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Mathematical Problems in Engineering
Volume 2011, Article ID 165032, 8 pages
http://dx.doi.org/10.1155/2011/165032
Research Article

The Iterative Solution for Electromagnetic Field Coupling to Buried Wires

College of Science, National University of Defense Technology, Changsha 410073, China

Received 9 January 2011; Revised 12 March 2011; Accepted 1 April 2011

Academic Editor: Ben T. Nohara

Copyright © 2011 Xufeng Zhang et al. 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.

Abstract

By integrating the electric field integral equation and the transmission line equation, an iterative solution for the electromagnetic field coupling to buried wires is obtained. At first we establish the integral equation which the difference between solutions of the integral equation and the telegraph equation satisfies. Then the solution of the telegraph equation is used to approximate the solution of this integral equation. Every following step of iteration is an improvement on the transmission line solution, and with several iterations, a well approximation to the solution of electric field integral equation can be obtained.

1. Introduction

The electromagnetic field coupling to buried wires plays an important role in many engineering applications, such as geophysical probes, power or communications cables, and grounding systems (see [17]). The integral equation approach or transmission line (TL) model (see [16]) can be used to resolve this problem. Actually the transmission line model is an approximation of the integral equation approach. This method is very efficient for infinite or at least very long buried wires, but if the line is of finite length, this method cannot give an accurate solution (see [13]). So if the buried wires of the finite length are of interest, the integral equation approach has to be used. However, the integral equation usually should be resolved by numerical methods, such as method of moment (MoM) (see [1, 8]) and boundary element method (BEM) (see [26]), which may cost much computation time for long lines.

This paper presents an iterative solution for the electromagnetic field coupling to buried wires. At first we establish the integral equation which the difference between solutions of the electric field integral equation and the telegraph equation satisfies. Then the solution of the telegraph equation is used to approximate the solution of this integral equation. By this way, the iterative solution is obtained. Taking the solution of the telegraph equation as initial value, a satisfying result can be obtained just after several iterations. Finally, several numerical examples are presented, which indicates that the results obtained only after several steps iteration are in good agreement with the results obtained by MoM.

Actually, the iterative method is a semianalytic method, which requires only one integration along the line length for every space point in every step of iteration; so compared with numerical methods, such as MoM or BEM, this method is easier to implement numerical computation.

2. Integral Equation and Telegraph Equation of Buried Wires

Figure 1 illustrates a finite wire of length 𝐿 and radius 𝑎, buried in a lossy ground at depth 𝑑, illuminated by a plane-wave from above ground. The upper half space is free-space, the lower half space is lossy ground, and the interface is at 𝑧=0.

165032.fig.001
Figure 1: Thin wire buried in a lossy ground.
2.1. Electric Field Integral Equation

Taking into account the influence of interface via the plane-wave Fresnel reflection coefficient, we obtained the following integral equation for the unknown current distribution induced along the buried wire:𝐸exc𝑥1=d𝑗𝜔𝜀2d𝑥2𝛾2𝐿0𝐼𝑥𝑔𝑥,𝑥d𝑥,(2.1) where 𝜔 is angular frequency of incidence plane-wave, and 𝑗 is imaginary unit. 𝜀 is the complex permittivity of the lossy ground:𝜀=𝜀𝑟𝜀0𝜎𝑗𝜔.(2.2)𝜀𝑟 and 𝜎 are relative permittivity and conductivity of the ground. According to the image theory, the Green function 𝑔(𝑥,𝑥) is given in [9] as𝑔𝑥,𝑥=𝑒𝛾𝑅14𝜋𝑅1Γref𝑒𝛾𝑅24𝜋𝑅2,(2.3) where 𝛾 is propagation constant of the ground:𝛾=𝑗𝜔𝜇𝜎𝜔2𝜇𝜀,(2.4) and 𝑅1, 𝑅2 are given by𝑅1=(𝑥𝑥)2+𝑎2𝑅2=(𝑥𝑥)2+4𝑑2.(2.5)Γref is Fresnel reflection coefficient (see [3]):Γref=1/𝑛0cos𝜃1/𝑛0sin2𝜃1/𝑛0cos𝜃+1/𝑛0sin2𝜃,||𝜃=arctan𝑥𝑥||2𝑑,𝑛0=𝜀𝜀0.(2.6) The excitation electric field on the wire for normal incidence plane-wave can be expressed as follows (see [7]):𝐸exc𝑥=𝐸0ΓTM𝑒𝛾𝑑,(2.7) where ΓTM denotes Fresnel transmission coefficient at the interface (see [7]):ΓTM=21+𝑛0.(2.8)

2.2. Transmission Line Telegraph Equation

The transmission line telegraph equation of electromagnetic field coupling to buried wire can be expressed as follows:d𝑉(𝑥)d𝑥+𝑍𝐼(𝑥)=𝐸exc𝑥,d𝐼(𝑥)d𝑥+𝑌𝑉(𝑥)=0,(2.9) where 𝑉(𝑥) and 𝐼(𝑥) are line voltage and current, and 𝑍 and 𝑌 are impedance and admittance of the ground, respectively (see [10]),𝑍=𝑗𝜔𝜇02𝜋ln1+𝛾𝑎+𝛾𝑎2𝑒2𝑑|𝛾|4+𝛾2𝑎2,𝛾𝑌=2𝑍.(2.10)

Eliminating 𝑉(𝑥) from (2.9), we haved2𝐼(𝑥)d𝑥2𝛾2𝐼(𝑥)+𝑌𝐸exc𝑥=0.(2.11)

Actually the solution of (2.11) is an approximation of (2.1). The transmission line telegraph equation can be resolved analytically (see [1]).

3. The Iterative Solution of Integral Equation

A simple iterative approach to correct the results obtained using TL approximation for an overhead line of finite length is proposed in [11, 12], where two additional source terms representing the correction to the TL approximation are added to the classical transmission line telegrapher’s equations. Resorting to the integral equation and transmission line telegraph equation, we will present a similar approach to obtain the iterative solution of (2.1).

Suggesting that 𝐼(𝑥) and 𝐼0(𝑥) are solutions of (2.1) and (2.11), and 𝐼(𝑥)=𝐼(𝑥)𝐼0(𝑥), the integral equation which 𝐼(𝑥) satisfies can be derived from (2.1) and (2.11):𝑌d𝑗𝜔𝜀2d𝑥2𝛾2𝐿0𝐼𝑥𝑔𝑥,𝑥d𝑥=d2d𝑥2𝛾2𝐼0𝑌(𝑥)𝑗𝜔𝜀𝐿0𝐼0𝑥𝑔𝑥,𝑥d𝑥.(3.1)

We can get an approximate solution of this equation as before:d2d𝑥2𝛾2𝐼1d(𝑥)=2d𝑥2𝛾2𝐼0𝑌(𝑥)𝑗𝜔𝜀𝐿0𝐼0𝑥𝑔𝑥,𝑥d𝑥.(3.2)

Repeating this process, an iterative solution can be established:𝐼(𝑥)=𝐼0(𝑥)+𝐼1d(𝑥)+,(3.3)2d𝑥2𝛾2𝐼𝑛d(𝑥)=2d𝑥2𝛾2𝐼𝑛1𝑌(𝑥)𝑗𝜔𝜀𝐿0𝐼𝑛1𝑥𝑔𝑥,𝑥d𝑥.(3.4) Denoting𝐹(𝑥)=𝐼𝑛1𝑌(𝑥)𝑗𝜔𝜀𝐿0𝐼𝑛1𝑥𝑔𝑥,𝑥d𝑥,(3.5)  (3.4) is equivalent tod2d𝑥2𝛾2𝐼𝑛(𝑥)𝐹(𝑥)=0.(3.6)

The solution of this equation with boundary condition 𝐼𝑛(0)=𝐼𝑛(𝐿)=0 is given by [11] 𝐼𝑛(𝑥)=𝐹(0)𝑒2𝛾𝐿𝐹(𝐿)𝑒𝛾𝐿1𝑒2𝛾𝐿𝑒𝛾𝑥+𝐹(𝐿)𝑒𝛾𝐿𝐹(0)1𝑒2𝛾𝐿𝑒𝛾𝑥+𝐹(𝑥).(3.7)

The convergence of the iterative solution is discussed in [13]. Since the solution of (2.11) is an approximation of (2.1), an excellent approximation to the solution of (2.1) can be obtained only after several iterations.

4. Numerical Results

The parameters of our example are as follows: conductor radius 𝑎=1 cm, depth 𝑑=2.5 m, the excitation field is a normally incident plane wave with a frequency of 50 M and magnitude 𝐸0=1 V/m, the relative dielectric constant of the ground is 𝜀𝑟=10, and the ground conductivity is 𝜎=0.01 S/m.

Figures 2, 3, and 4 show the real and imaginary parts of the current distribution induced along a wire with different lengths 𝐿=5 m, 𝐿=15 m, and 𝐿=25 m. The results obtained by MoM, TL, and two steps iteration are given in these figures. Evidently the results obtained using the TL approximation are sufficient for longer wire, but it fails to give an accurate current distribution for relatively shorter wire lengths. However the results obtained after two-steps iteration are in good agreement with the results obtained by MoM.

fig2
Figure 2: Induced current distribution along the thin wire (𝐿=5 m). (a) Real part. (b) Image part.
fig3
Figure 3: Induced current distribution along the thin wire (𝐿=15 m). (a) Real part. (b) Image part.
fig4
Figure 4: Induced current distribution along the thin wire (𝐿=25 m). (a) Real part. (b) Image part.

Keeping other parameters as forenamed, Figure 5 shows the influence of the buried depths to the induced current distribution for 𝐿=5 m; Figure 6 shows the influence of the ground conductivities to the induced current distribution for 𝐿=5 m and 𝑑=0.5 m. Obviously, induced current distribution magnitude decreases as the conductivity or buried depth increases, which is due to the increasing of the loss.

165032.fig.005
Figure 5: Induced current distribution magnitude along the thin wire for different depths.
165032.fig.006
Figure 6: Induced current distribution magnitude along the thin wire for different conductivities.

5. Conclusion

The problem of electromagnetic coupling on the buried thin wires is analyzed with the iterative method; the solution of the telegraph equation is used to approximate the solution of the integral equation in every step of iteration. And we can usually get satisfying result by just several iterations while taking the solution of the telegraph equation as initial value. This method may be applied to the multiple buried thin wires.

Acknowledgments

The authors would like to express their thanks to the reviewers for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (no. 10871231).

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