Boundary Elements and Other Mesh Reduction Methods XXXV 13 Wire antenna model of the vertical roundin electrode D. Poljak & S. Sesnic University of Split, FESB, Split, Croatia Abstract A straiht wire antenna model of the vertical roundin electrode has been presented in the paper. The formulation is based on the homoeneous interal equation of Pocklinton type for half-space problems. The influence of a finitely conductin half-space is taken into account via the correspondin reflection coicients. The Pocklinton equation is solved both numerically and analytically. The numerical solution is carried out via the Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM). Some illustrative computational examples are presented at the end. Keywords: vertical roundin electrode, antenna model, Pocklinton interal equation, boundary element method, analytical solution. 1 Introduction Vertical roundin electrode is considered to be one of the key components of practical roundin systems, particularly for the protection of wind turbines [1] as they are extremely vulnerable to lihtnin strikes due to their special shape and isolated locations, mainly in hih altitude areas. A typical realistic roundin system for wind turbines is rather complex and usually composed from rins, horizontal and vertical electrodes, respectively [1]. A frequency domain numerical/analytical antenna theory model of horizontal electrodes and rin electrodes as individual elements, has been carried out in [] (Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM) + analytical solution) and [3] (GB-IBEM), respectively. Vertical electrode, on the other hand, has not been analyzed as an individual element, to a reater extent, usin antenna model and related numerical and analytical solution procedures. doi:1.495/bem1311
14 Boundary Elements and Other Mesh Reduction Methods XXXV Thouh an electromanetic transient analysis of complex roundin systems for wind turbines has been already reported elsewhere, e.. in [1] where a combination of the rin, vertical rods and horizontal electrodes is considered, a detailed study of vertical electrode behavior, as an individual element, still remains rather important. Thus, of particular importance is the analytical analysis, as the complex practical confiurations consistin of straiht electrodes and rin incorporated in concrete blocks are rather difficult to deal with. Therefore, an analytical solution at least for the part of such eometry is of enineerin interest. Moreover, a trade-off between analytical and numerical approach clearly stressin out strenth and weaknesses of both approaches is necessary. The key parameter of the wire model is the current distribution induced alon the electrode bein excited by the equivalent current enerator. The wire antenna model of the electrode is based on the homoeneous Pocklinton interodifferential equation. The scattered voltae alon the electrode is obtained by interatin the scattered electric field enerated by the induced current. The impact of a round-air interface is taken into account via the correspondin Fresnel reflection coicient (within the numerical solution approach) and simplified reflection coicient approach arisin from the modified imae theory (within the analytical solution approach) [4]. The intero-differential relationships arisin from the proposed wire antenna model are numerically treated by means of the GB-IBEM and analytically. The scattered voltae alon the electrode is obtained by interatin the current derivative alon the electrode [5, 6]. Formulation Vertical roundin electrode of lenth L and radius a, buried in a lossy medium at depth d and excited by an equivalent current source is shown in Fi 1. Fiure 1: Vertical roundin electrode excited by the current source.
Boundary Elements and Other Mesh Reduction Methods XXXV 15 The overnin equations for the current and scattered voltae induced alon the vertical electrode can be readily derived from Maxwell s equations by enforcin the continuity conditions for the tanential components of the electric field alon the wire in the very similar manner as it was undertaken for the case of vertical electrode in [5] and horizontal electrode in [6]. A riorous approach to account for the influence of lossy medium to the frequency response of the roundin electrode is related to the solution of Sommerfeld interals, which is considered to be rather demandin and computationally expensive task [4 9]..1 Interal equation for the current alon the electrode The related homoeneous intero-differential equation of the Pocklinton type for the current distribution induced alon the electrode is iven by µ jω I ( z ) ( z, z ) dz 4 π L ( ) 1 I z ( z, z ) dz + Zs ( z) I ( z) = j4πω z z L (1) where the complex permittivity of the lossy round is iven by σ = r + () jω with r and σ bein the soil permittivity and conductivity, respectively. Furthermore, the Green function is of the form while ( ), and (, ) i (, ') (, ') (, ') zz = zz Γ zz (3) zz denotes the lossy medium Green function ( zz, ') zz is, accordin to the imae theory, iven by i ( zz, ') The propaation constant of the lossy round is ref γ jωµσ ω µ r 1 i γ R 1 e = (4) R γ R e = (5) R = (6) and R 1, R are the distances from the source and the imae to the observation point, respectively.
16 Boundary Elements and Other Mesh Reduction Methods XXXV The alternative approach, on the other hand, is related to the approximate reflection coicient approach [7, 8], which is used in this paper as well. Within the numerical solution the Fresnel reflection coicient (RC) is used [] 1 1 cosθ sin θ MIT n n x x Γ ref = ; θ = arct ; n = 1 1 d cosθ + sin θ n n (7) Furthermore, a simplified reflection coicient arisin from modified imae theory is used within the analytical solution [ 4] Γ MIT ref = + (8) The electrode is excited via an equivalent ideal current source with one terminal connected to the electrode and the other one rounded at infinity, as presented in Fi 1. This current source is incorporated into the Pocklinton equation formulation via the followin conditions at the electrode ends [5] ( ) ( ) I d = I, I L = (9) where I denotes the impressed unit current source. It is worth emphasizin that the knowlede of the electrode current provides the assessment of the scattered voltae.. The scattered voltae alon the electrode The scattered voltae alon the vertical electrode is defined by a line interal of the horizontal component of the scattered field (normal to the electrode and tanential to the round-air interface) from the remote soil to the electrode surface [5] a sct sct V ( z) Ex ( x, z) dx = (1) As the horizontal field component can be expressed in terms of the scalar potential radient sct E x φ = (11) x the scattered voltae alon the electrode can be written as follows a a ϕ (, ) ϕ (, ) (1) sct xz d V ( z) = dx = x z dx x dz
Boundary Elements and Other Mesh Reduction Methods XXXV 17 where ( xz, ) ϕ is defined by the followin particular interal 1 ϕ ( z) = q ( z ') ( z, z ') dz ' 4π (13) L and the chare density is linked to current distribution throuh the continuity equation [4] Substitutin continuity equation (14) into (13) yields 1 di q = jω dz (14) ( ') 1 I z ϕ ( z) = ( z, z ') dz ' j4 πω (15) z' L Insertin (15) into (1) and assumin the scalar potential in the remote soil to be zero [1], ives the scattered voltae alon the electrode ( ) sct 1 I z V ( z) = ( z, z ) dz j4πω (16) z L Note that the tedious interation from infinity to the electrode surface is avoided by takin the advantae of the very definition of voltae alon the electrode (1). 3 Solution The Pocklinton equation (1) is solved numerically and analytically, respectively. The numerical solution is undertaken via the GB-IBEM [4]. 3.1 Numerical solution The unknown current I e (z ) alon the straiht wire sement can be written as follows T ( ') { } { } e I z = f I (17) Assemblin the contributions from each sement, the Pocklinton equation (1) is transferred into the matrix equation M [ Z ] { I} =, and j = 1,,..., M (18) ji i j= 1 where M is the total number of sements and [Z] ji iven by
18 Boundary Elements and Other Mesh Reduction Methods XXXV T T Fr [ ] = { } { } ( ) { } { } ref i ( ) ji j + Γ i j i Z D D ' z, z ' dz ' D D ' z, z ' dz ' + γ lj li lj li T { } { } ( ) j i lj f f z, z ' dz ' dz li (19) is the mutual impedance matrix representin the interaction of the i-th source with the j-th observation sement, respectively. The matrices {f} and {f } contain the shape functions while {D} and {D } contain their derivatives, and l i, l j are the widths of i-th and j-th boundary elements. A linear approximation over a wire sement is used in this work, as it was shown to be optimal in various EMC problems includin thin wire confiurations. More details on the method can be found elsewhere, e.. in [4]. 3. Analytical solution The Pocklinton equation (1) can be solved analytically under certain conditions. It is convenient to write (1) in a followin way ( ') I z, ( ) ( ) () 1 γ ( z z ) dz + Zs z I z = j4πω z z L The first step in the analytical solution of () is to write the interal on the left-hand side of (1) as follows I( z' ) ( z, z' ) dz' = I( z) ( z, z' ) dz' + I( z' ) I( z) ( z, z' ) dz' (1) L L L The interal on the left hand side of (1) is now approximated by the first term on the riht hand side of (1), i.e. the second interal on the riht hand side of (1) is nelected. Furthermore, the solution of characteristic interal from (1) is L L L h+ h+ h+ jkr jkr e e L L ψ =, ' ' = ' +Γ ' = ln Γ ln R R a d i MIT MIT ( z z ) dz dz dz () ref ref L L L i h h h Consequently, the Pocklinton equation () simplifies into the partial differential equation of the form z γ I( z) = (3) The analytical solution of (3) is sh γ eq ( d + L z) I( z) = I (4) sh γ L ( eq )
Boundary Elements and Other Mesh Reduction Methods XXXV 19 where = + Z (5) γ eq γ j4πω ψ Furthermore, the scattered voltae alon the electrode (16) is iven by ( γ eq ) ( ) sct 1 I z V ( z) = ( z, z ) dz j4πω z L γ eq I = chγ eq ( d + L z) ( z, z ) dz ' j4πω sh L L and can be evaluated by means of numerical interation procedures. 4 Results s (6) The first set of the results is related to the absolute value of spatial distribution of induced current computed numerically and analytically, while the second set deals with the results for the scattered voltae induced alon the electrode and computed via both approaches, as well. The results are obtained for the followin parameters of the electrode: L=1 m, d=.3 m, a=5 mm. The excitation is the ideal current source I =1 A. Ground permittivity is r =1. All Fiures show the absolute value of the current distribution alon the electrode. Fiure shows the current distribution alon the electrode at the operatin frequency f=1 MHz for the round conductivity σ=.1 S/m. The results computed analytically and via GB-IBEM aree rather satisfactorily. Fiure : Current distribution (absolute value) alon the vertical electrode for f=1 MHz and σ=.1 S/m.
11 Boundary Elements and Other Mesh Reduction Methods XXXV Fiure 3 shows the current distribution alon the same electrode at the frequency f=1 MHz for the round conductivity σ=.1 S/m. The waveforms obtained via different approaches are still alike. However for the lower value of conductivity (σ=.1 S/m) differences in the results for the current distribution are slihtly hiher. Fiures 4 and 5 show the distribution of the scattered voltae induced alon the same electrode for different values of round conductivity at f=1 MHz. Fiure 3: Current distribution (absolute value) alon the vertical electrode for f=1 MHz and σ=.1 S/m. Fiure 4: Scattered voltae distribution (absolute value) alon the vertical electrode for f=1 MHz and σ=.1 S/m.
Boundary Elements and Other Mesh Reduction Methods XXXV 111 Fiure 5: Scattered voltae distribution (absolute value) alon the vertical electrode for f=1 MHz and σ=.1 S/m. A satisfactory areement between analytical and numerical results is achieved. Aain, certain differences are noticeable at f=1 MHz for lower value of round conductivity. It is worth notin that the problem of the vertical electrode transient response in principle does not have a solution in the closed form. Consequently, the numerical solution is considered to be exact one, or at least more realistic in this case contrary to the usual cases (if analytical solution is available) where the oal of comparin numerical with analytical solutions is to show converence of the numerical towards analytical results. 5 Conclusion The paper deals with an analysis of vertical roundin electrode by means of straiht wire antenna model featurin both numerical and analytical approach, respectively. The formulation is based on the solution of the homoeneous Pocklinton intero-differential equation for half-space problems in the frequency domain. The presence of a lossy round is taken into account via correspondin reflection coicients. The numerical solution is carried out via the Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB- IBEM). The same problem is also treated via the approximate analytical solution. Some illustrative computational examples are iven in the paper.
11 Boundary Elements and Other Mesh Reduction Methods XXXV References [1] D. Cavka, D. Poljak, V. Doric, R. Goic, Transient Analysis of Wind Turbines, Renewable Enery 43, pp. 84-91, 1. [] D. Poljak, S. Sesnic, and R. Goic, Analytical versus boundary element modellin of horizontal round electrode, Enineerin Analysis with Boundary Elements, 34, pp. 37-314, 1. [3] D. Cavka, D. Poljak, R. Goic, Transient Analysis of Rin Shaped Groundin Electrode usin Isoparametric Quadratic Elements, Proc. ICLP 1. [4] D. Poljak, Advanced Modelin in Computational Electromanetic Compatibility. New Jersey, USA: Wiley-Interscience, 7. [5] D. Poljak, V. Doric, Wire antenna Model for Transient Analysis of Simple Groundin systems, Part I: The Vertical Groundin Electrode, Proress in Electromanetics Research, PIER 64, pp. 149-166, 6. [6] D. Poljak, V. Doric, Wire antenna Model for Transient Analysis of Simple Groundin systems, Part II: The Horizontal Groundin Electrode, Proress in Electromanetics Research, PIER 64, pp. 149-166, 6. [7] E. K. Miller, A. J. Poio, G. J. Burke, and E. S. Selden, Analysis of wire Antennas in the Presence of a Conductin Half-Space. Part I. The Vertical Antenna in Free Space, Canadian Journal of Physics, 5, pp. 33-341, 197. [8] E. K. Miller, A. J. Poio, G. J. Burke, and E. S. Selden, Analysis of wire Antennas in the Presence of a Conductin Half-Space. Part II. The Horizontal Antenna in Free Space, Canadian Journal of Physics, 5, pp. 614-67, 197. [9] R. G. Olsen, M. C. Willis, A Comparison of Exact and Quasi-static Methods for Evaluatin Groundin Systems at Hih Frequencies, IEEE Trans. Power Delivery, 11 (), pp. 171-181, April 1996. [1] F. M. Tesche, M. Ianoz, and T. Karlsson, EMC Analysis Methods and Computational Models. New York, USA: John Wiley & Sons, Inc., 1997.