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International Journal of Antennas and Propagation
Volume 2012 (2012), Article ID 730145, 18 pages
Review Article

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

1Department of Electronics, University of Split, Split, Croatia
2Pascal Institute, Blaise Pascal University, Clermont-Ferrand, France

Received 14 August 2011; Accepted 3 December 2011

Academic Editor: Sergey V. Tkachenko

Copyright © 2012 Dragan Poljak and Khalil El Khamlichi Drissi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 [111], 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 [16]. 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 𝐸exc and scattered field 𝐸sct vanishes [1, 8]: 𝑒𝑥𝐸exc+𝐸sct=0onthewiresurface.(1) Starting from Maxwell’s equations and Lorentz gauge the scattered electric field can be expressed in terms of the vector potential 𝐴: 𝐸sct1=𝑗𝜔𝐴+𝐴𝑗𝜔𝜇𝜀.(2) The vector potential is defined by the particular integral over a given path C (considered conductive wire structure): 𝜇𝐴(𝑠)=4𝜋𝐶𝐼𝑠𝑔0𝑠,𝑠,𝑠𝑠𝑑𝑠,(3) where 𝐼(𝑠) is the induced current along the line and 𝑔0(𝑠,𝑠) denotes the lossless medium Green function:𝑔0𝑠,𝑠=𝑒𝑗𝑘𝑅𝑅,(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: 𝐸sct=1𝑗4𝜋𝜔𝜀0𝐶𝐼𝑠𝑠𝑘2𝑔+0𝑠,𝑠𝑑𝑠.(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: 𝐸exctan(1𝑠)=𝑗4𝜋𝜔𝜀0𝐶𝐼𝑠𝑠𝑠𝑘2𝑔+0𝑠,𝑠𝑑𝑠,(7) where 𝐸exctan 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 𝐸exc is now composed from incident and reflected field, respectively,𝐸exc=𝐸inc+𝐸ref.(8) Performing certain mathematical manipulations the Pocklington integro-differential equation for a curved wire above a lossy ground becomes [12]𝐸exc𝑠𝑗(𝑠)=4𝜋𝜔𝜀0𝐿0𝑘2𝑒𝑠𝑒𝑠𝜕2𝜕𝑠𝜕𝑠𝑔0𝑠,𝑠+𝑅TM𝑘2𝑒𝑠𝑒𝑠𝜕2𝜕𝑠𝜕𝑠𝑔𝑖𝑠,𝑠+𝑅TE𝑅TM𝑒𝑠𝑒𝑝𝑘2𝑒𝑝𝑒𝑠𝜕2𝜕𝑝𝜕𝑠𝑔𝑖𝑠,𝑠𝐼𝑠𝑑𝑠,(9) where 𝑒𝑝 is the unit vector normal to the incident plane, while 𝑔𝑖(𝑠,𝑠) arises from the image theory and is given by 𝑔𝑖𝑠,𝑠=𝑒𝑗𝑘𝑅𝑅,(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]𝐸exc𝑠𝑚=𝑗(𝑠)4𝜋𝜔𝜀0𝑁𝑤𝑛=1𝐿𝑛0𝑘𝑒𝑠𝑚𝑒𝑠𝑛𝜕2𝜕𝑠𝑚𝜕𝑠𝑛𝑔0𝑛𝑠𝑚,𝑠𝑛+𝑅TM𝑘2𝑒𝑠𝑚𝑒𝑠𝑛𝜕2𝜕𝑠𝑚𝜕𝑠𝑛𝑔𝑖𝑛𝑠𝑚,𝑠𝑛+𝑅TE𝑅TM𝑒𝑠𝑚𝑒𝑝𝑘2𝑒𝑝𝑒𝑠𝜕2𝜕𝑝𝜕𝑠𝑔𝑖𝑠𝑚,𝑠𝑛𝐼𝑠𝑛𝑑𝑠,(11) where 𝑁𝑤 is the total number of wires and 𝐼𝑛(𝑠𝑛) is the unknown current distribution induced on the nth wire. Furthermore, 𝑔0𝑚𝑛(𝑥,𝑥) and 𝑔𝑖𝑚𝑛(𝑠,𝑠)are the Green functions of the form 𝑔0𝑚𝑛𝑠𝑚,𝑠𝑛=𝑒𝑗𝑘𝑅1𝑚𝑛𝑅1𝑚𝑛,𝑔𝑖𝑚𝑛𝑠𝑚,𝑠𝑛=𝑒𝑗𝑘𝑅2𝑚𝑛𝑅2𝑚𝑛,(12) where 𝑅1𝑚𝑛 and 𝑅2𝑚𝑛 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], 𝑅TM=𝑛cos𝜃𝑛sin2𝜃𝑛cos𝜃+𝑛sin2𝑅𝜃,(13)TE=cos𝜃𝑛sin2𝜃cos𝜃+𝑛sin2𝜃,(14) where θ is the angle of incidence and 𝑛 is given by, 𝑛=𝜀e𝜀0,𝜀e=𝜀𝑟𝜀0𝜎𝑗𝜔,(15) and 𝜀e 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 𝐸exc𝑥𝜇=𝑗𝜔4𝜋𝐿0𝐼𝑥𝑔𝑥,𝑥𝑑𝑥1𝜕𝑗4𝜋𝜔𝜀𝜕𝑥𝐿0𝑥𝜕𝐼𝜕𝑥𝑔𝑥,𝑥𝑑𝑥,(16) where 𝐼(𝑥) is the induced current along the horizontal wire and 𝑔(𝑥,𝑥) denotes the Green’s function given by 𝑔𝑥,𝑥=𝑔0𝑥,𝑥𝑅TM𝑔𝑖𝑥,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 𝐸exc𝑥𝑚1=𝑗4𝜋𝜔𝜀0𝑁𝑤𝑛=1𝐿𝑛0𝜕2𝜕𝑥2+𝑘21𝑔𝑚𝑛𝑥,𝑥𝐼𝑛𝑥𝑑𝑥,𝑚=1,2,,𝑀,(18) where 𝐼𝑛(𝑥) is the unknown current distribution induced along nth wire, 𝐸exc𝑥𝑚 is the known excitation field tangential to the jth wire surface, and 𝑔𝑚𝑛 is the corresponding Green function:𝑔𝑚𝑛𝑥,𝑥=𝑔0𝑚𝑛𝑥,𝑥𝑅TM𝑔𝑖𝑚𝑛𝑥,𝑥𝑚=1,2,,𝑀.(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] 𝐸=𝑁𝑤𝑛=1𝐸0𝑛+𝑅TM𝐸𝑖𝑛+𝑅TE𝑅TM𝐸𝑖𝑛𝑒𝑝𝑒𝑝,(20) where 𝐸0𝑛=1𝑗4𝜋𝜔𝜀0𝑘21𝐿𝑛0𝑒𝑠𝑛𝐼𝑠𝑛𝑔0𝑛𝑟𝑟,𝑑𝑠𝑛+𝐿0𝑠𝜕𝐼𝑛𝜕𝑠𝑛𝑔0𝑛𝑟𝑟,𝑑𝑠𝑛,𝐸𝑖𝑛=1𝑗4𝜋𝜔𝜀0𝑘21𝐿𝑛0𝑒𝑠𝑛𝐼𝑠𝑛𝑔𝑖𝑛𝑟,𝑟𝑑𝑤𝐿𝑛0𝑠𝜕𝐼𝜕𝑠𝑛𝑔𝑖𝑛𝑟,𝑟𝑑𝑠.(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]𝐸𝑥=1𝑗4𝜋𝜔𝜀0𝐿0𝑥𝜕𝐼𝜕𝑥𝜕𝑔𝑥,𝑥𝜕𝑥𝑑𝑥+𝑘2𝐿0𝑔𝑥,𝑥𝐼𝑥𝑑𝑥,𝐸(22)𝑦=1𝑗4𝜋𝜔𝜀0𝐿0𝑥𝜕𝐼𝜕𝑥𝑥𝜕𝑔,𝑦𝜕𝑦𝑑𝑥𝐸,(23)𝑧=1𝑗4𝜋𝜔𝜀0𝐿0𝑥𝜕𝐼𝜕𝑥𝑥𝜕𝑔,𝑧𝜕𝑧𝑑𝑥.(24) For the case of multiple horizontal wires the expressions for electric field are given by [15]𝐸𝑥=1𝑗4𝜋𝜔𝜀0𝑁𝑤𝑛=1𝐿𝑛0𝜕𝐼𝑛𝑥𝜕𝑥𝜕𝑔𝑛𝑚𝑥,𝑥𝜕𝑥𝑑𝑥+𝑘2𝐿𝑛0𝑔𝑛𝑚𝑥,𝑥𝐼𝑛𝑥𝑑𝑥,(25)𝐸𝑦=1𝑗4𝜋𝜔𝜀0𝑁𝑤𝑛=1𝐿𝑛0𝜕𝐼𝑛𝑥𝜕𝑥𝜕𝑔𝑛𝑚𝑥,𝑥𝜕𝑦𝑑𝑥,(26)𝐸𝑧=1𝑗4𝜋𝜔𝜀0𝑁𝑤𝑛=1𝐿𝑛0𝜕𝐼𝑛𝑥𝜕𝑥𝜕𝑔𝑛𝑚𝑥,𝑥𝜕𝑧𝑑𝑥.(27) The radiated magnetic field of the curved wire system can be written as follows [13, 14]:𝐻=𝑁𝑤𝑛=1𝐻𝑆𝑛+𝑅TE𝐻𝐼𝑛+𝑅TM𝑅TE𝐻𝐼𝑛𝑒𝑝𝑒𝑝,(28) where𝐻𝑆𝑛1=4𝜋𝐿𝑛0𝐼𝑠𝑛𝑒𝑠×𝑔0𝑛𝑟𝑟,𝑑𝑠,𝐻𝐼𝑛1=4𝜋𝐿𝑛0𝐼𝑠𝑛𝑒𝑠×𝑔𝑖𝑛𝑟,𝑟𝑑𝑠.(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] 𝑀𝑁𝑛=1𝑛𝑖=1[𝑍]𝑒𝑗𝑖{𝐼}𝑒𝑖={𝑉}𝑒𝑗,(30) where the mutual impedance matrix is given by [13]: [𝑍]𝑒𝑖𝑗=11{𝐷}𝑗𝐷𝑇𝑖𝑔0𝑛𝑚𝑠𝑛,𝑠𝑚𝑑s𝑚𝑑𝜉𝑑𝜉𝑑𝑠𝑛𝑑𝜉𝑑𝜉+𝑘21𝑒𝑠𝑛𝑒𝑠𝑚11{𝑓}𝑗𝑓𝑇𝑖𝑔0𝑛𝑚𝑠𝑛,𝑠𝑚𝑑𝑠𝑚𝑑𝜉𝑑𝜉𝑑𝑠𝑛𝑑𝜉𝑑𝜉𝑅TM11{𝐷}𝑗𝐷𝑇𝑖𝑔𝑖𝑛𝑚𝑠𝑛,𝑠𝑚𝑑𝑠𝑚𝑑𝜉𝑑𝜉𝑑𝑠𝑛𝑑𝜉𝑑𝜉+𝑅TM𝑘21𝑒𝑠𝑛𝑒s𝑚×11{𝑓}𝑗𝑓𝑇𝑖𝑔𝑖𝑛𝑚𝑠𝑛,𝑠𝑚𝑑𝑠𝑚𝑑𝜉𝑑𝜉𝑑𝑠𝑛+𝑗𝑑𝜉𝑑𝜉4𝜋𝜔𝜀011𝑍𝑇{𝑓}𝑗{𝑓}𝑇𝑗𝑑𝑠𝑛𝑑𝜉𝑑𝜉,(31) while the voltage vector is given by [13] {𝑉}𝑛𝑗=𝑗4𝜋𝜔𝜀011𝐸exc𝑠𝑛𝑠𝑛𝑓𝑗𝑛𝑠𝑛𝑑𝑠𝑛𝑑𝜉𝑑𝜉𝑛.(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 𝐸=𝑁𝑘=1𝐸𝑒𝑆𝑘+𝑅TM𝐸𝑒𝐼𝑘+𝑅TE𝑅TM𝐸𝑒𝐼𝑘𝑒𝑝𝑒𝑝,(33) where the field components due to a wire segment radiation are given by𝐸𝑒𝑆𝑘=1𝑗4𝜋𝜔𝜀0𝑛𝑖=1𝑘211𝑒𝑘𝑠𝐼𝑒𝑖𝑘𝑓𝑖(𝜉)𝑔0𝑘𝑟𝑟,𝑑𝑠𝑘+𝑑𝜉𝑑𝜉11𝐼𝑒𝑖𝑘𝜕𝑓𝑖(𝜉)𝜕𝜉𝑔0𝑘𝑟𝑟,𝑑𝑠𝑘,𝐸𝑑𝜉𝑑𝜉𝑒𝐼=1𝑗4𝜋𝜔𝜀0𝑛𝑖=1𝑘211𝑒𝑘𝑠𝐼𝑒𝑖𝑘𝑓𝑖(𝜉)𝑔𝑖𝑘𝑟,𝑟𝑑𝑠𝑘𝑑𝜉𝑑𝜉11𝐼𝑒𝑖𝑘𝜕𝑓𝑖(𝜉)𝜕𝜉𝑔𝑖𝑘𝑟,𝑟𝑑𝑠𝑘.𝑑𝜉𝑑𝜉(34) The total magnetic field is given by [13]𝐻=𝑁𝑘=1𝐻𝑒𝑆𝑘+𝑅TE𝐻𝑒𝐼𝑘+𝑅TM𝑅TE𝐻𝑒𝐼𝑘𝑒𝑝𝑒𝑝,(35) while the magnetic field components are given by [13]𝐻𝑒𝑆𝑘1=4𝜋𝑛𝑖=111𝐼𝑖𝑘𝑓𝑖(𝜉)𝑒𝑠𝑘×𝑔0𝑘𝑟𝑟,𝑑𝑠𝑘𝐻𝑑𝜉𝑑𝜉,𝑒𝐼𝑘1=4𝜋𝑛𝑖=111𝐼𝑒𝑖𝑘𝑓𝑖(𝜉)𝑒𝑘𝑠×𝑔𝑖𝑘𝑟,𝑟𝑑𝑠𝑘𝑑𝜉𝑑𝜉.(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]: 𝑑𝑉+𝑍𝐼𝑑𝑥(𝑥)(𝑥)=𝑗𝜔𝜇00𝐻exc𝑦𝑑(𝑥,𝑧)𝑑𝑧,+𝑌𝑑𝑥𝐼(𝑥)𝑉(𝑥)=𝑗𝜔𝜇00𝐸exc𝑧(𝑥,𝑧)𝑑𝑧,(37) where the longitudinal impedance matrix is given by𝑍[𝐿]+𝑍=𝑗𝜔𝑤+𝑍𝑔,(38) and the transversal admittance matrix can be written as 𝑌[𝐶]+[𝐺]=𝑗𝜔,(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, 𝑍𝑤 is the per-unit length internal impedance matrix of the conductors, and 𝑍𝑔 is the per-unit length ground impedance matrix. Finally, [𝐻exc𝑦(𝑥,𝑧)] and [𝐸𝑧𝑒𝑥𝑐(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 =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, , as shown in Figure 7. The imperfectly conducting ground is characterized with electrical permeability εr and conductivity σ.

The conductors are modeled as thin wire antennas excited by the voltage generator 𝑉𝑔 at one end and terminated by the load impedance 𝑍𝐿 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: 𝐸inc𝑠+𝐸sct𝑠=𝑍𝐿𝐼(𝑠),(40) where 𝑍𝐿 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 𝐸inc1(𝑠)=𝑗4𝜋𝜔𝜀0𝐿0𝑘21𝑒𝑠𝑒𝑠𝜕2𝜕𝑠𝜕𝑠𝑔0𝑠,𝑠+𝑅𝑇𝑀𝑘21𝑒𝑠𝑒𝑠𝜕2𝜕𝑠𝜕𝑠𝑔𝑖𝑠,𝑠𝑠×𝐼𝑑𝑠+𝑍𝐿𝐼(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 1=10m and 2=11m. The maximum sag of the conductor is assumed to be 𝑠=2m. Ground parameters are 𝜀𝑟=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 𝑍𝐿 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 𝑍𝐿 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,𝐸inc𝑥+𝐸sct𝑥=0,(42) where 𝐸inc𝑥 is the incident and 𝐸sct𝑥 scattered field on the metallic wire surface. Starting from Maxwell equations and obeying the Lorentz gauge one obtains a time domain version of (2):𝜕2𝐴𝜕𝑡21𝐴||||𝜇𝜀tan=𝜕𝐸inc||||𝜕𝑡tan,(43) where 𝐴 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 𝜕2𝐴𝑥𝜕𝑥21𝑐2𝜕2𝐴𝑥𝜕𝑡21=𝑐2𝜕𝐸inc𝑥𝜕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:𝐴𝑥(𝑥,𝑡)=𝐴𝑥(𝑥,𝑡)+𝐴𝑝𝑥(𝑥,𝑡).(45) The solution of the homogeneous wave equation is given as a superposition of incident and reflected wave [8]:𝐴𝑥(𝑥,𝑡)=𝐹1𝑥𝑡𝑐+𝐹2𝑥𝑡+𝑐,(46) while the particular solution is given by the integral [8]:𝐴𝑝𝑥1(𝑥,𝑡)=2𝑍0L0𝐸inc𝑥𝑥||,𝑡𝑥𝑥||𝑐𝑑𝑥,(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:𝐴𝑥𝜇(𝑥,𝑡)=4𝜋𝑆𝐼𝑥,𝑡𝑅/𝑐𝑅𝑑𝑥.(48)

Combining (45)–(48) yields the space-time Hallen equation:𝐿0𝐼𝑥,𝑡𝑅/𝑐4𝜋𝑅𝑑𝑥=𝐹0𝑥𝑡𝑐+𝐹𝐿𝑡𝐿𝑥𝑐+12𝑍0𝐿0𝐸inc𝑥𝑥||,𝑡𝑥𝑥||𝑐𝑑𝑥,(49) where 𝐼(𝑥) is the equivalent axial current to be determined, 𝐸inc𝑥 is the known tangential incident field, 𝑅=[(𝑥𝑥)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 𝐹𝐿(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:𝐿0𝐼𝑥𝑒,𝑠𝑠𝑅/𝑐4𝜋𝑅𝑑𝑥=𝐹0(𝑠)𝑒𝑠𝑥/𝑐+𝐹𝐿(𝑠)𝑒𝑠((𝐿𝑥)/𝑐)+12𝑍0𝐿0𝐸inc𝑥𝑥𝑒,𝑠𝑠|𝑥𝑥|/𝑐𝑑𝑥,(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 𝑅TM(𝜃,𝑠) for TM polarization. The integral equation in the frequency domain is given by𝐿0𝐼(𝑥,𝑠)𝑒𝑠𝑅/𝑐4𝜋𝑅𝑑𝑥𝐿0𝑅TM(𝜃,𝑠)𝐼(𝑥,𝑠)𝑒𝑠𝑅/𝑐4𝜋𝑅𝑑𝑥=𝐹0(𝑠)𝑒𝑠𝑥/𝑐+𝐹𝐿(𝑠)𝑒𝑠((𝐿𝑥)/𝑐)+12𝑍0𝐿0𝐸exc𝑥𝑥𝑒,𝑠𝑠(|𝑥𝑥|/𝑐)𝑑𝑥,(51) where 𝑅=(𝑥𝑥)2+42 and 𝑅TM(𝜃,𝑠) is determined by the expression [1]𝑅TM𝜃=𝜀,𝑠𝑟(1+𝜎/𝜀𝑠)cos𝜃𝜀𝑟(1+𝜎/𝜀𝑠)sin2𝜃𝜀𝑟(1+𝜎/𝜀𝑠)cos𝜃+𝜀𝑟(1+𝜎/𝜀𝑠)sin2𝜃,(52) where 𝜎 and 𝜀 are the lossy medium conductivity and permittivity, respectively, and𝜃=arct𝑔(|𝑥𝑥|/2).

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 form𝐿0𝐼𝑥,𝑡𝑅/𝑐4𝜋𝑅𝑑𝑥𝑡𝐿0𝐼𝑥𝑟(𝜃,𝜏),𝑡𝑅/𝑐𝜏4𝜋𝑅=1𝑑𝑥𝑑𝜏2𝑍0𝐿0𝐸exc𝑥𝑥||,𝑡𝑥𝑥||𝑐𝑑𝑥+𝐹0𝑥𝑡𝑐+𝐹𝐿𝑡𝐿𝑥𝑐,(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)=4𝛽1𝛽2𝑒𝛼𝑡𝑡𝑛=1(1)𝑛+1𝑛𝐾𝑛𝐼𝑛𝜎(𝛼𝑡),𝜏=𝜀0𝜀𝑟,𝛽=𝜀𝑟sin2𝜃𝜀𝑟𝜏cos𝜃,𝛾=1sin2𝜃/𝜀𝑟,||||𝜃=arct𝑔𝑥𝑥2,𝐾=1𝛽𝜏1+𝛽,𝛼=2.(55)

Note that 𝐼𝑛 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 𝐸exc𝑥(𝑡)=𝐸inc𝑥(𝑡)𝐸ref𝑥𝑡,(56) where 𝑡=𝑡𝑅/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]𝐸ref𝑥(𝑡)=𝑡𝐸inc𝑥(𝑡𝜏)𝑟(𝜃=0,𝜏)𝑑𝜏(57) and the integral equation (53) becomes 𝐿0𝐼𝑥,𝑡𝑅/𝑐4𝜋𝑅𝑑𝑥𝑡𝐿0𝐼𝑥𝑟(𝜃,𝜏),𝑡𝑅/𝑐𝜏4𝜋𝑅𝑑𝑥𝑑𝜏=𝐹0𝑥𝑡𝑐+𝐹𝐿𝑡𝐿𝑥𝑐+12𝑍0𝐿0𝐸inc𝑥||||𝑥,𝑡𝑥𝑥𝑐1𝑑𝑥2𝑍0𝑡𝐿0𝐸inc𝑥𝑥||,𝑡𝑥𝑥||𝑐𝜏×𝑟(𝜃=0,𝜏)𝑑𝑥𝑑𝜏.(58)

The unknown time functions 𝐹0(𝑡),𝐹𝐿(𝑡),𝐹0(𝑡(𝐿/𝑐)), and 𝐹𝐿(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 𝐾𝐿(𝑡) [8]:𝐹0(𝑡)=𝑛=0𝐾0𝑡2𝑛𝐿𝑐𝑛=0𝐾𝐿𝑡(2𝑛+1)𝐿𝑐,𝐹𝐿(𝑡)=𝑛=0𝐾𝐿𝑡2𝑛𝐿𝑐𝑛=0𝐾0𝑡(2𝑛+1)𝐿𝑐,(59) where𝐾𝐿(𝑡)=𝐿0𝐼𝑥,𝑡𝑅0/𝑐4𝜋𝑅0𝑑𝑥𝑡𝐿0𝑟𝜃𝐼𝑥,𝜏,𝑡𝑅0/𝑐𝜏4𝜋𝑅01𝑑𝑥𝑑𝜏2𝑍0𝐿0𝐸inc𝑥𝑥,𝑡𝑥𝑐𝑑𝑥,𝐾𝐿(𝑡)=𝐿0𝐼𝑥,𝑡𝑅𝐿/𝑐4𝜋𝑅𝐿𝑑𝑥𝑡𝐿0𝑟𝜃𝐼𝑥,𝜏,𝑡𝑅𝐿/𝑐𝜏4𝜋𝑅𝐿1𝑑𝑥𝑑𝜏2𝑍0𝐿0𝐸exc𝑥𝑥,𝑡𝐿𝑥𝑐𝑑𝑥,(60) while 𝑅0 and 𝑅𝐿 are the distances from the wire ends to the source point and 𝑅0, 𝑅𝐿 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 into𝐿0𝐼𝑥,𝑡𝑅/𝑐4𝜋𝑅𝑑𝑥𝐿0𝐼𝑥𝑟(𝜃),𝑡𝑅/𝑐4𝜋𝑅𝑑𝑥=𝐹0𝑥𝑡𝑐+𝐹𝐿𝑡𝐿𝑥𝑐+12𝑍0𝐿0𝐸exc𝑥||||𝑥,𝑡𝑥𝑥𝑐𝑑𝑥.(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]:𝑀𝑠=1𝑥𝐿𝑠𝑥0𝑠𝐼𝑠𝑥,𝑡𝑅𝑣𝑠/𝑐4𝜋𝑅𝑣𝑠𝑑𝑥𝑀𝑠=1𝑡𝑥𝐿𝑠𝑥0𝑠𝑟𝑣𝑠𝐼(𝜃,𝜏)𝑠𝑥,𝑡𝑅𝑣𝑠/𝑐𝜏4𝜋𝑅𝑣𝑠𝑑𝑥𝑑𝜏=𝐹0𝑣𝑡𝑥𝑥0𝑣𝑐+𝐹𝐿𝑣𝑥𝑡𝐿𝑣𝑥𝑐+12𝑍0𝑥𝐿𝑣𝑥0𝑣𝐸exc𝑥𝑣𝑥||,𝑡𝑥𝑥||𝑐𝑑𝑥,(62) where 𝑣,𝑠=1,2,,𝑀 denote the index of the observed and source wire, respectively. Furthermore, 𝐿𝑠 and 𝐿𝑣 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 by𝑅𝑣𝑠=(𝑥𝑥)2+(𝑦𝑦)2+(𝑧𝑧)2𝑣𝑠(𝑥𝑥)2+𝑎2𝑅𝑣=𝑠,𝑣𝑠=(𝑥𝑥)2+(𝑦𝑦)2+(𝑧+𝑧)2,(63) where asterisk is related to source points are located on the image wire.

Unknown time signals 𝐹0𝑣(𝑡) and 𝐹𝐿𝑣(𝑡) account for the multiple reflections of transient currents at the wire open ends and can be written in the form𝐹0𝑣(𝑡)=𝑛=0𝐾0𝑣𝑡2𝑛𝐿𝑣𝑐𝑛=0𝐾𝐿𝑣𝑡(2𝑛+1)𝐿𝑣𝑐,(64)𝐹𝐿𝑣(𝑡)=𝑛=0𝐾𝐿𝑣𝑡2𝑛𝐿𝑣𝑐𝑛=0𝐾0𝑣𝑡(2𝑛+1)𝐿𝑣𝑐,(65) while the auxiliary functions 𝐾 are defined, as follows:𝐾0𝑣=(𝑡)𝑀𝑠=1𝑥𝐿𝑠𝑥0𝑠𝐼𝑠𝑥,𝑡𝑅(0)𝑣𝑠/𝑐4𝜋𝑅(0)𝑣𝑠𝑑𝑥𝑀𝑠=1𝑡𝑥𝐿𝑠𝑥0𝑠𝑟𝑣𝑠𝐼(𝜃,𝜏)𝑠𝑥,𝑡𝑅(0)𝑣𝑠/𝑐𝜏4𝜋𝑅(0)𝑣𝑠1𝑑𝑥𝑑𝜏2𝑍0𝑥𝐿𝑣𝑥0𝑣𝐸exc𝑥𝑣𝑥||,𝑡𝑥𝑥||𝑐𝐾𝑑𝑥,(66)𝐿𝑣=(𝑡)𝑀𝑠=1𝑥𝐿𝑠𝑥0𝑠𝐼𝑠𝑥,𝑡𝑅(𝐿)𝑣𝑠/𝑐4𝜋𝑅(𝐿)𝑣𝑠𝑑𝑥𝑀𝑠=1𝑡𝑥𝐿𝑠𝑥0𝑠𝑟𝑣𝑠𝐼(𝜃,𝜏)𝑠𝑥,𝑡𝑅(𝐿)𝑣𝑠/𝑐𝜏4𝜋𝑅(𝐿)𝑣𝑠1𝑑𝑥𝑑𝜏2𝑍0𝑥𝐿𝑣𝑥0𝑣𝐸exc𝑥𝑣𝑥||,𝑡𝑥𝑥||𝑐𝑑𝑥,(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=𝑅𝑣𝑠||𝑥=𝑥0𝑣,𝑅(𝐿)𝑣𝑠=𝑅𝑣𝑠||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=𝑅𝑣𝑠||𝑥=𝑥0𝑣,𝑅(𝐿)𝑣𝑠=𝑅𝑣𝑠||𝑥=𝑥𝐿𝑣.(69)

The space-time reflection coefficient 𝑟𝑣𝑠(𝜃,𝑡) accounts for the influence of the interface and is given by [11] 𝑟𝑣𝑠𝜃𝑣𝑠,𝑡=𝐴𝛿(𝑡),(70) where𝐴=1𝛽1+𝛽,𝛽=𝜀𝑟sin2𝜃𝜀𝑟,𝜃cos𝜃𝑣𝑠=Arct𝑔(𝑥𝑥)2+(𝑦𝑦)2.𝑧+𝑧(71)

The angle 𝜃𝑣𝑠is 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) yields𝑀𝑠=1𝑥𝐿𝑠𝑥0𝑠𝐼𝑠𝑥,𝑡𝑅𝑣𝑠/𝑐4𝜋𝑅𝑣𝑠𝑑𝑥𝑀𝑠=1𝑥𝐿𝑠𝑥0𝑠𝑟𝑣𝑠𝐼(𝜃)𝑠𝑥,𝑡𝑅𝑣𝑠/𝑐4𝜋𝑅𝑣𝑠𝑑𝑥=𝐹0𝑣𝑡𝑥𝑥0𝑣𝑐+𝐹𝐿𝑣𝑥𝑡𝐿𝑣𝑥𝑐+12𝑍0𝑥𝐿𝑣𝑥0𝑣𝐸exc𝑥𝑣𝑥||,𝑡𝑥𝑥||𝑐𝑑𝑥,(72) where 𝐸exc𝑥𝑣 is the space-time-dependent tangential electric field on the vth wire.

For the case of normal incidence the total excitation field 𝐸exc𝑥𝑣(𝑥,𝑡) is given as the sum of the incident field 𝐸inc𝑥𝑣(𝑥,𝑡) and the field reflected from the interface 𝐸ref𝑥𝑣(𝑥,𝑡) [11],𝐸exc𝑥𝑣𝑥,𝑧,𝑡=𝐸inc𝑥𝑣𝑥,𝑡𝑇+𝐸ref𝑥𝑣𝑥,𝑡𝑇.(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 written𝑧𝑇=𝑈𝑧𝑐,𝑧𝑈𝑧=max1,𝑧2,,𝑧,,𝑧𝑀.(74)

The field reflected from the interface for the case of normal incidence is given by𝐸ref𝑥𝑣𝑥,𝑡𝑇=𝑟(𝜃=0)𝐸inc𝑥𝑣𝑥,𝑡𝑇2𝑧𝑐,(75) where 𝑡𝑇2𝑧/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 becomes𝑟𝑣𝑠𝜃𝑣𝑠,𝑡=1.(76)

Thus, the set of (72) simplifies into 𝑀𝑠=1𝑥𝐿𝑠𝑥0𝑠𝐼𝑠𝑥,𝑡𝑅𝑣𝑠/𝑐4𝜋𝑅𝑣𝑠𝑑𝑥𝑀𝑠=1𝑥𝐿𝑠𝑥0𝑠𝐼𝑠𝑥,𝑡𝑅𝑣𝑠/𝑐4𝜋𝑅𝑣𝑠𝑑𝑥=𝐹0𝑣𝑡𝑥𝑥0𝑣𝑐+𝐹𝐿𝑣𝑥𝑡𝐿𝑣𝑥𝑐+12𝑍0𝑥𝐿𝑣𝑥0𝑣𝐸exc𝑥𝑣𝑥||,𝑡𝑥𝑥||𝑐𝑑𝑥,(77) and the field reflected from PEC ground is simply given by𝐸ref𝑥𝑣𝑥,𝑡𝑇=𝐸inc𝑥𝑣𝑥,𝑡𝑇2𝑧/𝑐.(78)

Given the dielectric constant of the medium and the known excitation 𝐸inc𝑥𝑣(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 𝑡0.

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||𝑡𝑅/𝑐𝐴{𝐼}𝑖||𝑡𝑅/𝑐𝐴|||𝑡𝑅/𝑐=[𝐵]||{𝐸}𝑡|𝑥𝑥|/𝑐+[𝐶]𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅0/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)𝐶𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅0/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)𝑛=0𝐶𝑛|||||𝑡(𝑅0/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)[𝐵]𝑛=0𝐸𝑛|||||𝑡(𝑥/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)[𝐷]𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅𝐿/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+𝐷𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅𝐿/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+[𝐵]𝑛=0𝐸𝑛|||||𝑡((𝐿𝑥)/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+𝑛=0𝐷𝑛|||||𝑡(𝑅𝐿/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+[𝐷]𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅𝐿/𝑐)(2𝑛𝐿/𝑐)(𝐿𝑥/𝑐)𝐷𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅𝐿/𝑐)(2𝑛𝐿/𝑐)(𝐿𝑥/𝑐)+𝑛=0𝐷𝑛|||||𝑡(𝑅𝐿/𝑐)(2𝑛𝐿/𝑐)(𝐿𝑥/𝑐)[𝐵]𝑛=0𝐸𝑛|||||𝑡((𝐿𝑥)/𝑐)(2𝑛𝐿/𝑐)(𝐿𝑥/𝑐)[𝐶]𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅0/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/𝑐)+𝐶𝑛=0𝐼𝑛𝑖|||||𝑡(𝑅0/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/𝑐)+[𝐵]𝑛=0𝐸𝑛|||||𝑡(𝑥/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/𝑐)𝑛=0𝐶𝑛|||||𝑡(𝑅0/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/𝑐).(79)

The space-dependent local matrices representing the interaction between ith source and jth observation element are defined as follows:[𝐴]=Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖1[𝐵]=14𝜋𝑅𝑑𝑥𝑑𝑥,2𝑍0Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖[𝐶]=𝑑𝑥𝑑𝑥,Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖14𝜋𝑅0[𝐷]=𝑑𝑥𝑑𝑥,Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖14𝜋𝑅𝐿𝐴𝑑𝑥𝑑𝑥,=Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝑟(𝜃)4𝜋𝑅𝐶𝑑𝑥𝑑𝑥,=Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝑟(𝜃)4𝜋𝑅0𝐷𝑑𝑥𝑑𝑥,=Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝑟(𝜃)4𝜋𝑅𝐿𝑑𝑥𝑑𝑥,(80) where {𝑓} stands for the shape functions, while additional time dependent vectors are given by 𝐴=𝑡𝑅0/𝑐Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝐻1𝑑𝑥𝑑𝑥{𝐼(𝜏)}𝑖𝐶𝑑𝜏𝑛=𝑡(𝑅00/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)×Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝐻2𝑑𝑥𝑑𝑥{𝐼(𝜏)}𝑖𝐷𝑑𝜏𝑛=𝑡(𝑅𝐿0/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)×Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝐻3𝑑𝑥𝑑𝑥{𝐼(𝜏)}𝑖𝑑𝜏,(81) where

𝐻1=𝑟𝑅𝜃,𝑡/𝑐𝜏4𝜋𝑅,𝐻2=𝑟𝑅𝜃,𝑡0/𝑐(2𝑛𝐿/𝑐)(𝑥/𝑐)𝜏4𝜋𝑅0,𝐻3=𝑟𝑅𝜃,𝑡𝐿/𝑐((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)𝜏4𝜋𝑅0.(82) Assembling the local matrices and vectors into the global ones the following global matrix system is formed:[𝐴]||{𝐼}𝑡𝑅/𝑐||={𝑔}previoustimeinstants||+{̂𝑔}previoustimeinstants,(83) where 𝐴{𝑔}=||{𝐼}𝑡𝑅/𝑐+[𝐵]||{𝐸}𝑡|𝑥𝑥|/𝑐+[𝐶]𝑛=0𝐼𝑛|||||𝑡(𝑅0/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)𝐶𝑛=0𝐼𝑛|||||𝑡(𝑅0/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)[𝐵]𝑛=0𝐸𝑛|||||𝑡(𝑥/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)[𝐷]𝑛=0𝐼𝑛|||||𝑡(𝑅𝐿/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+𝐷𝑛=0𝐼𝑛|||||𝑡(𝑅𝐿/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+[𝐵]𝑛=0𝐸𝑛|||||𝑡((𝐿𝑥)/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+[𝐷]𝑛=0𝐼𝑛|||||𝑡(𝑅𝐿/𝑐)(2𝑛𝐿/𝑐)(𝐿𝑥/𝑐)𝐷𝑛=0𝐼𝑛|||||𝑡(𝑅𝐿/𝑐)(2𝑛𝐿/𝑐)(𝐿𝑥/𝑐)[𝐵]𝑛=0𝐸𝑛|||||𝑡(𝐿𝑥/𝑐)(2𝑛𝐿/c)(𝐿𝑥/𝑐)[𝐶]𝑛=0𝐼𝑛|||||𝑡(𝑅0/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/𝑐)+𝐶𝑛=0𝐼𝑛|||||𝑡(𝑅0/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/𝑐)+[𝐵]𝑛=0𝐸𝑛|||||𝑡(𝑥/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/𝑐),𝐴|||(84){̂𝑔}=𝑡(𝑅/𝑐)𝑛=0𝐶𝑛|||||𝑡(𝑅0/𝑐)(2𝑛𝐿/𝑐)(𝑥/𝑐)+𝑛=0𝐷𝑛|||||𝑡(𝑅𝐿/𝑐)((2𝑛+1)𝐿/𝑐)(𝑥/𝑐)+𝑛=0𝐷𝑛|||||𝑡(𝑅𝐿/𝑐)(2𝑛𝐿/𝑐)(𝐿𝑥/𝑐)𝑛=0𝐶𝑛|||||𝑡(𝑅0/𝑐)((2𝑛+1)𝐿/𝑐)(𝐿𝑥/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:𝐼𝑗||𝑡𝑘=𝑁𝑠𝑖=1𝑎𝑗𝑖𝐼𝑗||𝑡𝑘𝑅/𝑐𝑔𝑗||previoustimeinstantŝ𝑔𝑗||previoustimeinstants𝑎𝑗𝑗,(86) where 𝐼𝑗|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:𝑀𝑠=1Δ𝑙𝑖Δ𝑙𝑗14𝜋𝑅𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝑑𝑥𝐼𝑑𝑥𝑠||||𝑡𝑅𝑣𝑠/𝑐Δ𝑙𝑖Δ𝑙𝑗𝑟𝑣𝑠(𝜃)4𝜋𝑅𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝐼𝑑𝑥𝑑𝑥𝑠||||𝑡𝑅𝑣𝑠/𝑐=12𝑍0Δ𝑙𝑖Δ𝑙𝑗𝐸exc𝑥𝑣𝑥||,𝑡𝑥𝑥||𝑐{𝑓}𝑗+𝑑𝑥𝑑𝑥Δ𝑙𝑗𝐹0𝑡𝑥𝑥0𝑣𝑐{𝑓}𝑗+𝑑𝑥Δ𝑙𝑗𝐹𝐿𝑥𝑡𝐿𝑣𝑥𝑐{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:𝑀𝑠=1𝐴𝑣𝑠𝐼𝑠|||||𝑡𝑅𝑣𝑠/𝑐𝑀𝑠=1𝐴𝑣𝑠𝐼𝑠|||||𝑡𝑅𝑣𝑠/𝑐=𝐵𝑣𝐸𝑣|𝑡|𝑥𝑥|/𝑐+𝑀𝑠=1𝐶𝑣𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝑥0𝑣)/𝑐)((2𝑛/𝑐)𝐿𝑣)(𝑅(0)𝑣𝑠/𝑐)𝑀𝑠=1𝐶𝑣𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝑥0𝑣)/𝑐)((2𝑛/𝑐)𝐿𝑣)(𝑅(0)𝑣𝑠/𝑐)𝐷𝑣𝑛=0𝐸𝑛𝑣|||||𝑡((𝑥𝑥0𝑣)/c)((2𝑛/𝑐)𝐿𝑣)(|𝑥𝑥0𝑣|/𝑐)𝑀𝑠=1𝐸𝑣𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝑥0𝑣)/𝑐)((2𝑛+1/𝑐)𝐿𝑣)𝑅(𝐿)𝑣𝑠/𝑐+𝑀𝑠=1𝐸𝑣𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝑥0𝑣)/𝑐)((2𝑛+1/𝑐)𝐿𝑣)𝑅(𝐿)𝑣𝑠/𝑐+𝐷𝑣𝑛=0𝐸𝑛𝑣|||||𝑡((𝑥𝑥0𝑣)/c)((2𝑛+1/𝑐)𝐿𝑣)(|𝑥𝐿𝑣𝑥|/𝑐)+𝑀𝑠=1𝐸𝑣𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝐿𝑣𝑥)/𝑐)((2𝑛/𝑐)𝐿𝑣)(𝑅(𝐿)𝑣𝑠/𝑐)𝑀𝑠=1𝐸𝑣𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝐿𝑣𝑥)/𝑐)((2𝑛/𝑐)𝐿𝑣)(𝑅(𝐿)𝑣𝑠/𝑐)𝐷𝑣𝑛=0𝐸n𝑣|||||𝑡((𝑥𝐿𝑣𝑥)/𝑐)((2𝑛/𝑐)𝐿𝑣)(|𝑥𝐿𝑣𝑥|/𝑐)𝑀𝑠=1𝐶𝑣𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝐿𝑣𝑥)/𝑐)((2𝑛+1/𝑐)𝐿𝑣)(𝑅(0)𝑣𝑠/𝑐)+𝑀𝑠=1𝐶𝑜𝑠𝑛=0𝐼𝑛𝑠|||||𝑡((𝑥𝐿𝑜𝑥)/𝑐)((2𝑛+1/𝑐)𝐿𝑜)(𝑅(0)𝑜𝑠/𝑐)+𝐷𝑜𝑛=0E𝑛𝑜|||||𝑡((𝑥𝐿𝑜𝑥)/𝑐)((2𝑛+1/𝑐)𝐿𝑜)(|𝑥𝑥0𝑜|/𝑐),(88) where {𝐸} denotes excitation vector and space-dependent matrices are of the form𝐴𝑣𝑠=Δ𝑙𝑗Δ𝑙𝑖14𝜋𝑅𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝐴𝑑𝑥𝑑𝑥,𝑣𝑠=Δ𝑙𝑗Δ𝑙𝑖𝑟𝑣𝑠(𝜃)4𝜋𝑅𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝐵𝑑𝑥𝑑𝑥,𝑣=12𝑍0Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝐶𝑑𝑥𝑑𝑥,𝑣𝑠=Δ𝑙𝑗Δ𝑙𝑖14𝜋𝑅(0)𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝐶𝑑𝑥𝑑𝑥,𝑣𝑠=Δ𝑙𝑗Δ𝑙𝑖𝑟𝑣𝑠(𝜃)4𝜋𝑅(0)𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝑑𝑥𝐷𝑑𝑥,𝑣=12𝑍0Δ𝑙𝑗Δ𝑙𝑖{𝑓}𝑗{𝑓}𝑇𝑖𝐸𝑑𝑥𝑑𝑥,𝑣𝑠=Δ𝑙𝑗Δ𝑙𝑖14𝜋𝑅(𝐿)𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝐸𝑑𝑥𝑑𝑥,𝑣𝑠=Δ𝑙𝑗Δ𝑙𝑖𝑟𝑣𝑠(𝜃)4𝜋𝑅(𝐿)𝑣𝑠{𝑓}𝑗{𝑓}𝑇𝑖𝑑𝑥𝑑𝑥.(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}𝑡𝑅𝑣𝑠/𝑐𝐴||{𝐼}𝑡𝑅𝑣𝑠/𝑐={𝑔}.(90)

The time-domain solution on the ith boundary element is given by𝐼𝑖(𝑡)=𝑁𝑡𝑘=1𝐼𝑘𝑖𝑇𝑘𝑡,(91) where 𝐼𝑘𝑖 are unknown coefficients, 𝑇𝑘 are the linear time-domain shape functions, and 𝑁𝑡 is the total number of time samples.

Applying the weighted residual approach to (90) leads to the expression𝑡𝑘𝑡+Δ𝑡𝑘[𝐴]||{𝐼}𝑡𝑅𝑣𝑠/𝑐𝐴||{𝐼}𝑡𝑅𝑣𝑠/𝑐{𝑔}𝜃𝑘𝑑𝑡=0,𝑘=1,2,,𝑁𝑡,(92) where 𝜃𝑘 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}𝑡𝑘𝑅𝑣𝑠/𝑐𝐴||{𝐼}𝑡𝑘𝑅𝑣𝑠/𝑐||=𝑔allpreviousdiscreteinstants.(93)

If the space-time discretization is performed by satisfying the Courant condition, Δ𝑥𝑐Δ𝑡, 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 𝑡𝑘 in (93) yields𝐴𝑗𝑗𝐼𝑗||𝑡𝑘+𝐴|||{𝐼}𝑡𝑘𝑅𝑣𝑠/𝑐𝐴||{𝐼}𝑡𝑘𝑅𝑣𝑠/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 forms𝐼𝑗||𝑡𝑘=𝑁𝑖=1𝐴𝑗𝑖𝐼𝑖|||𝑡𝑘𝑅𝑣𝑠/𝑐+𝐴𝑗𝑖𝐼𝑖||𝑡𝑘𝑅𝑣𝑠/𝑐+𝑔𝑗||allpreviousdiscreteinstants𝐴𝑗𝑗,(95)

where 𝑁 is total number of space elements, 𝑘=1,2,,𝑁𝑡 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] 𝜕[]+[𝑅]+[𝐿]𝜕𝜕𝑥𝑉(𝑥,𝑡)][𝐼(𝑥,𝑡)[]=𝐸𝜕𝑡𝐼(𝑥,𝑡)𝐹[][𝐼],𝜕(𝑥,𝑡)𝑧(𝑡)(𝑥,𝑡)(96)[]+[𝐺][]+[𝐶]𝜕𝜕𝑥𝐼(𝑥,𝑡)𝑉(𝑥,𝑡)[]=𝐻𝜕𝑡𝑉(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 [𝑍𝑤𝑍(𝑠)+𝑔(𝑠)], and 𝑠=𝑗𝜔 is the Laplace variable.

[𝐸𝐹(𝑥,𝑡)]and [𝐻𝐹(𝑥,𝑡)] are the excitation terms, given by [11]𝐸𝐹𝜕(𝑥,𝑡)=𝑉𝜕𝑥𝑇+𝐸(𝑥,𝑡)𝐿,𝐻(𝑥,𝑡)𝐹[𝐺]𝑉(𝑥,𝑡)=𝑇[𝐶]𝜕(𝑥,𝑡)𝑉𝜕𝑡𝑇,(𝑥,𝑡)(98) where [𝑉𝑇(𝑥,𝑡)] is the transverse voltage derived from the transverse incident field excitation [11], and [𝐸𝐿(𝑥,𝑡)] represents the longitudinal electrical field excitation.

The classical per-unit-length inductances of [𝐿] matrix have been used: