||+𝐹₀𝑐𝑑𝑥′𝑡−𝑥𝑐+𝐹^𝐿𝑡−𝐿−𝑐𝑥,(53) where r(𝜃,𝑡) is the space-time reflection coefficient which, for convenience, can be written in the form [18] 𝑟(𝜃,𝜏)=𝑟^(𝜃,𝜏)+𝑟^(𝜃,𝜏),(54) where𝑟𝑟^(𝜃,𝑡)=𝐾𝛿(𝑡),_(𝜃,𝑡)=4𝛽1−𝛽²𝑒^−𝛼𝑡𝑡^∞_𝑛=1𝜏=𝜎(−1)^𝑛+1𝑛𝐾^𝑛𝐼_𝑛(𝛼𝑡),𝜀₀𝜀_𝑟,𝛽=√𝜀_𝑟−sin²𝜃𝜃=arct𝑔||𝑥−𝑥𝜀′||^𝑟cos𝜃,𝛾=1−si𝜏n²𝜃/𝜀_𝑟,2ℎ,𝐾=11+−𝛽𝛽,𝛼=𝜏2.(55)

Note that 𝐼_𝑛 is the modified Bessel function of the first order and nth degree.

For the case of normal incidence, which is considered for the sake of simplicity, the excitation term is given by 𝐸^exc_𝑥(𝑡)=𝐸^inc_𝑥(𝑡)−𝐸^ref_𝑥𝑡^∗,(56) where 𝑡^∗=𝑡−𝑅^∗/𝑐.

The transient ground-reflected field is obtained as the convolution of the incident field and the space-time reflection coefficient for the angle of incidence 𝜃=0 (in accordance with the parallel incidence of the electric field), as is proposed in [20]𝐸^ref_𝑥(𝑡)=^𝑡−∞𝐸^inc_𝑥(𝑡−𝜏)𝑟(𝜃=0,𝜏)𝑑𝜏(57) and the integral equation (53) becomes ^𝐿0𝐼𝑥^,𝑡−𝑅/𝑐−4𝜋𝑅𝑑𝑥′_𝑡−∞^𝐿0𝑟(𝜃,𝜏)𝐼𝑥^,𝑡−𝑅^∗/𝑐−𝜏4𝜋𝑅^∗=𝐹₀𝑑𝑥′𝑑𝜏+1𝑡−𝑥𝑐+𝐹^𝐿𝑡−𝐿−𝑐𝑥2𝑍₀^𝐿0𝐸^inc_𝑥−1𝑥′,𝑡−||𝑥−𝑐𝑥′||𝑑𝑥′2𝑍₀^𝑡−∞^𝐿0𝐸^inc_𝑥𝑥^,𝑡−||𝑥−𝑥^||×𝑟(𝜃=0,𝜏)𝑑𝑥′𝑑𝜏.𝑐−𝜏(58)

The unknown time functions 𝐹₀(𝑡),𝐹_𝐿(𝑡),𝐹₀(𝑡−(𝐿/𝑐)), and 𝐹_𝐿(𝑡−(𝐿/𝑐)) can be obtained in the same manner, as in the case of free space, in terms of as auxilliary functions 𝐾₀(𝑡) and 𝐾_𝐿(𝑡) [8]:𝐹₀(𝑡)=^∞_𝑛=0𝐾₀𝑡−2𝑛𝐿𝑐−^∞_𝑛=0𝐾_𝐿𝐹_𝐿𝑡−(2𝑛+𝑐1)𝐿,(𝑡)=^∞_𝑛=0𝐾_𝐿𝑡−2𝑛𝐿𝑐−^∞_𝑛=0𝐾₀𝑡−(2𝑛+𝑐1)𝐿,(59) where𝐾_𝐿(𝑡)=^𝐿0𝐼𝑥^,𝑡−𝑅₀/𝑐4𝜋𝑅₀−𝑑𝑥′_𝑡−∞^𝐿0𝑟𝜃^,𝜏𝐼𝑥^,𝑡−𝑅^∗0/𝑐−𝜏4𝜋𝑅^∗0−1𝑑𝑥′𝑑𝜏2𝑍₀^𝐿0𝐸^inc_𝑥𝑥′,𝑡−𝑥′𝑐𝑑𝑥^𝐾_𝐿,(𝑡)=^𝐿0𝐼𝑥^,𝑡−𝑅_𝐿/𝑐4𝜋𝑅_𝐿−𝑑𝑥′_𝑡−∞^𝐿0𝑟𝜃^,𝜏𝐼𝑥^,𝑡−𝑅^∗𝐿/𝑐−𝜏4𝜋𝑅^∗𝐿−1𝑑𝑥′𝑑𝜏2𝑍₀^𝐿0𝐸^exc_𝑥𝑥′,𝑡−𝐿−𝑥′𝑐𝑑𝑥^,(60) while 𝑅₀ and 𝑅_𝐿 are the distances from the wire ends to the source point and 𝑅^∗0, 𝑅^∗𝐿 are the distances from the image wire ends to the image source point.

If the case of a perfect (ideal) dielectric half-space is considered, the Hallen equation (53) simplifies into^𝐿0𝐼𝑥^,𝑡−𝑅/𝑐4𝜋𝑅𝑑𝑥′−^𝐿0𝑟(𝜃)𝐼𝑥^,𝑡−𝑅^∗/𝑐4𝜋𝑅^∗=𝐹₀𝑑𝑥′+1𝑡−𝑥𝑐+𝐹^𝐿𝑡−𝐿−𝑐𝑥2𝑍₀^𝐿0𝐸^exc_𝑥𝑥′,𝑡−||𝑥−𝑐𝑥′||𝑑𝑥′.(61) Space-time integral equation (53) or (61), respectively, can be solved assuming the zero current at the free ends of the wire and with the initial conditions requiring the wire not to be excited before the certain instant 𝑡=𝑡₀ [8].

The transient behavior of 𝑀 straight horizontal thin wires located at different heights above an infinite ground plane is determined by a set of the coupled space-time integral equations of the Hallen type [11]:^𝑀_𝑠=1^𝑥𝐿𝑠_𝑥0𝑠𝐼_𝑠𝑥^,𝑡−𝑅_𝑣𝑠/𝑐4𝜋𝑅_𝑣𝑠−𝑑𝑥′_𝑀_𝑠=1^𝑡−∞^𝑥𝐿𝑠_𝑥0𝑠𝑟_𝑣𝑠(𝜃,𝜏)𝐼^𝑠𝑥^,𝑡−𝑅^∗𝑣𝑠/𝑐−𝜏4𝜋𝑅^∗𝑣𝑠=𝐹_0𝑣𝑑𝑥′𝑑𝜏𝑡−𝑥−𝑐𝑥^0𝑣+𝐹^𝐿𝑣𝑡−𝑥^𝐿𝑣−𝑥+1𝑐2𝑍₀^𝑥𝐿𝑣_𝑥0𝑣𝐸^exc_𝑥𝑣𝑥^,𝑡−||𝑥−𝑥^||𝑐𝑑𝑥′,(62) where 𝑣,𝑠=1,2,…,𝑀 denote the index of the observed and source wire, respectively. Furthermore, 𝐿_𝑠 and 𝐿_𝑣 are the lengths of the sth and vth wire, and 𝑥^, 𝑥 are the x-coordinates of the source and observation points on respective wires. The distances between observation point (𝑥,𝑦,𝑧) on the wire v and source point (𝑥^,𝑦^,𝑧^) on the wire 𝑠 are given by𝑅_𝑣𝑠=√√(𝑥−𝑥^)²+(𝑦−𝑦^)²+(𝑧−𝑧^)²∶𝑣≠𝑠𝑅(𝑥−𝑥^)²+𝑎²∶𝑣=𝑠,_∗𝑣𝑠=(𝑥−𝑥^)²+(𝑦−𝑦^)²+(𝑧+𝑧^)²,(63) where asterisk is related to source points are located on the image wire.

Unknown time signals 𝐹_0𝑣(𝑡) and 𝐹_𝐿𝑣(𝑡) account for the multiple reflections of transient currents at the wire open ends and can be written in the form𝐹_0𝑣(𝑡)=^∞_𝑛=0𝐾_0𝑣𝑡−2𝑛𝑐𝐿^𝑣−^∞_𝑛=0𝐾_𝐿𝑣𝑡−(2𝑛+𝑐1)𝐿^𝑣,(64)𝐹_𝐿𝑣(𝑡)=^∞_𝑛=0𝐾_𝐿𝑣𝑡−2𝑛𝑐𝐿^𝑣−^∞_𝑛=0𝐾_0𝑣𝑡−(2𝑛+𝑐1)𝐿^𝑣,(65) while the auxiliary functions 𝐾 are defined, as follows:𝐾=_0𝑣(𝑡)_𝑀_𝑠=1^𝑥𝐿𝑠_𝑥0𝑠𝐼_𝑠𝑥^,𝑡−𝑅^(0)𝑣𝑠/𝑐4𝜋𝑅^(0)𝑣𝑠−𝑑𝑥′_𝑀_𝑠=1^𝑡−∞^𝑥𝐿𝑠_𝑥0𝑠𝑟_𝑣𝑠(𝜃,𝜏)𝐼^𝑠𝑥^,𝑡−𝑅^{∗(0)𝑣𝑠}/𝑐−𝜏4𝜋𝑅^{∗(0)𝑣𝑠}−1𝑑𝑥′𝑑𝜏2𝑍₀^𝑥𝐿𝑣_𝑥0𝑣𝐸^exc_𝑥𝑣𝑥^,𝑡−||𝑥−𝑥^||𝐾𝑐𝑑𝑥′,(66)_𝐿𝑣=(𝑡)_𝑀_𝑠=1^𝑥𝐿𝑠_𝑥0𝑠𝐼_𝑠𝑥^,𝑡−𝑅^{(𝐿)𝑣𝑠}/𝑐4𝜋𝑅^{(𝐿)𝑣𝑠}−𝑑𝑥′_𝑀_𝑠=1^𝑡−∞^𝑥𝐿𝑠_𝑥0𝑠𝑟_𝑣𝑠(𝜃,𝜏)𝐼^𝑠𝑥^,𝑡−𝑅^{∗(𝐿)𝑣𝑠}/𝑐−𝜏4𝜋𝑅^{∗(𝐿)𝑣𝑠}−1𝑑𝑥′𝑑𝜏2𝑍₀^𝑥𝐿𝑣_𝑥0𝑣𝐸^exc_𝑥𝑣𝑥^,𝑡−||𝑥−𝑥^||𝑐𝑑𝑥′,(67) where 𝑅^(0)𝑣𝑠 and 𝑅^{(𝐿)𝑣𝑠} are distances from considered source point on each wire 𝑠 to a corresponding observation point at the ends of the wire 𝑣:𝑅^(0)𝑣𝑠=𝑅_𝑣𝑠||_{𝑥=𝑥0𝑣},𝑅^{(𝐿)𝑣𝑠}=𝑅_𝑣𝑠||_{𝑥=𝑥𝐿𝑣},(68) while 𝑅^{∗(0)𝑣𝑠} and 𝑅^{∗(𝐿)𝑣𝑠} are distances between the source point at the image of the wire 𝑠and observation point located at the ends of the wire 𝑣:𝑅^{∗(0)𝑣𝑠}=𝑅^∗𝑣𝑠||_{𝑥=𝑥0𝑣},𝑅^{∗(𝐿)𝑣𝑠}=𝑅^∗𝑣𝑠||_{𝑥=𝑥𝐿𝑣}.(69)

The space-time reflection coefficient 𝑟_𝑣𝑠(𝜃,𝑡) accounts for the influence of the interface and is given by [11] 𝑟_𝑣𝑠𝜃^𝑣𝑠,𝑡=𝐴𝛿(𝑡),(70) where𝐴=1−𝛽1+𝛽,𝛽=√𝜀_𝑟−sin²𝜃′𝜃𝜀_𝑟cos𝜃′_𝑣𝑠,=Arct𝑔√(𝑥^−𝑥)²+(𝑦^−𝑦)²𝑧′+𝑧.(71)

The angle 𝜃^𝑣𝑠is the angle between the source point on the image of the wire 𝑠(𝑥^,𝑦^,−𝑧^) and the observation point (𝑥,𝑦,𝑧) on wire v.

Substituting (70) into (62) yields^𝑀_𝑠=1^𝑥𝐿𝑠_𝑥0𝑠𝐼_𝑠𝑥^,𝑡−𝑅_𝑣𝑠/𝑐4𝜋𝑅_𝑣𝑠𝑑𝑥′−^𝑀_𝑠=1^𝑥𝐿𝑠_𝑥0𝑠𝑟_𝑣𝑠(𝜃)𝐼^𝑠𝑥^,𝑡−𝑅^∗𝑣𝑠/𝑐4𝜋𝑅^∗𝑣𝑠=𝐹_0𝑣𝑑𝑥′𝑡−𝑥−𝑐𝑥^0𝑣+𝐹^𝐿𝑣𝑡−𝑥^𝐿𝑣−𝑥+1𝑐2𝑍₀^𝑥𝐿𝑣_𝑥0𝑣𝐸^exc_𝑥𝑣𝑥^,𝑡−||𝑥−𝑥^||𝑐𝑑𝑥′,(72) where 𝐸^exc_𝑥𝑣 is the space-time-dependent tangential electric field on the vth wire.

For the case of normal incidence the total excitation field 𝐸^exc_𝑥𝑣(𝑥^,𝑡) is given as the sum of the incident field 𝐸^inc_𝑥𝑣(𝑥^,𝑡) and the field reflected from the interface 𝐸^ref_𝑥𝑣(𝑥^,𝑡) [11],𝐸^exc_𝑥𝑣𝑥^,𝑧,𝑡=𝐸^inc_𝑥𝑣𝑥^,𝑡−𝑇+𝐸^ref_𝑥𝑣𝑥^,𝑡−𝑇.(73)

The time shift 𝑇 represents the time required for the wave to travel from the highest wire to the height 𝑧 of the observed vth wire. Assigning the highest wire with index 𝑈, it can be written𝑇=𝑧^𝑈−𝑧𝑐,𝑧^𝑈=max𝑧₁,𝑧₂,…,𝑧,…,𝑧_𝑀.(74)

The field reflected from the interface for the case of normal incidence is given by𝐸^ref_𝑥𝑣𝑥^,𝑡−𝑇=𝑟(𝜃=0)⋅𝐸^inc_𝑥𝑣𝑥^,𝑡−𝑇−2𝑧𝑐,(75) where 𝑡−𝑇−2𝑧/𝑐 is the time needed for the wave to travel from observed vth wire to the interface.

For the case of PEC ground plane, the space-time reflection coefficient (54) simply becomes𝑟_𝑣𝑠𝜃^𝑣𝑠,𝑡=1.(76)

Thus, the set of (72) simplifies into ^𝑀_𝑠=1^𝑥𝐿𝑠_𝑥0𝑠𝐼_𝑠𝑥^,𝑡−𝑅_𝑣𝑠/𝑐4𝜋𝑅_𝑣𝑠−𝑑𝑥′_𝑀_𝑠=1^𝑥𝐿𝑠_𝑥0𝑠𝐼_𝑠𝑥^,𝑡−𝑅^∗𝑣𝑠/𝑐4𝜋𝑅^∗𝑣𝑠=𝐹_0𝑣𝑑𝑥′𝑡−𝑥−𝑐𝑥^0𝑣+𝐹^𝐿𝑣𝑡−𝑥^𝐿𝑣−𝑥+1𝑐2𝑍₀^𝑥𝐿𝑣_𝑥0𝑣𝐸^exc_𝑥𝑣𝑥^,𝑡−||𝑥−𝑥^||𝑐𝑑𝑥′,(77) and the field reflected from PEC ground is simply given by𝐸^ref_𝑥𝑣𝑥^,𝑡−𝑇=𝐸^inc_𝑥𝑣𝑥^,𝑡−𝑇−2𝑧/𝑐.(78)

Given the dielectric constant of the medium and the known excitation 𝐸^inc_𝑥𝑣(𝑥^,𝑡), a system of 𝑀 coupled Hallen integral equations can be solved using time domain version of GB-IBEM and by taking into account appropriate boundary and initial conditions. Boundary conditions assume zero currents at the end of each wire, while initial conditions assume all the currents to be zero for 𝑡≤0.

3.2. The Numerical Solution

First, numerical procedure for single wire Hallen equation is outlined. Applying the weighted residual approach in the spatial domain and GB-IBEM procedure [8], the following local matrix system is obtained: [𝐴]{𝐼}_𝑖||_{𝑡−𝑅/𝑐}−𝐴^∗{𝐼}_𝑖||_{𝑡−𝑅∗/𝑐}−𝐴|||_{𝑡−𝑅∗/𝑐}=[𝐵]{𝐸}||_{𝑡−|𝑥−𝑥′|/𝑐}+[𝐶]^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅0/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}−𝐶^∗^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅∗0/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}−^∞_𝑛=0𝐶^𝑛|||||_{𝑡−(𝑅∗0/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}−[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−(𝑥/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}−[𝐷]^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅𝐿/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+𝐷^∗^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅∗𝐿/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−((𝐿−𝑥′)/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+^∞_𝑛=0𝐷^𝑛|||||_{𝑡−(𝑅∗𝐿/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+[𝐷]^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅𝐿/𝑐)−(2𝑛𝐿/𝑐)−(𝐿−𝑥/𝑐)}−𝐷^∗^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅∗𝐿/𝑐)−(2𝑛𝐿/𝑐)−(𝐿−𝑥/𝑐)}+^∞_𝑛=0𝐷^𝑛|||||_{𝑡−(𝑅∗𝐿/𝑐)−(2𝑛𝐿/𝑐)−(𝐿−𝑥/𝑐)}−[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−((𝐿−𝑥′)/𝑐)−(2𝑛𝐿/𝑐)−(𝐿−𝑥/𝑐)}−[𝐶]^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅0/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥/𝑐)}+𝐶^∗^∞_𝑛=0𝐼^𝑛_𝑖|||||_{𝑡−(𝑅∗0/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥/𝑐)}+[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−(𝑥/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥/𝑐)}−^∞_𝑛=0𝐶^𝑛|||||_{𝑡−(𝑅∗0/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥/𝑐)}.(79)

The space-dependent local matrices representing the interaction between ith source and jth observation element are defined as follows:[𝐴]=_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖1[𝐵]=14𝜋𝑅𝑑𝑥′𝑑𝑥,2𝑍₀_Δ𝑙𝑗_Δ𝑙𝑖[𝐶]={𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥,_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖14𝜋𝑅₀[𝐷]=𝑑𝑥′𝑑𝑥,_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖14𝜋𝑅_𝐿𝐴𝑑𝑥′𝑑𝑥,_∗=_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖𝑟(𝜃)4𝜋𝑅^∗𝐶𝑑𝑥′𝑑𝑥,_∗=_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖𝑟(𝜃)4𝜋𝑅^∗0𝐷𝑑𝑥′𝑑𝑥,_∗=_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖𝑟(𝜃)4𝜋𝑅^∗𝐿𝑑𝑥^𝑑𝑥,(80) where {𝑓} stands for the shape functions, while additional time dependent vectors are given by 𝐶𝐴=^{𝑡−𝑅∗0/𝑐}_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖𝐻₁𝑑𝑥′𝑑𝑥{𝐼(𝜏)}_𝑖𝑑𝜏_𝑛=^{𝑡−(𝑅∗00/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}×_Δ𝑙𝑗_Δ𝑙𝑖𝐷{𝑓}_𝑗{𝑓}^𝑇𝑖𝐻₂𝑑𝑥′𝑑𝑥{𝐼(𝜏)}_𝑖𝑑𝜏_𝑛=^{𝑡−(𝑅∗𝐿0/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}×_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖𝐻₃𝑑𝑥′𝑑𝑥{𝐼(𝜏)}_𝑖𝑑𝜏,(81) where

𝐻₁=𝑟^𝜃,𝑡−𝑅^∗/𝑐−𝜏4𝜋𝑅^∗𝐻₂,=𝑟^𝜃,𝑡−𝑅^∗0/𝑐−(2𝑛𝐿/𝑐)−(𝑥/𝑐)−𝜏4𝜋𝑅^∗0𝐻₃,=𝑟^𝜃,𝑡−𝑅^∗𝐿/𝑐−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)−𝜏4𝜋𝑅^∗0.(82) Assembling the local matrices and vectors into the global ones the following global matrix system is formed:[𝐴]{𝐼}||_{𝑡−𝑅/𝑐}={𝑔}||_{previoustimeinstants}+{𝑔̂}||_{previoustimeinstants},(83) where {𝑔}=𝐴^∗{𝐼}||_𝑡−𝑅^∗_/𝑐+[𝐵]{𝐸}||_{𝑡−|𝑥−𝑥|/𝑐}+[𝐶]^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅0/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}−𝐶^∗^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅∗0/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}−[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−(𝑥′/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}−[𝐷]^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅𝐿/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+𝐷^∗^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅∗𝐿/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−((𝐿−𝑥)/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+[𝐷]^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅𝐿/𝑐)−(2𝑛𝐿/𝑐)−(𝐿−𝑥/𝑐)}−𝐷^∗^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅∗𝐿/𝑐)−(2𝑛𝐿/𝑐)−(𝐿−𝑥/𝑐)}−[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−(𝐿−𝑥′/𝑐)−(2𝑛𝐿/c)−(𝐿−𝑥/𝑐)}−[𝐶]^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅0/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥/𝑐)}+𝐶^∗^∞_𝑛=0𝐼^𝑛|||||_{𝑡−(𝑅∗0/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥/𝑐)}+[𝐵]^∞_𝑛=0𝐸^𝑛|||||_{𝑡−(𝑥′/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥/𝑐)}{𝑔̂}=𝐴|||_{𝑡−(𝑅∗/𝑐)},(84)−^∞_𝑛=0𝐶^𝑛|||||_{𝑡−(𝑅∗0/𝑐)−(2𝑛𝐿/𝑐)−(𝑥/𝑐)}+^∞_𝑛=0𝐷^𝑛|||||_{𝑡−(𝑅∗𝐿/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝑥/𝑐)}+^∞_𝑛=0𝐷^𝑛|||||_{𝑡−(𝑅∗𝐿/𝑐)−(2𝑛𝐿/𝑐)−(𝐿−𝑥′/𝑐)}−^∞_𝑛=0𝐶^𝑛|||||_{𝑡−(𝑅∗0/𝑐)−((2𝑛+1)𝐿/𝑐)−(𝐿−𝑥′/𝑐)}.(85)

Applying the weighted residual approach in the time domain and using the Dirac impulses as weight functions, the time sampling is provided, and the following recurrent formula is obtained:𝐼_𝑗||_𝑡𝑘=∑^{𝑁𝑠𝑖=1}𝑎_𝑗𝑖𝐼_𝑗||_{𝑡𝑘−𝑅/𝑐}−𝑔_𝑗||_{previoustimeinstants}−𝑔̂_𝑗||_{previoustimeinstants}𝑎_𝑗𝑗,(86) where 𝐼_𝑗|_𝑡𝑘 is current for the jth space node at kth time instant, 𝑁 is total number of space segments, while the overbar indicates the absence of diagonal members.

It is worth noting that the numerical calculation of convolution integrals is rather tedious task leading to tremendously large computational time of the overall method. The main advantage of the method, on the other hand, is its unconditional stability.

Time domain GB-BEM procedure for the set of Hallen equations is undertaken in a similar manner as in the case of a single wire.

The solution of (72) and (77), respectively, is also carried out using the GB-IBEM technique.

Applying the boundary element discretisation to (72) and (77), respectively, leads to a local system of linear equations for the vth observed wire:^𝑀_𝑠=1_Δ𝑙𝑖_Δ𝑙𝑗14𝜋𝑅_𝑣𝑠{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥^𝑑𝑥𝐼_𝑠||||_{𝑡−𝑅𝑣𝑠/𝑐}−_Δ𝑙𝑖_Δ𝑙𝑗𝑟_𝑣𝑠(𝜃)4𝜋𝑅^∗𝑣𝑠{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥𝐼_𝑠||||_{𝑡−𝑅∗𝑣𝑠/𝑐}=12𝑍₀_Δ𝑙𝑖_Δ𝑙𝑗𝐸^exc_𝑥𝑣𝑥^,𝑡−||𝑥−𝑥^||𝑐{𝑓}^𝑗+𝑑𝑥′𝑑𝑥_Δ𝑙𝑗𝐹₀𝑡−𝑥−𝑥^0𝑣𝑐{𝑓}^𝑗+𝑑𝑥_Δ𝑙𝑗𝐹_𝐿𝑡−𝑥^𝐿𝑣−𝑥𝑐{𝑓}^𝑗𝑑𝑥,(87) where 𝑖,𝑗=1,2,…,𝑁denotes the index of the elements located on the sth source wire and the vth observed wire, respectively, with 𝑁 as the total number of space segments, while 𝑀 is the actual number of wires.

Finally, substituting (64)–(67) into (87), the following local matrix system is obtained:^𝑀_𝑠=1𝐴_𝑣𝑠𝐼_𝑠|||||_{𝑡−𝑅𝑣𝑠/𝑐}−^𝑀_𝑠=1𝐴^∗𝑣𝑠𝐼_𝑠|||||_{𝑡−𝑅∗𝑣𝑠/𝑐}=𝐵_𝑣𝐸_𝑣|_{𝑡−|𝑥−𝑥|/𝑐}+^𝑀_𝑠=1𝐶_𝑣𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥−𝑥0𝑣)/𝑐)−((2𝑛/𝑐)𝐿𝑣)−(𝑅(0)𝑣𝑠/𝑐)}−^𝑀_𝑠=1𝐶^∗𝑣𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥−𝑥0𝑣)/𝑐)−((2𝑛/𝑐)𝐿𝑣)−(𝑅∗(0)𝑣𝑠/𝑐)}−𝐷_𝑣^∞_𝑛=0𝐸^𝑛𝑣|||||_{𝑡−((𝑥−𝑥0𝑣)/c)−((2𝑛/𝑐)𝐿𝑣)−(|𝑥′−𝑥0𝑣|/𝑐)}−^𝑀_𝑠=1𝐸_𝑣𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥−𝑥0𝑣)/𝑐)−((2𝑛+1/𝑐)𝐿𝑣)−𝑅(𝐿)𝑣𝑠/𝑐}+^𝑀_𝑠=1𝐸^∗𝑣𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥−𝑥0𝑣)/𝑐)−((2𝑛+1/𝑐)𝐿𝑣)−𝑅∗(𝐿)𝑣𝑠/𝑐}+𝐷_𝑣^∞_𝑛=0𝐸^𝑛𝑣|||||_{𝑡−((𝑥−𝑥0𝑣)/c)−((2𝑛+1/𝑐)𝐿𝑣)−(|𝑥𝐿𝑣−𝑥′|/𝑐)}+^𝑀_𝑠=1𝐸_𝑣𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥𝐿𝑣−𝑥)/𝑐)−((2𝑛/𝑐)𝐿𝑣)−(𝑅(𝐿)𝑣𝑠/𝑐)}−^𝑀_𝑠=1𝐸^∗𝑣𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥𝐿𝑣−𝑥)/𝑐)−((2𝑛/𝑐)𝐿𝑣)−(𝑅∗(𝐿)𝑣𝑠/𝑐)}−𝐷_𝑣^∞_𝑛=0𝐸ⁿ_𝑣|||||_{𝑡−((𝑥𝐿𝑣−𝑥)/𝑐)−((2𝑛/𝑐)𝐿𝑣)−(|𝑥𝐿𝑣−𝑥′|/𝑐)}−^𝑀_𝑠=1𝐶_𝑣𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥𝐿𝑣−𝑥)/𝑐)−((2𝑛+1/𝑐)𝐿𝑣)−(𝑅(0)𝑣𝑠/𝑐)}+^𝑀_𝑠=1𝐶^∗𝑜𝑠^∞_𝑛=0𝐼^𝑛𝑠|||||_{𝑡−((𝑥𝐿𝑜−𝑥)/𝑐)−((2𝑛+1/𝑐)𝐿𝑜)−(𝑅∗(0)𝑜𝑠/𝑐)}+𝐷_𝑜^∞_𝑛=0E^𝑛𝑜|||||_{𝑡−((𝑥𝐿𝑜−𝑥)/𝑐)−((2𝑛+1/𝑐)𝐿𝑜)−(|𝑥′−𝑥0𝑜|/𝑐)},(88) where {𝐸} denotes excitation vector and space-dependent matrices are of the form𝐴_𝑣𝑠=_Δ𝑙𝑗_Δ𝑙𝑖14𝜋𝑅_𝑣𝑠𝐴{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥,_∗𝑣𝑠=_Δ𝑙𝑗_Δ𝑙𝑖𝑟_𝑣𝑠(𝜃)4𝜋𝑅^∗𝑣𝑠𝐵{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥,_𝑣𝐶=2𝑍1₀_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥,_𝑣𝑠=_Δ𝑙𝑗_Δ𝑙𝑖14𝜋𝑅^(0)𝑣𝑠𝐶{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥,_∗𝑣𝑠=_Δ𝑙𝑗_Δ𝑙𝑖𝑟_𝑣𝑠(𝜃)4𝜋𝑅^{∗(0)𝑣𝑠}𝐷{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥^𝑑𝑥,_𝑣𝐸=2𝑍1₀_Δ𝑙𝑗_Δ𝑙𝑖{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥,_𝑣𝑠=_Δ𝑙𝑗_Δ𝑙𝑖14𝜋𝑅^{(𝐿)𝑣𝑠}𝐸{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥,_∗𝑣𝑠=_Δ𝑙𝑗_Δ𝑙𝑖𝑟_𝑣𝑠(𝜃)4𝜋𝑅^{∗(𝐿)𝑣𝑠}{𝑓}_𝑗{𝑓}^𝑇𝑖𝑑𝑥′𝑑𝑥.(89)

Relations containing summation from 𝑛=0 to infinity pertain to the reflections of transient current from the wire ends. Note as the observed time interval is always finite, only a finite number of reflections occurs within a given observation interval. A shorter observed interval requires smaller number of summands and vice versa.

According to GB-IBEM, a global matrix system is assembled from the local matrix systems for all wires 𝑣=1,2,…,𝑀. Finally, the resulting global matrix system can be written as follows:[𝐴]{𝐼}||_{𝑡−𝑅𝑣𝑠/𝑐}−𝐴^∗{𝐼}||_𝑡−𝑅^∗𝑣𝑠_/𝑐={𝑔}.(90)

The time-domain solution on the ith boundary element is given by𝐼_𝑖(𝑡)=^𝑁𝑡_𝑘=1𝐼^𝑘𝑖𝑇^𝑘𝑡^,(91) where 𝐼^𝑘𝑖 are unknown coefficients, 𝑇_𝑘 are the linear time-domain shape functions, and 𝑁_𝑡 is the total number of time samples.

Applying the weighted residual approach to (90) leads to the expression^{𝑡𝑘𝑡+Δ𝑡}_𝑘[𝐴]{𝐼}||^𝑡−𝑅_𝑣𝑠^/𝑐−𝐴^∗{𝐼}||_𝑡−𝑅^∗𝑣𝑠_/𝑐−𝑘{=𝑔}1,𝜃2^𝑘,…𝑑𝑡,=𝑁0_𝑡,,(92) where 𝜃_𝑘 denotes the set of time-domain weights.

Using the set of Dirac impulses for the test functions, time sampling is ensured and (92) becomes[𝐴]{𝐼}||_{𝑡𝑘−𝑅𝑣𝑠/𝑐}−𝐴^∗{𝐼}||_{𝑡𝑘−𝑅}^∗𝑣𝑠_/𝑐=𝑔||_{allpreviousdiscreteinstants}.(93)

If the space-time discretization is performed by satisfying the Courant condition, Δ𝑥≥𝑐Δ𝑡, the transient current for a jth space node and kth time node can be obtained from a recurrence formula. Separating the terms relating to the current induced at the instant 𝑡_𝑘 in (93) yields𝐴_𝑗𝑗𝐼_𝑗||_𝑡𝑘+𝐴{𝐼}|||_{𝑡𝑘−𝑅𝑣𝑠/𝑐}−𝐴^∗{𝐼}||_{𝑡𝑘−𝑅}^∗𝑣𝑠_/𝑐={𝑔}||_{allpreviousdiscreteinstants},(94) where overbar indicates the absence of diagonal terms.

The first term in (94) pertains to the current at the jth space node and kth time node, that is, the present instant. Other terms are related to all previous instants. Finally, the recurrence formula for the transient current at jth space node and kth time node is obtained in the forms𝐼_𝑗||_𝑡𝑘=−∑^𝑁𝑖=1𝐴_𝑗𝑖𝐼_𝑖|||_{𝑡𝑘−𝑅𝑣𝑠/𝑐}+𝐴^∗𝑗𝑖𝐼_𝑖||_{𝑡𝑘−𝑅∗𝑣𝑠/𝑐}+𝑔^𝑗||_{allpreviousdiscreteinstants}𝐴_𝑗𝑗,(95)

where 𝑁 is total number of space elements, 𝑘=1,2,…,𝑁_𝑡 is the index of the kth time instant.

3.3. The Transmission Line Model

The time-domain field-to-transmission line coupling equations can be written in the matrix form [11] 𝜕𝜕𝑥[𝑉=(𝑥𝐸,𝑡)]+[𝑅][𝐼(𝑥,𝑡)]+[𝐿]⋅𝜕𝜕𝑡[𝐼(𝑥,𝑡)_𝐹]𝜕(𝑥,𝑡)−[𝑧′(𝑡)]∗[𝐼(𝑥,𝑡)],(96)𝜕𝑥[𝐼=(𝑥𝐻,𝑡)]+[𝐺]⋅[𝑉(𝑥,𝑡)]+[𝐶]⋅𝜕𝜕𝑡[𝑉(𝑥,𝑡)_𝐹](𝑥,𝑡),(97) where “*” stands for the convolution product, [𝑧′(𝑡)] is the transient inverse Fourier transform of the ground, conductors matrix [𝑍_𝑤(𝑠)+𝑍_𝑔(𝑠)], and 𝑠=𝑗𝜔 is the Laplace variable.

[𝐸_𝐹(𝑥,𝑡)]and [𝐻_𝐹(𝑥,𝑡)] are the excitation terms, given by [11]𝐻𝐸_𝐹(𝑥,𝑡)=−𝜕𝜕𝑥𝑉^𝑇(𝑥,𝑡)+𝐸_𝐿(𝑥,𝑡)_𝐹,(𝑥,𝑡)=−[𝐺]𝑉_𝑇(𝑥,𝑡)−[𝐶]𝜕𝜕𝑡𝑉^𝑇(𝑥,𝑡),(98) where [𝑉_𝑇(𝑥,𝑡)] is the transverse voltage derived from the transverse incident field excitation [11], and [𝐸_𝐿(𝑥,𝑡)] represents the longitudinal electrical field excitation.

The classical per-unit-length inductances of [𝐿] matrix have been used:𝐿_𝑖𝑖=𝜇⁰2𝜋ln2𝑎ℎ^𝑖_𝑖,𝐿^𝑖𝑗=𝜇⁰2𝜋ln𝐷𝑑^𝑖𝑗_𝑖𝑗,(99) where ℎ_𝑖,𝑎_𝑖 are, respectively, the height and the radius of the ith conductor. 𝐷_𝑖𝑗 is the distance between the ith conductor and the image of jth conductor and 𝑑_𝑖𝑗 is the distance between the ith conductor and the jth conductor.

The capacitance matrix ⌈𝐶⌉ is derived from the inductance matrix ⌈𝐿⌉ by [𝐶]=𝜀₀𝜇₀[𝐿]⁻¹.

In case of PEC ground, the transient ground matrix [𝑧^(𝑡)] is equal to [𝑍_𝑊(𝑡)] while if a dielectric half-space is of interest, the ground matrix [𝑍_𝑔(𝑠)] is given by Carson integral [11].

It is worth emphasizing that, for both cases, the conductivity matrix is neglected.

3.4. The Solution of Transmission Lineequations via FDTD

Discretizing each conductor of the multiconductor transmission line (MTL) into 𝑁_𝑥 sections each of length Δ𝑥 and discretizing the entire time interval into increments of duration Δ𝑡, the FDTD method is applied to (88) and (97).

The solutions of (91) and (99) for lossy MTL by FDTD are given by 𝑍⁺𝐼^𝑘𝑛+1/2=+−[𝑉𝑍⁻]𝐼^{𝑘𝑛−1/2}_{𝑛𝑇,𝑘−1/2}+𝑉^{𝑛𝑘−1/2}−𝑉_{𝑛𝑇,𝑘+1/2}−𝑉^𝑛𝑘+1/2for0≤+𝑘≤𝑁𝐸^{𝑛𝐿,𝑘}_𝑥−𝑆^𝑛𝑘Δ𝑥−1,𝑛≥1,(100) where 𝑍^±=[𝑅2]±[𝐿]+𝑍⁰(1)Δ𝑡𝑌Δ𝑡,(101)₊=−[𝑌𝑉^{𝑛+1𝑘+1/2}₋+𝑉^{𝑛+1𝑇,𝑘+1/2}−𝐼]V^𝑛𝑘+1/2+𝑉^{𝑛𝑇,𝑘+1/2}_{𝑛+1/2𝑘+1}for0≤𝑘+≤𝐼𝑁^𝑘𝑛+1/2_𝑥𝑌−1,𝑛≥1,(102)_±=[𝐺2]±[Δ𝐶𝑡].(103)

The expressions for [𝑉],[𝐼],[𝑉_𝑇],[𝐸_𝐿],[𝑆] are given for time 𝑛≥1 and space 𝑘≥0:𝑉^𝑛𝑘+1/2=[𝑉((𝑘+1/2)Δ𝑥,𝑛Δ𝑡)],0≤𝑘≤𝑁_𝑥−1.(104)

Voltage and adjacent current are interlaced in time and space, respectively, by Δ𝑡/2 and Δ𝑥/2, then𝑉𝐼^{𝑘𝑛−1/2}=𝐼𝑘Δ𝑥,𝑛−21Δ𝑡_{𝑛𝑇,𝑘+1/2},𝐸=𝑉^𝑇𝑘+12Δ𝑥,𝑛Δ𝑡_{𝑛𝐿,𝑘},𝑆=𝐸^𝐿(𝑘Δ𝑥,𝑛Δ𝑡)_𝑛𝑘,=[𝑆(𝑘Δ𝑥,𝑛Δ𝑡)],0≤𝑘≤𝑁_𝑥.(105)

The convolution product [𝑆^𝑛𝑘] appearing in (100) can be written as follows:𝑆^𝑛𝑘=−𝑍𝑍₀(₀2)𝐼^{𝑘𝑛−1/2}+^𝑛−1_𝑙=1𝑍₀(𝑙+1)−𝑍₀(𝑙)𝐼^{𝑘𝑛−𝑙+1/2}(𝑛)𝐼^𝑘1/2,(106) where 𝑍₀(𝑙)==𝑍_0𝑤(𝑙)+𝑍_0𝑔(𝑙)_𝑙𝑙−1𝑍_𝑤+𝑍_𝑔(𝑢Δ𝑡)𝑑𝑢,1≤𝑙≤𝑛,(107)𝑡=𝑛Δ𝑡, 𝑁_𝑥 is the number of space steps, and 𝑛 is the number of time steps.

The corresponding components of the 𝑀×1 vectors [𝐸^{𝑛𝑇,𝑘}], [𝐸^{𝑛𝐿,𝑘}] are as follows:𝑉^{𝑛𝑇,𝑘}_𝑖=+𝑦_𝑖𝑧𝐸^inc_𝑦𝑖𝑥_𝑖=𝑘Δ𝑥,𝑦_𝑖,𝑧_𝑖,𝑛Δ𝑡𝐸𝐸^inc_𝑧x_𝑖=𝑘Δ𝑥,𝑦_𝑖,𝑧_𝑖,𝑛Δ𝑡,(108)_{𝑛𝐿,𝑘}_𝑖=−𝐸𝐸^inc_𝑥inc𝑧𝑥_𝑖=𝑘Δ𝑥,𝑦_𝑖,𝑧_𝑖,𝑛Δ𝑡𝑥_𝑖=𝑘Δ𝑥,0,0,𝑛Δ𝑡.(109)

Equations (107) and (108) are valid for 1≤𝑖≤𝑀, where 𝑀 is the number of conductors, and 𝑦_𝑖,𝑧_𝑖 are the positions of ith conductor. 𝐸^inc_𝑥,𝐸^inc_𝑦,𝐸^inc_𝑧 are the components of the incident electromagnetic field evaluated in the absence of conductors. [𝑅],[𝐿],[𝐺],and[𝐶] are, respectively, the per-unit-length resistance, inductance, conductance, and capacitance matrices of dimension 𝑀×𝑀.

[𝑍_𝑔(𝑡)] is the transient ground resistance and is equal to the inverse Fourier of [𝑍_𝑔(𝑠)/𝑠]:𝑍_𝑔𝑖𝑗(𝑡)=𝐹⁻¹𝑍^𝑔𝑖𝑗(𝑠)𝑠,(110) where the ground impedance in frequency domain is given by Carson formula:𝑍_𝑔𝑖𝑗(𝑠)𝑠=𝜇𝜋⁰^∞0𝑒^{−(ℎ𝑖+ℎ𝑗)𝜆}𝜆²+𝛾^2𝑔+𝜆⋅cos𝑑^𝑖𝑗𝜆𝑑𝜆.(111)[𝑍_𝑤(𝑡)] is a diagonal matrix and corresponds to the transient conductor resistance, where each element is given by𝑍_𝑤𝑖(𝑡)=1𝜋𝜎_𝑤𝑖𝑎^2𝑖∞_𝑚=1𝑒^{−𝑥2𝑚(𝑡/𝜏𝑤𝑖)},where𝜏_𝑤𝑖=𝜇₀𝜎_𝑤𝑖𝑎^2𝑖,(112) and ℎ_𝑖,ℎ_𝑗,𝑑_𝑖𝑗 are the corresponding heights of the two conductors 𝑖,𝑗 and the distance between the two conductors in the horizontal plane.

Terms 𝑥_𝑚 stand for the zeros of 𝐽₁, Bessel function of first kind.

Finally, 𝛾_𝑔 is the propagation constant defined as𝛾^2𝑔=𝑠𝜇₀𝜎_𝑔+𝑠𝜀₀𝜀_𝑟𝑔,(113) where 𝜎_𝑔 and 𝜀_𝑟𝑔 are, respectively, ground conductivity and permittivity.

For the cases considered in this paper, all conductors are in open circuit at both ends, so the currents vanishes at near and far ends.

By using the boundary formulation [11], voltages at both ends are simply expressed as follows:𝑉^𝑛0=𝐼𝑉^𝑛(1/2),𝑉^𝑛𝑁_𝑥=𝑉^𝑛𝑁_𝑥^−(1/2),(114)_𝑛0=[0],𝐼^𝑛𝑁_𝑥=[0].(115)

4. Numerical Results

Figure 11 shows the transient response at the centre of the straight wire 𝐿=20 m, 𝑎=0.005m, located at height ℎ=1m above a dielectric half-space (𝜀_𝑟=10) excited by the electromagnetic pulse (EMP):𝐸^inc_𝑥=𝐸₀𝑒^−𝑎𝑡−𝑒^−𝑏𝑡,(116) with 𝐸₀=1.1V/m, 𝑎=7.92∗10⁴s⁻¹, 𝑏=4∗10⁴s⁻¹.

Figure 11 

The transient current induced at the center of the line above dielectric half-space (𝜀𝑟=10).

The next example is related to a transient scattering from a straight thin wire of length is 𝐿=1m, radius 𝑎=2mm, located at heightℎ=0.25m above ground with permittivity 𝜀_𝑟=10, while the conductivity is varied. The wire is illuminated by the tangential electromagnetic pulse (EMP) plane wave with 𝐸₀=1V/m, 𝑎=4⋅10⁷s⁻¹, 𝑏=6⋅10⁸s⁻¹.

Figure 12 shows the transient current induced at the wire center for different ground conductivities.

Figure 12 

Transient current at the wire center, 𝐿=1 m, 𝑎=2mm, ℎ=0.25, 𝜀𝑟=10.

The influence of the ground conductivity to the transient response is particularly visible from around 0.1 S/m to 1 S/m.

The time domain results obtained via different approaches are found to agree satisfactorily.

Next set of examples is related to a two-wire array above a PEC ground Figure 13, (Geometry No. 1) and dielectric-half space (𝜀_𝑟=10), Figure 14 (Geometry No. 2), respectively.

Figure 13 

Geometry No. 1: Two-wire array above a PEC ground (𝑎=2cm, 𝐿=10m, 𝑑=1m, ℎ1=1m, and ℎ2=2m).

Figure 14 

Geometry No. 2: Two-wire array above a dielectric half-space (𝜀𝑟=10, 𝑎=2cm, 𝐿=10m, 𝑑=1m, ℎ1=1m, and ℎ2=2m).

Figures 15 and 16 show the transient current induced at the center of wire 2 for the case of Geometry No. 1 and No. 2, respectively, obtained via TD GB-IBEM, TL, and NEC 2 combined with inverse fast Fourier transform (IFFT).

Figure 15 

Transient current induced at the center of wire 2 (Geometry No. 1)—comparison between IFFT-NEC2, TD GB-IBEM and TL results.

Figure 16 

Transient current induced at the center of wire 2 (Geometry No. 2)—comparison between IFFT-NEC2, TD GB-IBEM and TL results.

Generally, the results calculated via different approaches are in relatively acceptable agreement. Nevertheless, some discrepancies can be noticed, in particular for the case of PEC ground. In this analysis, the applied TL model accounts not only for classical propagation effect but also for skin effects and for a correction resistance representing the radiation effect.

In order to include the radiation effect in TL model, a small DC resistance (1 Ω/m) has been used to to represent the attenuation effect.

It is known that TL model accounts for coupling between transverse cells only while AT takes into account mutual effects. This phenomenon is assumed to be the source of the differences in propagation velocities which are observed.

5. Conclusion

The paper reviews the models and methods used for the analysis of electromagnetic field coupling to overhead wires in the frequency and time domain, respectively, using the wire antenna theory and the transmission line approximation, respectively. The frequency domain wire antenna model is based on the space-time Pocklington integral equations, while the transmission line model is based on the frequency domain Telegrapher's equations. The time domain wire antenna model is based on a set of the space-time Hallen integral equations, while the transmission line model is based on the time domain Telegrapher's equations. The set of Pocklington equations is solved via the Galerkin-Bubnov variant of the Indirect Boundary Element Method (GB-IBEM), while the frequency domain transmission line equations are treated using the chain matrix method and modal equation to derive per-unit-length parameters. A number of illustrative computational examples for the frequency response of several configurations of overhead wires, obtained via different approaches, are given in this paper.

The coupled space-time Hallen integral equations are numerically solved via the time domain Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM), while the time-domain transmission line equations are solved by the finite difference time domain (FDTD) method.

Some numerical results pertaining to the transient behaviour of overhead wires, obtained via different approaches, are given in this paper.