TIME DOMAIN ANALYTICAL MODELING OF A STRAIGHT THIN WIRE BURIED IN A LOSSY MEDIUM. Otto-von-Guericke University Magdeburg, Magdeburg D-39106, Germany

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1 Progress In Electromagnetics Research, Vol., , TIME DOMAIN ANALYTICAL MODELING OF A STRAIGHT THIN WIRE BURIED IN A LOSSY MEDIUM S. Sesnić, *, D. Poljak, and S. Tkachenko Faculty of Electrical Engineering, Mechanical Engineering and Naal Architecture, Uniersity of Split, Rud era Boškoića 3, Split, Croatia Department of Electrical Engineering and Information Technology, Otto-on-Guericke Uniersity Magdeburg, Magdeburg D-396, Germany Abstract This paper deals with an analytical solution of the time domain Pocklington equation for a straight thin wire of finite length, buried in a lossy half-space and excited ia the electromagnetic pulse EMP) excitation. Presence of the earth-air interface is taken into account ia the simplified reflection coefficient arising from the Modified Image Theory MIT). The analytical solution is carried out using the Laplace transform and the Cauchy residue theorem. The EMP excitation is treated ia numerical conolution. The obtained analytical results are compared to those calculated using the numerical solution of the frequency domain Pocklington equation combined with the Inerse Fast Fourier Transform IFFT).. INTRODUCTION Transient analysis of thin wire scatterers has been a subject of considerable interest of many prominent researchers for more than fifty years. There are numerous applications of these studies in antenna theory and propagation, as well as in electromagnetic compatibility EMC), such as buried power and telecommunication cables, respectiely, grounding, target identification, stimulation of biological tissue, etc. [ ]. The electromagnetic field coupling to thin wire scatterers can be treated ia different approaches. Transient response can be obtained by means of direct time domain modeling, or ia an indirect approach Receied July, Accepted 6 September, Scheduled 4 Noember * Corresponding author: Silestar Sesnić ssesnic@fesb.hr).

2 486 Sesnić, Poljak, and Tkachenko in the frequency domain, where the time domain solution is obtained using certain inerse transform procedures [, ]. The principal adantage of the indirect approach is the relatie simplicity of both the formulation and the selected numerical treatment. On the other hand, direct time modeling ensures better physical insight, accurate modeling of highly resonant structures, possibility of calculating only early time period and easier implementation of nonlinearities [, ]. The formulation of the problem in thin wire transient analysis is usually goerned by some ariants of integral or integro-differential equations Hallen or Pocklington type), respectiely. The solution of such equations is, in most cases, undertaken using some ariant of numerical methods. Numerical modeling is widely used for soling arious complex problems. On the other hand, analytical solution can be obtained when dealing with canonical problems, using a carefully chosen set of approximations [ 5]. These approximations are usually posed by limiting parameters for which the solution is alid or by using the approximation procedures to simplify the goerning equations. The adantage of analytical solutions oer numerical ones is the ability to follow up the procedure with the complete control of adopted approximations. In this way, the insight into the physical characteristics of the problem is ensured, which is, when using numerical methods, rather complex task. Also, analytical solutions are readily implemented for benchmark purposes, as well as some fast engineering estimation of phenomena. Furthermore, analytical solutions can be used within some hybrid approaches for modeling complex structures, where computational time can be significantly reduced []. Valuable contributions in the area of analytical solutions of integral equations in electromagnetics are gien by R. W. P. King et al. [, 3]. In [, 3] analytical solutions of different expressions, arising from the antenna analysis, in either transmitting or receiing mode, are deried primarily in the frequency domain. S. Tkachenko deries the analytical solution for the current induced along the wire aboe perfectly conducting PEC) ground using the transmission line modeling TLM) for LF excitations [4]. This model is extended to the case of high frequencies in [5]. Time domain analytical modeling is not inestigated to a great extent and papers on the subject are rather scarce. Such papers usually deal with a narrow set of parameters for which the model is alid free space, homogeneous medium) [6 9]. Hoorfar and Chang gie the solution for transient response of thin wire in free space using singularity expansion method [6]. Velazquez and Mukhedar derie analytical solution for the current induced along a grounding electrode,

3 Progress In Electromagnetics Research, Vol., 487 based on the TL model [7]. Chen gies the solution for transient response of infinite antenna in a lossy medium, drien by a oltage generator [8]. Analytical solution for transient response based on the full wae model is needed, as TL approximation fails to account for radiation effects. For the simple case of a straight thin wire embedded in a homogeneous lossy medium analytical solution has been reported by the authors in []. In the second part of this paper, Pocklington integro-differential formulation for current induced on a thin wire buried in a lossy halfspace, illuminated by a transmitted electromagnetic wae is posed in the time domain. The influence of the boundary is taken into account ia simplified reflection coefficient arising from Modified Image Theory. Thin wire approximation is used throughout the paper [, ]. In the third part, the corresponding Pocklington equation is soled using the approximation of the unknown current function, Laplace transform and Cauchy residue theorem. Results obtained with these relations are compared, in the fourth section, with the results obtained ia numerical solution of corresponding Pocklington equation in the frequency domain and subsequent Inerse Fast Fourier Transform IFFT) []. The results obtained ia different approaches agree satisfactorily.. TIME DOMAIN FORMULATION A perfectly conducting thin wire of length L and radius a, is horizontally buried in a lossy medium at depth d. The electrical properties of a medium are permittiity ε and conductiity σ. The wire is illuminated by a transmitted part of a transient electromagnetic EM) wae. For the sake of simplicity, only normal incidence is considered. The geometry of the problem is shown in Figure. The goerning Pocklington integro-differential equation for halfspace is deried gradually. First, thin wire in a homogeneous lossy medium is considered. A time dependant electric field can be expressed in terms of electric scalar ϕ and magnetic ector potential A, respectiely [, ] E = A ϕ. ) t Performing certain mathematical manipulations, the following relation is obtained [] µ A A A σ t ε t = µ A + µσϕ + µε ϕ ) J t i, )

4 488 Sesnić, Poljak, and Tkachenko z y H E ε, µ x d ε, µ, σ L a Figure. Horizontal straight thin wire buried in a lossy medium. where J i represents current density produced by the external source. Adopting the Lorentz gauge [] ) becomes A + µσϕ + µε ϕ t =, 3) A A A µσ t µε t = µ J i, 4) representing the wae equation for magnetic ector potential. Combining the Equations ) 4), the electric field can be expressed as follows µε t ) + µσ E = A A A µσ t µε t. 5) The solution of 4) is the particular integral [] t A r, t) = µ V Ji r, t ) g r, r, t, t ) dv dt, 6) assuming the current density J i to be the only source in the considered domain. g r, r, t, t ) represents the Green s function of a homogeneous lossy medium that can be obtained by soling the following differential equation µσ ) µε t t g r, r, t, t ) = δ r, r, t, t ). 7)

5 Progress In Electromagnetics Research, Vol., 489 The solution of 7) can be written in the form [] g r, r, t, t ) = e R δ t t R ) + σ t t I u) 4πR 6πε e τg u, 8) where time constant is defined as τ g = ε 9) σ and the distance from the source to obseration point is gien with R = r r, ) while I u) represents the modified Bessel function of the first kind and first order with the argument u defined as u = ) R t t τ ). ) g The second term in 8) can be neglected if the condition σ 6πε is satisfied [], so the Green s function can be defined as g r, r, t, t ) = e R δ t t R ). ) 4πR Combining Equations 6) and ) and taking into account the thin wire approximation [], the axial component of ector potential is obtained [] A x x, t) = µ 4π I x, t R ) e R R dx, 3) where the distance from the source point in the axis of the wire to the obseration point on the wire surface is defined as R = x x ) + a. 4) For the case of dissipatie half-space, shown in Figure, the expression for magnetic ector potential 3) has to be expanded to account for the ground-air interface, as shown in Figure. In the frequency domain, the additional term due to the image wire in the air is obtained by multiplying the reflection coefficient with the corresponding Green s function. To obtain the expression for a lossy half-space, first the relation 3) is transformed into frequency domain applying the Laplace transform. The following equation is obtained A x x, s) = µ 4π R Ix, s)e R s e R dx, 5)

6 49 Sesnić, Poljak, and Tkachenko z L x' d y θ ε, µ, σ a x d x L a Figure. Wire and its image due to image theory. where s = jω is the Laplace ariable. Now, the expression for the magnetic ector potential can be written as follows A x x, s) = = µ 4π µ 4π R Ix, s)e R s e R Γ MIT ref dx s)ix, s)e R s e τg R R dx 6) where Γ MIT ref s) is the reflection coefficient due to earth-air interface. The reflection coefficient arises from the Modified Image Theory and is gien by [3] Γ MIT ref s) = sτ + sτ +, 7) where τ = ε ε r ), σ τ = ε 8) ε r + ). σ Applying the inerse Laplace transform to Equation 7), a time domain counterpart for the reflection coefficient 7) is obtained in the form [ Γ MIT τ ref t) = δt) + τ ) ] e t τ. 9) τ τ τ

7 Progress In Electromagnetics Research, Vol., 49 The distance from the source point on the image wire in the air to the obseration point on the original wire in the ground is gien as R = x x ) + 4d. ) Applying the inerse Laplace transform to Equation 6), the following time domain expression is obtained A x x, t) = µ 4π µ 4π I t x, t R ) e R R dx ) Γ MIT ref τ)i x, t R e R τ R dx dτ, ) where the conolution integral is applied instead of multiplication in the Laplace domain. Combining the relations for the electric field 5) and for magnetic ector potential ), the deried time domain integro-differential Pocklington equation is obtained µε t ) + µσ Ex tr t) = x µσ ) µε t t µ t 4π µ 4π I x, t R ) Γ MIT ref τ) I x, t R e τ τg R R ) e R R dx dx dτ. ) Soling the integro-differential Equation ), the space-time dependant current is obtained. Knowing the current distribution, other parameters of interest can be determined. Apart from this special case of Pocklington equation, it is worth mentioning some general information about the equation itself. The Pocklington equation is used for obtaining the current induced along the wire. The current can be induced either by incident electromagnetic wae or by impressed oltage source. It can be used both in frequency and time domain and it is, generally speaking, easier to sole numerically) in frequency domain []. Pocklington equation can be written for different electromagnetic problems, such as: thin wire, thick wire, cured wire, free space, lossy half-space, multilayered medium, wire array, arbitrary wire configurations, to name just a few. As opposed to differential formulations that are, generally, used to sole domain problems, integral formulations are intended for soling

8 49 Sesnić, Poljak, and Tkachenko problems with un)known sources, as is in this case, induced current. Pocklington equation is obtained directly from Maxwell s equations, using magnetic ector potential as auxiliary ariable. Because of this, Pocklington equation approach is considered to be a full wae approach, since only used assumption is well known Lorentz gauge 3) that is needed to define the diergence of magnetic ector potential []. In this way, no generality is lost and results obtained by soling Pocklington equations are considered to be most accurate. 3. ANALYTICAL SOLUTION OF POCKLINGTON EQUATION Under some conditions it is possible to handle differential operator and integral operator from Equation ) separately. First, the solution of the integral operator is acquired, using certain approximations. The first integral from the right-hand side of Equation ) can be, generally, written as I [ x, t ] x x ) + a e x x ) +a x x ) + a dx = fx, t) 3) The accurate analytical integration of kernel in expression 3) is not possible to obtain [4, 5]. On the other hand, the solution is possible to acquire with the aid of numerical methods and those solutions are well known in literature [,, 6]. Using certain approximations, the analytical solution of 3) can be obtained. In 938, Hallen used an approximation to sole corresponding integral equation in the frequency domain [7]. Adjustment of that approximation has recently been used to obtain the solution of integral equation in time domain [9, 8]. To handle the integral operator in Equation 3), addition and subtraction technique is applied = I x, t R ) e R R dx [ I x, t a ) +I x, t R ) I x, t a ) ] e R R dx. 4)

9 Progress In Electromagnetics Research, Vol., 493 Now, the right hand side of the Equation 4) can be written as = + [ I x, t a ) + I x, t R ) I x, t a ) ] e R R I x, t a ) e R R dx dx [ I x, t R ) I x, t a ) ] e R R dx. 5) Adopting the following assumption [7, 8] I x, t R ) I x, t a ) Equation 4) becomes I x, t R ) e R R dx = I x, t a ) 6) e R R dx. 7) The approximation 6) has proen to be alid in papers by Tijhuis et al. [9, 8]. The apparent adantage of this approximation is replacing the integral equation with corresponding ordinary differential equation with the unknown induced current. The drawback of this approximation is that time retardation is assumed to be a/ instead of R/, which results in the loss of the information on radiated energy due to current reflection on the wire free ends. This information is significant when dealing with free space. In this paper, howeer, the primary concern is to deal with a lossy medium where the approximation is alid. Substituting 7) into ) yields µε t ) + µσ Ex tr t) = x µσ ) µε t t µ x, 4π I t a ) e R R dx µ t 4π Γ MIT ref τ)i x, t a ) τ e R R dx dτ. 8)

10 494 Sesnić, Poljak, and Tkachenko The next step in soling the differential Equation 8) is to apply the Laplace transform. Taking into account following assumptions E x t) =, t I x, t a ) =, t I x, t a ), 9) =, t < t and performing the Laplace transform, the following equation is obtained µεs + µσ)ex tr s) = µ ) µσs µεs 4π x Ix, s)e a s Satisfying the condition e R R σ a dx Γ MIT ref s) e R R dx. 3) ε µ, 3) which is achieed for the medium conductiity of order ms/m, integrals in 3) can be soled analytically as follows [4, 8] e R R dx e R R dx Imposing Equation 7), it can be written Ψs) = = e R R dx ln L a ; 3) R dx ln L d. 33) dx Γ MIT ref s) R ln L a + sτ + sτ + ln L d e R R dx ). 34) Now, relation 3) can be written as µεs + µσ)ex tr s) = µ ) µσs µεs Ix, s)e a s Ψs). 35) 4π x

11 Progress In Electromagnetics Research, Vol., 495 Adopting the expression for the propagation constant of a lossy medium γ = µε s + σ ) ε s, 36) the following differential equation is obtained Ix, s) x γ Ix, s) = 4π µsψs) e a s γ E tr x s). 37) The solution of 37) can be readily obtained, prescribing the boundary conditions at the wire ends I, s) = IL, s) =. 38) The solution of 37) can be written as [ Ix, s) = 4πe a s µsψs) Etr x s) cosh γ L x)) ] cosh γ L ). 39) To obtain the solution for the current distribution in time domain, inerse Laplace transform has to be performed featuring the Cauchy residue theorem [9] x+jy n ft) = lim e ts F s)ds = Ress k ), 4) y jπ x jy where residues of the function are defined as k= Ress k ) = lim s sk s s k )e ts F s), 4) while s k denotes the poles of the function F s). The excitation function in the Laplace domain is of the form E tr x s) = [V/m] 4) which corresponds to impulse excitation in time domain. Calculating all the residues of the function 39) and undertaking the inerse transform as in 4), the following expression is obtained Ix, t) [ = 4π Rs Ψ ) coshγ Ψ L x)) L coshγ Ψ ) µ π µεl n= ] e t+ a )s Ψ n ± n )πx sin b 4c ns,n Ψs,n ) L e t+ a )s,n, 43)

12 496 Sesnić, Poljak, and Tkachenko where coefficients Rs Ψ ) and s Ψ represent physical properties of the system, taking into account properties of the medium, as well as the dimensions of the wire and the distance from the interface Rs Ψ ) = ln L d s Ψ s Ψ τ + τ τ s Ψ τ + s Ψ τ + ), 44) s Ψ = ln L a + ln L d τ ln L a + τ ln L. d Furthermore, other coefficients in relation 43) are gien as follows γ Ψ = µε s Ψ + bs ) Ψ s,n = b ± ) b 4c n b = σ. 45) ε c n = n ) π µεl, n =,, 3,... Expression 43) represents the space-time distribution of the current along the straight wire buried in a lossy medium excited by an impulse excitation, i.e., it represents an impulse response. The part with infinite series can be truncated after first ten terms achieing ery good conergence. Furthermore, the response to an arbitrary excitation can be obtained performing the corresponding conolution. The excitation function is plane wae in the form of double exponential electromagnetic pulse tangential to the wire [3] E x t) = E e αt e βt). 46) The transmitted electric field in the Laplace frequency) domain can be written as follows [] Ex tr s) = Γ tr s)e x s)e γd 47) where Γ tr s) represents the Fresnel transmission coefficient defined as [] Γ tr s) = sε sε + σ + sε. 48) The expression for the transmitted field 47) is too complex for the use of analytical conolution, therefore the numerical conolution is applied to obtain the results for the induced current. Numerical conolution is carried out with the discrete samples obtained from inerse Fourier transform of expression 47) and the discrete samples of Equation 43).

13 Progress In Electromagnetics Research, Vol., RESULTS In this section some illustratie results obtained ia the proposed method are presented. The analytical results are obtained using the presented method of conolution and are compared to the results obtained by numerical solution of the corresponding frequency domain Pocklington equation using the Galerkin-Bubno Indirect Boundary Element Method GB-IBEM) and the Inerse Fast Fourier Transform IFFT) []. Eidently, Pocklington equation is used in both models, in that way making both a full wae model. The main difference lies in the method of soling integro-differential equation. The approximations used for analytical solution hae been stated and explained. Howeer, frequency domain model used for comparison is soled numerically in the frequency domain. Generally, numerical solution is considered to be more accurate, because fewer approximations are made in calculation procedure, so these results are considered to be a benchmark. After extensie numerical tests, it has been found that optimal parameters for numerical conolution outlined at the end of the third part, are: number of samples n t = 6 and period of obseration T = 33 µs. With these parameters, accurate results are obtained in reasonable time period. All results for the current response are calculated using the standard EMP excitation [3] E = V/m, α = 4 6 /s, β = /s. 49) t [s] -7 Figure 3. Transient current at the center of the straight wire, L = m, d = 3 cm, σ = ms/m.

14 498 Sesnić, Poljak, and Tkachenko The first geometry of interest to be analyzed is a wire of length L = m, radius a = 5 mm and the burial depth d = 3 cm. Relatie electric permittiity of a medium is gien as ε r = and a conductiity is aried with the respectie alues σ = ms/m, ms/m and ms/m. The results calculated ia different methods are shown in Figures 3 to 7. The largest discrepancies of the results can be seen in Figure 3. But een there, a relatiely good agreement between the results for the t [s] -7 Figure 4. Transient current at the center of the straight wire, L = m, d = 3 cm, σ = ms/m t [s] -7 Figure 5. Transient current at the center of the straight wire, L = m, d = 3 cm, σ = ms/m t = ns t = 5 ns t = ns x [m] Figure 6. Current along the wire at different time instants, L = m, d = 3 cm, σ = ms/m t = 5 ns t = ns t = ns x [m] Figure 7. Current along the wire at different time instants, L = m, d = 3 cm, σ = ms/m.

15 Progress In Electromagnetics Research, Vol., 499 early time behaior t < ns) is obious. The discrepancies at a later time instants can be readily explained taking into account the nature of the approximations used throughout the formulation. Namely, it is reasonable not to expect approximation 6) to proide alid results for ery short wire L < m) immersed in a low conducting medium σ < ms/m). Howeer, een for a short wire, for higher conductiities of a medium Figures 4 and 5) relatiely good agreement between the results is achieed. This can be explained with larger dissipation of energy through more conductie medium. In Figures 6 and 7, current distribution along the wire at different time instants is shown. The parameters correspond to geometries shown in Figures 3 and 5, respectiely. Smaller discrepancies can be obsered, but oerall agreement is ery good. Figures 8 to show the transient current induced at the center of straight longer wires buried in a lossy medium with σ = ms/m. The length of the wire is L = m and m, respectiely, while burial depth aries as d = 4 m, m, 5 m, respectiely. As it can be seen in Figure 8, for a m-long wire the agreement between the results is rather satisfactorily. In Figures 9 and, another interesting phenomenon can be obsered. Namely, for m- long wire, a relatiely low discrepancy between the results can be seen, due to the implementation of simplified approximation for the reflection coefficient arising from Modified Image Theory 9). The discrepancy that can be obsered for smaller burial depths is caused by the icinity of the earth-air interface. As the burial depth increases, as in Figure, it can be seen that the results agree satisfactorily because t [s] -6 Figure 8. Transient current at the center of the straight wire, L = m, d = 4 m, σ = ms/m.

16 5 Sesnić, Poljak, and Tkachenko t [s] -6 Figure 9. Transient current at the center of the straight wire, L = m, d = m, σ = ms/m t [s] -6 Figure. Transient current at the center of the straight wire, L = m, d = 5 m, σ = ms/m. -3 t = ns t = 3 ns t = 3 ns 4 3 t = 4 ns - -4 t = ns x [m] Figure. Current along the wire at different time instants, L = m, d = 4 m, σ = ms/m. t = 6 ns x [m] Figure. Current along the wire at different time instants, L = m, d = 5 m, σ = ms/m. of the diminishing influence of the two media interface. In Figures and, also, a current distribution along the wire is shown. The results are obtained for different time instants. The parameters of the geometries correspond to those shown in Figures 8 and, respectiely. The oerall agreement between the methods is ery good. From examples gien in Figures 3 to, it can be concluded that two ariables affect the agreement of the results the most; wire length and medium conductiity. Generally speaking, the longer the wire, the

17 Progress In Electromagnetics Research, Vol., t [s] -6 Figure 3. Transient current at the center of the straight wire, L = m, d = m, σ =.833 ms/m. better is the agreement and the higher the conductiity, the better is the agreement. The lower limit for wire length is about m, and for medium conductiity is about ms/m. Howeer, these limits are not to be taken strictly, because the influence of these two parameters is coupled. If the wire is longer, the results are in better agreement een for a lower conductiity of the medium, and ice ersa. The last example is related to Figure 3 representing the transient current at the center of a m-long wire, buried at depth d = m, with the electrical properties of the media ε r = 9 and σ =.833 ms/m. These alues correspond to realistic configuration of power cable layout. It can be seen in the Figure 3, that there is a small discrepancy in the early time behaior of the induced current which manifests in a smaller maximum alue. Namely, the conductiity of the medium is ery low and the cable is buried relatiely close to the boundary, so, taking into account the adopted approximations, the obtained results agree rather satisfactorily. 5. CONCLUSION In this paper, the direct analytical solution of the time domain Pocklington integro-differential equation for straight buried thin wire is presented. The goerning equation is posed in time domain and subsequently soled using the approximation of the unknown current function, Laplace transform and Cauchy residue theorem, thus obtaining analytical expression for space-time arying induced current. The numerical conolution of the impulse response with the incident transmitted wae is performed to obtain the transient response for

18 5 Sesnić, Poljak, and Tkachenko actual excitation. The results are compared with the results obtained ia numerical solution of corresponding Pocklington equation in the frequency domain and subsequent IFFT. The results obtained ia different approaches agree satisfactorily. REFERENCES. Rao, S. M., Time Domain Electromagnetics, S. M. Rao ed.), Academic Press, San Diego, USA, Poljak, D., Adanced Modeling in Computational Electromagnetic Compatibility, Wiley-Interscience, New Jersey, USA, Miller, E. K., A. J. Poggio, and G. J. Burke, An integrodifferential equation technique for the time-domain analysis of thin wire structures. I. The numerical method, Journal of Computational Physics, Vol., No., 4 48, May Bridges, G. E., Transient plane wae coupling to bare and insulated cables buried in a lossy half-space, IEEE Transactions on Electromagnetic Compatibility, Vol. 37, No., 6 7, Feb Lorentzou, M. I., N. D. Hatziargyriou, and B. C. Papadias, Time domain analysis of grounding electrodes impulse response, IEEE Transactions on Power Deliery, Vol. 8, No., 57 54, Apr Poljak, D. and V. Doric, Wire antenna model for transient analysis of simple grounding systems, part II: The horizontal grounding electrode, Progress In Electromagnetics Research, Vol. 64, 67 89, Chen, H. T., J.-X. Luo, and D.-K. Zhang, An analytic formula of the current distribution for the VLF horizontal wire antenna aboe lossy half-space, Progress In Electromagnetics Research Letters, Vol., 49 58, Zhang, G.-H., M. Xia, and X.-M. Jiang, Transient analysis of wire structures using time domain integral equation method with exact matrix elements, Progress In Electromagnetics Research, Vol. 9, 8 98, Jin, X. and M. Ali, Embedded antennas in dry and saturated concrete for application in wireless sensors, Progress In Electromagnetics Research, Vol., 97,.. Seo, D.-W., J.-H. Yoo, K. Kwon, and N.-H. Myung, A hybrid method for computing the RCS of wire scatterers with an arbitrary orientation, Progress In Electromagnetics Research B, Vol. 3, 55 68,.

19 Progress In Electromagnetics Research, Vol., 53. Poljak, D. and N. Koac, Time domain modeling of a thin wire in a two-media configuration featuring a simplified reflection/transmission coefficient approach, Engineering Analysis with Boundary Elements, Vol. 33, 83 93, 9.. King, R. W. P., G. J. Fikioris, and R. B. Mack, Cylindrical Antennas and Arrays, Cambridge Uniersity Press, Cambridge, UK,. 3. King, R. W. P., Embedded bare and insulated antennas, IEEE Transactions on Biomedical Engineering, Vol. 4, No. 3, 53 6, May Tkatchenko, S., F. Rachidi, and M. Ianoz, Electromagnetic field coupling to a line of finite length: Theory and fast iteratie solutions in frequency and time domains, IEEE Transactions on Electromagnetic Compatibility, Vol. 37, No. 4, 59 57, No Tkatchenko, S., F. Rachidi, and M. Ianoz, High-frequency electromagnetic field coupling to long terminated lines, IEEE Transactions on Electromagnetic Compatibility, Vol. 43, No., 7 9, May. 6. Hoorfar, A. and D. C. Chang, Analytic determination of the transient response of a thin-wire antenna based upon an SEM representation, IEEE Transactions on Antennas and Propagation, Vol. 3, No. 6, 45 5, No Velazquez, R. and D. Mukhedkar, Analytical modelling of grounding electrodes transient behaior, IEEE Transactions on Power Apparatus and Systems, Vol. 3, No. 6, 34 3, Jun Chen, K. C., Transient response of an infinite cylindrical antenna in a dissipatie medium, IEEE Transactions on Antennas and Propagation, Vol. 35, No. 5, , May Tijhuis, A. G., Z. Peng, and A. Rubio Bretones, Transient excitation of a straight thin-wire segment: A new look at an old problem, IEEE Transactions on Antennas and Propagation, Vol. 4, No., 3 46, Oct Sesnic, S. and D. Poljak, Time domain analytical modeling of finite length thin wire embedded in homogeneous lossy medium, SoftCOM 8th International Conference on Software, Telecomumnications & Computer Networks, Split-Bol,.. Stratton, J. A., Electromagnetic Theory, Wiley-Interscience, New Jersey, USA, 7.. Rothwell, E. J. and M. J. Cloud, Electromagnetics, CRC Press,

20 54 Sesnić, Poljak, and Tkachenko Boca Raton, London, New York, Washington, D.C.,. 3. Takashima, T., T. Nakae, and R. Ishibashi, Calculation of complex fields in conducting media, IEEE Transactions on Electrical Insulation, Vol. 5, No., 7, Feb Polyanin, A. D. and A. V. Manzhiro, Handbook of Integral Equations, CRC Press, Boca Raton, London, New York, Washington, D.C., Dyakin, V. V. and S. V. Marin, Initial-boundary alue problem and integrodifferential equations of electrodynamics for an inhomogeneous conductie body, Computational Mathematics and Mathematical Physics, Vol. 48, No., 75 83, Bost, F., L. Nicolas, and G. Rojat, A time-domain integral formulation for the scattering by thin wires, IEEE Transaction on Magnetics, Vol. 36, No. 4, , Jul.. 7. Hallen, E., Theoretical inestigations into the transmitting and receiing qualities of antennæ, Noa Acta Regiae Societatis Scientiarum Upsaliensis, Vol., No. 4, 3 44, No Bogerd, J. C., A. G. Tijhuis, and J. J. A. Klaasen, Electromagnetic excitation of a thin wire: A traeling-wae approach, IEEE Transactions on Antennas and Propagation, Vol. 46, No. 8,, Aug Schiff, J. L., The Laplace Transform Theory and Applications, Springer-Verlag, New York, USA, Tesche, F. M., M. Ianoz, and T. Karlsson, EMC Analysis Methods and Computational Models, John Wiley & Sons, Inc., New York, USA, 997.