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Computation of Lightning Horizontal Field Over the Two-Dimensional Rough Ground by Using the Three-Dimensional FDTD Dongshuai Li, Qilin Zhang, Member, IEEE, Zhenhui Wang, and Tao Liu
Abstract—A three-dimensional (3-D) finite-difference timedomain (FDTD) method is developed for simulating the lightningradiated electromagnetic field over the two-dimensional (2-D) rough ground. This 3-D FDTD method provides a new approach to analyze the propagation effect of 2-D rough ground on the lightning horizontal field. It is noted that the effect of the 2-D surface roughness on the horizontal field cannot be ignored even at a distance of 100 m from the lightning channel, and the increase of the land roughness results in a lower magnitude of the horizontal field waveform. When the root-mean-square height (RMSH) of the rough surface is 5 m, the magnitude of the horizontal field at a height of h = 10 m above the rough ground reduces to about half of its field peak value with respect to the smooth ground within distances of 100 m from the lightning channel. We also found that the horizontal field is more obviously affected by the land roughness when the land conductivity is smaller than 0.001 S/m. Index Terms—Lightning horizontal field, three-dimensional (3-D) finite-difference time-domain (FDTD), two-dimensional (2-D) rough surface.
Fig. 1. Propagation of the lightning electromagnetic fields over 2-D rough ground.
I. INTRODUCTION HE horizontal electric field is very important in estimating the lightning-induced over voltages on overhead line, and it is sensitive to the finitely conducting ground. There are several approximate approaches for calculating the lightning horizontal electric field. Among them, the most remarkable one is the Cooray–Rubinstein (C-R) simplified formula proposed by Cooray [1], [2] and Rubinstein [3]. For the homogeneous and finitely conducting ground, the C-R simplified formula has been extensively used for estimating the lightning horizontal electric fields [4], [5]. For a horizontally stratified conducting ground, Shoory et al. [6] took into account the propagation effect of the azimuthal magnetic field and presented a new formula for predicting the horizontal electric field in frequency domain. In fact, their new formula is similar to the C-R formula, which can be viewed as the generalization of the C-R formula both for
T
Manuscript received March 24, 2013; revised May 17, 2013 and May 30, 2013; accepted June 1, 2013. Date of publication July 3, 2013; date of current version January 27, 2014. This work was supported by the National Natural Science Foundation of China under Grant 41275009 and Grant 40975002, by a program for Postgraduates Research Innovation of Jiangsu Higher Education Institutions (CXZZ13_0512), and by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors are with Key Laboratory of Meteorological Disaster of Ministry of Education (KLME), Nanjing University of Information Science and Technology, Nanjing 210044, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/TEMC.2013.2266479
the homogeneous conducting ground and horizontally stratified conducting ground. For a vertically stratified ground (mixed path), Zhang et al. [7] have extended the C-R formula in a smooth and ocean-land mixed path, and found it has a satisfactory accuracy for land section conductivities ranging from 0.01 to 0.001 S/m when the fields propagate from the ocean surface to the land section. More recently, Zhang et al. [8] have estimated the lightning horizontal electric field over a mixed and 2-D rough ground surface by using their extended the C-R formula. It shows that the decrease of rough-land conductivity results in a more negative excursion peak value of the horizontal field waveform in the initial time. However, the results proposed by Zhang et al. [8] are based on the analytic and approximate solutions. Until now, no one has computed the lightning-radiated horizontal electric field over the rough ground without using any kind of approximate methods (e.g., finite-difference time-domain (FDTD) and method of moments (MoM) [5], [9], [10]). Therefore, in this paper, we will employ a three-dimensional (3-D) FDTD technique for analyzing the lightning horizontal field over 2-D rough ground. II. CALCULATION THEORY AND METHODS A. Introduction of Calculation Model The simulation domain of the 3-D FDTD technique is shown in Fig. 1. The working space is 211 m × 51 m × 3501 m, which is divided into square cells of Δx × Δy × Δz = 1 m × 1 m ×
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Fig. 2. 2-D rough ground surface is simulated by using a normalized 2-D band-limited Weierstrass fractal function. Fig. 3.
Treatment of material parameter on the 2-D rough surface.
Fig. 4.
Yee’s grid units and the lightning channel is set along E z component.
1 m, the time increment is set to 1.66 ns. The air and ground are both split by Yee’s grid units and 10 planes of the perfectly matching layer (PML) is adopted as the absorption boundary condition in order to avoid reflections there [11], [12]. The letters H and d are the height of the lightning channel and the total propagation distance, respectively, and h is the relative height of the observation with respect to the smooth surface. B. Two-Dimensional Rough Surface Since natural surfaces are generally neither purely random nor purely periodic and often anisotropic, the introduction of fractal geometry provided a new tool for describing natural rough structures [13]. A normalized 2-D band limited Weierstrass fractal function is employed to simulate the rough land surface [14], it possesses a very large derivative and a finite band of special frequency with self-affine fractal character, which is as follows: √ 1 2δ[1 − b(2D −6) ] 2 F (x, y) = 1 {M [b(2D −6)N 1 − b(2D −6)(N 2 +1) ]} 2 N2 M 2πm × b(D −3)n sin Kbn x cos M m =1 n =N 1 2πm + y sin (1) + φn m M where δ is the root-mean-square height (RMSH) of the rough surface, which is used to depict the rough characteristics of an irregular terrain and can be calculated by (2) [15] (2) δ = E[f 2 (x, y)] − {E[f (x, y)]}2 where f (x, y) is a randomly rough surface function, E[f (x, y)] and E[f 2 (x, y)] are the mathematical expectation values of the functions f (x, y) and f 2 (x, y), respectively. D (2 < D < 3) is the fractal dimension, K is the fundamental wavenumber, b is the fundamental spatial frequency, N and M are the numbers of harmonics, and φn m is a phase term that has a uniform distribution over the interval [−π, π]. In Fig. 2, the 2-D rough surface is simulated by using a normalized 2-D band-limited Weierstrass fractal function in (1), with N1 = 0, N2 = 6, M = 7, b = 1.6, D = 2.3, δ = 2, and K = 100.
In order to divide the ground boundary surface into square cells as shown in Fig. 3, on the interface between the air and ground the electric and magnetic parameters are taken as the linear average of both mediums as follows [16]: ε2 + ε1 2 σ2 + σ1 σf = 2 μf = μ1 = μ2 εf =
σm f = σm 1 = σm 2
(3) (4) (5) (6)
where the electromagnetic parameters of air and ground are ε1 , σ1 , μ1 , σm 1 and ε2 , σ2 , μ2 , σm 2 , respectively. εf , σf , μf , and σm f are the electromagnetic parameters on the 2-D rough ground boundary surface. C. Lightning Source Current In the FDTD calculations, the lightning channel is represented by a vertical array of current sources [17]. Each current source has a length of 1 m and is described by specifying the four magnetic-field vectors forming a square contour surrounding the cubic cell representing the current, as shown in Fig. 4.
LI et al.: COMPUTATION OF LIGHTNING HORIZONTAL FIELD OVER THE TWO-DIMENSIONAL ROUGH GROUND
Fig. 5.
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(a) Channel-base current waveforms and (b) their frequency spectrum.
Based on Ampere’s law, we can obtain the following equation: ∇×H=ε
∂E + σE + J ∂t
(7)
where J is the lightning current density. The lightning channel is set along the edge of Ez component in the Yee’s grid units as shown in Fig. 5. The return stroke discharging current includes two components—a breakdown current and a corona current. Each of the two components is calculated by using the analytical expression suggested by Heidler [18] (t/τ11)n 1 (t/τ21)n 2 I02 I01 e−t/τ 1 2 + e−t/τ 2 2 n η1 [(t/τ11) 1 + 1)] η2 [(t/τ21 )n 2 +1)]
1/n 1 τ12 τ11 η1 = exp − · n1 τ12 τ11
1/n 2 τ22 τ21 η2 = exp − · n2 (8) τ22 τ21
i(0, t) =
Where, I0 is the amplitude, τ1 the current rise-time constant, and τ2 is the current decay time constant. Table I shows the pa-
Fig. 6. Examination of the accuracy of the 3-D FDTD along a smooth land over a height of h = 10 m at different distances of from the lightning channel by using the Sommerfeld’s integral for [(a) and (b)] the subsequent return stroke and [(c) and (d)] the first return stroke.
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TABLE I PARAMETERS FOR THE FIRST AND SUBSEQUENT RETURN STROKE CURRENTS
rameters for the first return stroke and subsequent return stroke current waveforms [19]. Fig. 5 presents the channel-based current waveforms corresponding to the first and typical subsequent return stroke in time domain and frequency domain.
III. SIMULATED RESULTS AND ANALYSIS
Fig. 7. Effects of the RMSH = 5 m on the horizontal electric field for [(a) and (b)] the subsequent return stroke and [(c) and (d)] the first return stroke.
In the paper, we adopted the modified transmission line model with a linear decay of current with height (MTLL) [20], the height of the channel is assumed to be 7.5 km, the return-stroke speed is v = 1.5 × 108 m/s. First, we test the validity of our 3-D FDTD method by using Sommerfeld’s integrals [21], [22]. From Fig. 6, it is noted that the results predicted by our 3-D FDTD method are the same as those by Sommerfeld’s integrals within distances of 100 m both for the subsequent return stroke and for the first return stroke. Fig. 7 shows the simulated lightning horizontal electric field at a height of h = 10 m with the land conductivity of 0.1 S/m above the rough land at distances of 50 and 100 m from the lightning channel. Note that the effect of the 2-D surface roughness on the horizontal field cannot be ignored at a distance of 100 m from the channel. When the RMSH is 5 m, the magnitude of the horizontal field reduces to about half of the field peak value with respect to the smooth ground. Also, we find that the increase of the land roughness causes more attenuation of the magnitude for the subsequent return stroke than that for the first return stroke, because the first return stroke has relatively more low-frequency components that are less affected by the finitely conducting ground. From Fig. 8, we have studied the effect of some other different variables on the horizontal electric field over 2-D rough ground at a distance of 50 m from the lightning channel for the subsequent return stroke and the first return stroke, respectively. It is noted from Fig. 8(a) and (b) that the increase of the land roughness results in a lower magnitude of the horizontal field waveform. Also, we find that the increase of the land roughness causes more attenuation of the magnitude for the subsequent return stroke than that for the first return stroke. From Fig. 8(c) and (d), when the land conductivity is less than 0.001 S/m, the horizontal fields are more obviously affected by the land roughness. However, as shown in Fig. 8(e) and (f), although the temporal variation of the calculated horizontal electric field depends on many parameters such as current waveform, current decay as well as the return stroke velocity along the height along the channel, the effect of the rough land seems not to be affected by the engineering return stroke model.
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Fig. 8. Effects of different variables [(a,b) RMSH, (c,d) land conductivity, (e,f) return stroke model] on the horizontal electric field over 2-D rough ground at distances of 50 m from the lightning channel for [(a), (c), and (e)] the subsequent return stroke and [(b), (d), and (f)] the first return stroke.
IV. CONCLUSION In this paper, a 3-D FDTD model is developed to simulate lightning electromagnetic fields over a rough ground boundary. It is found that the effect of the 2-D surface roughness on the horizontal field cannot be ignored even within distances of 100 m from the channel, the increase of the land roughness results in a lower magnitude of the horizontal field waveform. When the RMSH of the rough surface is 5 m, the horizontal field peak at the initial time at a height of h = 10 m above the rough ground
reduces to about half of its field peak value with respect to the smooth ground within distances of 100 m from the lightning channel. When the land conductivity is smaller than 0.001 S/m, the horizontal field is more affected by the land roughness. Due to the limitations of computer memory and calculation time, we cannot simulate the horizontal electric fields beyond the distance of 100 m from the lightning channel. In the future, we will develop the parallel 3-D FDTD technique to analyze the field propagation.
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REFERENCES [1] V. Cooray, “Horizontal fields generated by return strokes,” Radio Sci., vol. 27, pp. 529–537, 1992. [2] V. Cooray, “Some considerations on the ‘Cooray–Rubinstein’ formulation used in deriving the horizontal electric field of lightning return strokes over finitely conducting ground,” IEEE Trans. Electromag. Compat., vol. 44, no. 4, pp. 560–566, Nov. 2002. [3] M. Rubinstein, “An approximate formula for the calculation of the horizontal electric field from lightning at close, intermediate, and long range,” IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 531–535, Aug. 1996. [4] V. Cooray, “Horizontal electric field above-and underground produced by lightning flashes,” IEEE Trans. Electromagn. Compat., vol. 52, no. 4, pp. 936–943, Nov. 2010. [5] A. Shoory, R. Moini, S. H. H. Sadeghi, and V. A. Rakov, “Analysis of lightning-radiated electromagnetic fields in the vicinity of lossy ground,” IEEE Trans. Electromagn. Compat., vol. 47, no. 1, pp. 131–145, Fab. 2005. [6] A. Shoory, F. Rachidi, F. Delfino, R. Procopio, and M. Rossi, “Lightning electromagnetic radiation over a stratified conducting ground: 2. Validity of simplified approaches,” J. Geophys. Res., vol. 116, no. D11, pp. D11115, Jun. 2011. [7] Q. Zhang, D. Li, Y. Fan, Y. Zhang, and J. Gao, “Examination of the coorayrubinstein (CR) formula for a mixed propagation path by using FDTD,” J. Geophys. Res., vol. 117, no. D15, pp. D15309, Aug. 2012. [8] Q. Zhang, D. Li, X. Tang, and Z. Wang, “Lightning-radiated horizontal electric field over a rough- and ocean-land mixed propagation path,” IEEE Trans. Electromagn. Compat., vol. PP, no. 99, pp. 1–6, Jan. 2013. [9] Y. Baba and V. A. Rakov, “Voltages induced on an overhead wire by lightning strikes to a nearby tall grounded object,” IEEE Trans. Electromagn. Compat., vol. 48, no. 1, pp. 212–224, Fab. 2006. [10] C. Yang and B. Zhou, “Calculation methods of electromagnetic fields very close to lightning,” IEEE Trans. Electromagn. Compat., vol. 46, no. 1, pp. 133–141, Fab. 2004. [11] J. P. Berenger, “Perfectly matched layer for the FDTD solution of wavestructure interaction problems,” IEEE Trans. Antennas Propag., vol. 44, no. 1, pp. 110–117, Jan. 1996. [12] K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, May 1966. [13] B. B. Mandelbrot, The Fractal Geometry of Nature, New York, USA: W. H. Freeman, 1982. [14] R. Xin-Cheng and G. Li-Xin, “Fractal characteristics investigation on electromagnetic scattering from 2-D weierstrass fractal dielectric rough surface,” Chin. Phys. B, vol. 17, no. 8, pp. 2956, Mar. 2008. [15] L. Guo, R. Wang, and Z. Wu, The Theory and Calculation Method of the Random Rough Surface. China: Science Press, 2009. [16] A. Taflove and S. C. Hagness, Computational Electrodynamics, vol. 160. Boston, London: Artech House, 2000. [17] Y. Baba and V. Rakov, “On the transmission line model for lightning return stroke representation,” Geophys. Res. Lett., vol. 30, no. 24, pp. 2294, Dec. 2003. [18] F. Heidler, “Traveling current source model for LEMP calculation,” in Proc. 6th Int. Zurich Symp. Electromagn. Compat., 1985, pp. 157–162. [19] F. Rachidi, W. Janischewskyj, A. M. Hussein, C. A. Nucci, S. Guerrieri, B. Kordi, and J.-S. Chang, “Current and electromagnetic field associated with lightning-return strokes to tall towers,” IEEE Trans. Electromagn. Compat., vol. 43, no. 3, pp. 356–367, Aug. 2001. [20] V. Rakov and A. Dulzon, “A modified transmission line model for lightning return stroke field calculations,” in Proc. 9th Int. Symp. Electromagn. Compat., 1991, pp. 229–235. [21] F. Delfino, R. Procopio, and M. Rossi, “Lightning return stroke current radiation in presence of a conducting ground: 1. Theory and numerical evaluation of the electromagnetic fields,” J. Geophys. Res., vol. 113, no. D05, pp. D05110, Mar. 2008. [22] F. Delfino, R. Procopio, M. Rossi, F. Rachidi, and C. A. Nucci, “Lightning return stroke current radiation in presence of a conducting ground: 2. Validity assessment of simplified approaches,” J. Geophys. Res., vol. 113, no. D05, pp. D05111, Mar. 2008.
Dongshuai Li was born in Inner Mongolia, China, in 1987. She received the B.E. degree in lightning protection science and technology in 2010 from Nanjing University of Information Science and Technology, Nanjing, China, where she is currently working toward the Ph.D. degree. Her current research interests include electromagnetic field theory, numerical methods of electromagnetics, global lightning activity, and Shumann resonance (SR).
Qinlin Zhang was born in Gansu, China, in 1971. He received the B.S. degree from the Department of Physics, Tianshui Normal University, Gansu, China, in 1995, and the M.S. and Ph.D. degrees in Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, China, in 2002 and 2007, respectively. In 2007, he joined the College of Atmospheric Physics, Nanjing University of Information Science and Technology, China, where he is currently a Professor. He has presided over and attended various national scientific projects. He is the author or coauthor of more than 40 scientific papers published in reviewed journals or presented at national and international conferences. His current research interests include lightning physics, electromagnetic field theory, numerical calculation of electromagnetic fields, global lightning activity, and Shumann resonance (SR).
Zhenhui Wang was born in Shandong, China, in 1955. He received the B.S. degree in the Department of Atmospheric Sounding, Nanjing University of Information Science and Technology (Nanjing Institute of Meteorology), Nanjing, China. He was engaged in atmospheric sounding and atmospheric remote sensing teaching for many years. He is currently a Professor at the College of Atmospheric Physics, Nanjing University of Information Science and Technology. He has been in charge of various national scientific projects, and is the editor of three books. His current research interests include cloud-derived wind, products of meteorological satellite, lightning monitoring, and warning.
Tao Liu was born in Sichuan, China, in 1988. He received the B.E. degree from the College of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing, China, where he is currently working toward the M.S. degree. His current research interests include lightning physical and the numerical methods of lightning electromagnetic field.
