Boundary element modelling of the metalic post protection zone

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1 Boundary element modelling of the metalic post protection zone B. 3ajac1, D. poljak2 &N. ~0vai5' I Department of Electrical Engineering, University of Split, Croatia 2 Department of Electronics, University of Split, Croatia Abstract The boundary element analysis of the protection zone of a metallic post, as one of the most commonly used lightning protection devices is presented in the paper. The calculation of induced charge density is carried out via the potential coefficients approach. Some illustrative numerical results related to the various engineering applications are also presented. 1 Introduction Electromagnetic interferences (EMI) on electrical equipment are caused by a direct lightning strike or by an indirect discharge in a close proximity of the lightning channel [l]-[5]. Such lightning flash surges can produce adverse effects on electrical equipement. The key-point in the analysis of lightning protection devices is the determination of the protection zone. The protection zone is a space in the vicinity of the lightning rod where the electric field intensity is much less with respect to the case where the lightning rod is absent. The geometry of the protection zone is determined by the lines of the associated electric field in the vicinity of the rod. The classical analytical technique of determination of the protection zone combined with the numerical approach for the correction of the protection zone of a single lightning rod has been presented in [6]. This paper deals with the exstension of the numerical method presented in [6] for a single metallic post to the system of lightning rods.

2 2 Theoretical background A simple lightning protection system in the form of metallic post is sited vertically on the perfectly conducting ground which is assumed to be on a zero potential. is a A negatively charged cloud (flat model) is located at a certain distance h above ground, Figure 1. Figure 1 : Geometry of the metallic post. The mathematical model is based on following assumptions: - excitation is an electrostatic field with constant value Eo, - thickness of the post is neglected, - the metallic post is assumed to be on the earth zero potential. The protection zone of the metallic post is determined from the electric field distribution in the vicinity of the post. The calculation procedure is divided in two steps: - the calculation of the induced charge along the post via Boundary Element Method (BEM), - the calculation of the electric field via the finite difference algorithm. The calculation of the induced charge along the post is carried out using the potential coefficients approach. Having performed a discretization of the post one comes up with a system of equations for potential p along the each segment of the wire structure:

3 where: g,, g;' is the induced charge on the source i-th segment and on the related image, qb aik are the potential coefficients to be determined, zi stands for the height of the i-th segment from the ground, E. denotes the value of the incident field due to the charged cloud. Since induced charges qi and qi' have the same values with opposite sign, the system of equations (1) is re-written as follows: where a;k = ci,k - a ik. The potential on the k-th wire of the lightning system can be written in the form: q k = Cqiakj - E Ozk 1 (3) l where akk denotes self and aik the mutual potential coefficients. 2.1 Boundary element calculation of seif and mutual coefficients by the mean potential approach Two wire segments dlk and L, respectively with associated charge distributions q k and qm are located in xy plane as it is shown in Figure 2. On the other hand, the mean value of the potential along the wire k, is given by: Consequently, the potential coefficients is simply given by:

4 222 Boutrdary Elcmatr~~ XXV Figure 2: Geometry of two wire segments. In addition, if i-th segment coupled with k-th element and its image k* is considered, Figure 3, it follows: where Rik and is the distance from the i-th rod to the k-th rod and to its image, respectively. Induced charges along the rods can be determined from (1) with calculated potential coefficients. 2.2 Finite difference calculation of the electric field The electric field at an arbitrary point in the vicinity of the lightning protection system can be obtained as a sum of the incident field Ei,, (due to the charged cloud) and the induced field Eind (due to the charge distribution along the lightning rods): where the induced field is simply given by: The equation (8) is computed by using the finite difference algorithm as was presented in [6].

5 Figure 3: Lightning rods segments and associated images. 3 Numerical results Firstly a single lightning rod with length L = 20 m and negligible radius is considered. An incident field due to a charged cloud is assumed to be E. = ~ V/m. Figure 4 shows the protection zone of single lightning rod in the yz plane. It can be concluded that the zone of very rare lightning strike is characterised by the field line of the 30% greater value than it is the field value causing the static electricity of the cloud. The discrepancy between protection zone determined by the classical approach (dashed line) [6] and numerical procedure (solid curve) is obvious. The corresponding zones in xy plane for the cases y = 5m, y = IOm and y = 20m are shown in Figures 5 to 7. Figures 8 and 11 show the protection zones of two lightning rods with same height (L = 20m) and different heights (L, = 15m, L2 = 28m) with separation between them d = 29.5m. The corresponding protection zones in xy plane for the cases y = 8m and y = 20m are shown in Figures 9 and 10. Finally, the corresponding protection zones in xy plane for the cases y = 6m, y = 15m and y = 25m are shown in Figures 12 to 14.

6 224 Boutrdary Elcmatr~~ XXV Figure 4: Protection zone of a single lightning rod. Figure 5: Protection zone of a single lightning rod (y = 5m). Figure 6: Protection zone of a single lightning rod (y = 10m).

7 Figure 7: Protection zone of a single lightning rod (y = 20m). Figure 8: Protection zone of two lightning rods of same heights (L = 20m) with separation d = 29.5m. Figure 9: Protection zone of two identical lightning rods (y = gm).

8 226 Boutrdary Elcmatr~~ XXV Figure 10: Protection zone of two identical lightning rods (y = 20m). Figure 11: Protection zone of two lightning rods with different heights (L1 = 15m, L2 = 28m) with separation d = 29.5m. Figure 12: Protection zone of two lightning rods with different heights (y = 6m).

9 Figure 13: Protection zone of two lightning rods with different heights (y =15m). Figure 14: Protection zone of two lightning rods with different heights (y =25m). 4 Conclusion The paper deals with a boundary element modelling of the protection zone of the lightning protection systems based on the potential coefficients approach. The mean potential boundary element scheme with piece-wise constant elements is used for the calculation of induced charge along the metallic structure. The corresponding electric field is determined via finite difference technique. Obtained numerical results clearly demonstrate the efficiency of the proposed numerical approach compared to the classical approach for the calculation of the protection zone of lightning protection system.

10 228 Boutrdary Elcmatr~~ XXV References [l] Poljak D., Jajac B., Lightning induced current on a metallic post - frequency domain analysis, 14" International Conference on Applied Electromagnetics and Communications, Dubrovnik, [2] Poljak D., Jajac B., KovaE N., Time domain Hallen integral equation approach to a lightning channel modelling", 4th European Symposium on Electromagnetic Compatibility, Brugge, Belgium, [3] Poljak D., Jajac B., On the use of monopole antenna model in lightning protection system analysis, International Symposium on Electromagnetic Compatibility, EMC Roma 98, Roma, [4] Cristina S., Orlandi A., EMC effects of the lightning protection system: Shielding properties of the roof-grid, Proc. IEEE Int. Symposium on EMC, [5] Orlandi A., Lightning induced transient voltages in presence of complex structures and nonlinear loads, IEEE Trans. on EMC, 38, May [6] Jajac B., Poljak D., Correction of the protection zone of the metallic post, Elektrotehnika, 41, (in Croatian).

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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