Transient radiation of a thin wire antenna buried in a dielectric half space

Transcription

1 Transient radiation of a thin wire antenna buried in a dielectric half space D. Poljak ', B. Jajac * & N.Kovai: ~Departmentof Electronics, 2Department ofelectrical Engineering, University of Split, Croatia Abstract Transient analysis of thin wire antenna buried in a dielectric medium is presented in the paper. The time domain formulation of the problem is based on the time domain Hallen integral equation. The Hallen integral equation is solved via time domain version of the Galerkin-Bubnov Integral Equation Method (GB BIEM). Some illustrative numerical results are presented. 1 Introduction Transient excitation of wire antennas buried in the earth is very important in geophysical exploration and electromagnetic stimulation of biological tissue. This work deals with a time domain analysis of athin wire buried in a dielectric half-space excited by the transient voltage source. The problem of the electromagnetic coupling to aboveground wires [l], [2]has been investigated to a much greater extent than the buried wires configurations In addition, the studies on the buried wires are mostly based on an approximate transmission line (TL) approach, and they are usually related to the infmite buried wires. Consequently, the effect of the air-earth interface is being neglacted featuring the assumption that the wire is buried at a very large depth (the homogeneous case) [3]. However, when the finite length wires are buried

2 near the air-earth interface the TL approximation cannot be applied. Therefore, the studies dealing with the transient excitation of finite length wires buried near the air-earth interface require an antenna theory approach. The transient analysis of a buried wire structure presented in this work is carried out directly in the time domain using the time domain Hallen integral equation. The influence of the lower medium to the transient response of the wire is taken into account via the reflection coefficient arising from the modified image theory [5].The transient current along the buried wire is obtained by solving the corresponding integral equation via the time domain (TD) scheme of the Galerkin-Bubnov Boundary Integral Equation Method (GB-BIEM) [6]-[ lo]. 2 Time Integral Equation Formulation A perfectly conducting thin wire of length L and radius a, buried in a dielectric medium at a depth d is considered. The wire is excited by a time-varying voltage source expressed by the strongly localised incident electric field different fiom zero only within the feed gap, Figure 1. air: EO, p0 liiiiiiiiiiiiiiiiiiiiiiiiiiiii ikiiiiiiiiiiiiiiiiiiiii dielectric medium: d 2a &O+g, PO V! < v(t) * Figure l : Geometry of the buried wire. The corresponding space-time domain integral equation can be obtained as an extension of the fi-ee-spacehallen integral equation [9] of the form: L

3 where I(x) is the equivalent axial current to be determined, Exine is the tangential incident field, c is the velocity of light, R is the distance from the source point to the observation point, and Zo is the wave impedance of a free space. The multiple reflections of the current at the free ends of the wire are taken into account by the unknown functions F&)and FL(~).The Hallen integral equation for the buried wire can be derived gradually. First, the equation (l ) is transferred into the frequency domain: 0.sR sx - S. - &= F, ($)e? +F,,(s)e c +- where s=jw is the Laplace variable. In addition, according to the modified image theory, the frequency domain fiee space integral equation (2) is extended by an image wire in the air term containing the reflection coefficient T,the veocity of light is replaced by the velocity of wave propagation in the ground v, and the free space impedance Zo is replaced by the corresponding ground impedance Zg Consequently, the frequency domain Hallen integral equation for the wire buried in the dielectric half-space becomes: where the reflection coefficient T is defined by the expression [5]: where Erg is the lower medium permitivity, R* is the distance from the image wire to the observation point, and Zg and v are given by: Finally, applying the convolution theorem, the time-domain counterpart of the equation (3) is obtained in the form:

4 452 Bou~iw-~~ Elmcnts XXIV 0 (5) V V V Where r(t)is the domain counterpart of the expression (4): where S(t) denotes the Dirac impulse. 3 Time Domain Galerkin-Bubnov Boundary Integral Equation Method (GB BIEM) The Hallen equation types are often solved by some variant of the pointunknown current cm be expressed in the matching technique combined with marching-on-in time procedure. However, this relatively simple method suffers from a poor convergence rate and late time spurious oscillations and requires an additional numerical filtering procedures. The time domain version of the Galerkin-Bubnov boundary integral equation method (GB BIEM) applied to the solution of various types of Hallen equation was shown to be stable for an arbitrary time interval [6]-[9], so it is applied to the buried wire antenna problem in this work. According to the usual discretisation procedure, the local approximation for form: where Qj is a vector containing shape functions, and {I) is the time-dependent solution vector. In addition, utilizing the weighted residual approach, the space boundary discretization leads to the local equation system for i-th source andj-th observation boundary element: J J w r & 4 q,{ N,N J v AI, AIr J~f~,~fK*41~ = r-- = J~](+~){f},dx+JF;(+L-x,if)+ix+~ 1 p~(x~j-~){,f},dx'& AIi V Mi V 2zg Alj Nj C

5 B o u d u r - yekwwmxxiv 453 Relation can be written inthe convinient matrix form given by: where: (12)

6 454 Bou~iw-~~ Elmcnts XXIV bj=j JfTlfJj{f);ilx'dx AIj AIi 4KR, and {E} vector denotes the excitation function. When the space discretisation procedure is performed, the weighted residual approach is used for the time discretisation procedure, as well. Assuming that soluton in time on the i-th boundary element can be expressed: I,(t')= -&;*yq k=l (17) where Iik are the unknown coefficients and Tk are the time domain shape functions, and choosing the Dirac impulses as test functions, the recurrence formula for the space-time varying current can be written as: N.. where Aji are the global matrix terms, g ~ *is the whole right ride of the expression (9) containing the excitation and the currents at previous instants, and the overbar line denotes that the self them is omitted. 4 Numerical results The numerical examples are related to the straight thin wire element of length L and radius a buried in the dielectric half-space at a certain depth d. Figure 2 shows the transient current at the feed point against time for the wire with dimensions: L=lm, a=5mm, buried at a depth d=0,5m in a dielelectric medium ~ ~ ~ The = 9 voltage. source excitation is given in the form of Gaussian pulse: v(t)=iqeg2@d with parameters: vo=l V,g=2*lO9s-1, t0=2ns. Figure 3 shows the transient current at the feed point against time for the wire with dimensions: L=.50m, a=2mm, buried at a depth d=o,5m in a dielectric

7 B o u d u r - yekww~tsxxiv 455 medium cy -10. The corresponding Gaussian pulse parameters are: Vo=lK g=2*109,-f;~=20~s , L ;... ;... &.... ;... ii 0, ;... j... ;... :... ;... 0, i... j...i....;... ;... i.. j.. i i ;... :... -0,004_ , t j... j... j Ysl (Times 1OE-8) Figure 2: Transient current at the feed point of the l-meter long wire. Figure 3: Transient current at the feed point of the 50-meter long wire.

8 456 Bou~iw-~~ Elmcnts XXIV 5 Concluding remarks The transient analysis of a thin wire antenna buried in a dielectric medium and excited by the time-varying voltage source is presented in the paper. The newly developed mathematical model is based on the time Hallen integral equation. The effect of a dielectric half-space has been taken into account via the reflection coefficient arising from the modified image theory. The corresponding integral equation is solved by the time domain (TD) Galerkin-Bubnov boundary integral equation method (GB-BIEM). References [l] Poljak, D., Roje, V. Induced Current and Voltages Along a Horizontal Wire Above a Lossy Ground, 21th International Conf. On Boundary Elements, BEM 21, pp , OxfordUK, Aug , [2] Poljak, D., Jajac, B., SmundiC, R. Current Induced Along Horizontal Wire Above an Imperfectly Conducting Half-space, Engineering Analysis with Boundary Elements 23, pp , [3] Bridges, G.E. Transient Plane Wave Coupling to Bare and Insulated Cables Buried in a Lossy Half-space, IEEE Trans. EMC, 37(1), pp , Feb [4] Burke, G.J., Miller, E.K. Modeling Antennas Near to and penetrating a Lossy Interface, IEEE Trans. AP, Vol. 32, pp , [5] Grcev, L.D., Menter, F.E. Transient electromagnetic Fields Near Large Earthing Systems, IEEE Trans. Magnetics, 32, pp ,May [6] Poljak, D. Transient Response of Resistively Loaded Straight Thin Wire m Half-space Configuration, Journ. Electromagn. Waves and Applic., 12(6), pp , [7] Poljak, D., Miller, E.K., Tham, CY., Roje, V. Time Domain Calculation of the Energy Stored in the Thin Wire Near Field, Millennium Conference on Antennas & Propagation, Davos, Switzerland, 9-14 April [S] Poljak, D., Ctham,Y., MCCowen, A., Roje, V. Transient Analysis of Two Coupled Horizontal Wires over a Real Ground, IEE Proc. Microw. Antennas Propag., 147(2),pp 87-94, April [9] Poljak, D., Tham, C.Y., Roje, V., SaroliC, A. Time Domain Analysis of the Human Body Exposed to the EMP Excitation Using the Human Equivalent Antenna Model, IEEE 2000 Antenna and Propagation Symposium and URSI National Radio Science meeting, Salt Lake City, Utah, USA, URSI Digest pp 159, July 16-21,2000. [lo]poljak, D. Electromagnetic Modelling of Wire Antenna Structures, WIT Press, Southempton,