IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 56, NO. 2, APRIL 2014
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Electromagnetic Fields of a Lightning Return Stroke in Presence of a Stratified Ground Abdenbi Mimouni, Farhad Rachidi, Fellow, IEEE, and Marcos Rubinstein, Senior Member, IEEE
Abstract—We present an analysis of the nearby electromagnetic fields generated by lightning discharges in the presence of a horizontally stratified, two-layer ground. To the best of our knowledge, this is the first time the effect of ground stratification on underground fields generated by lightning is analyzed. The analysis is performed by solving Maxwell’s equations using the finitedifference time-domain technique. The return stroke channel is modeled using the modified transmission line model with exponential decay. The effect of the soil stratification on both above-ground fields and the fields penetrating into the ground is illustrated and discussed for two different cases characterized, respectively, by an upper layer more conductive than the lower level, and vice versa. The analysis was carried out for close distances (10 m– 100 m from the channel). It is shown that, for these distances, the ground stratification does not significantly affect the electromagnetic fields above the ground. The above-ground vertical electric field and the azimuthal component of the magnetic field can be calculated assuming the ground as a perfectly conducting plane. The above-ground horizontal electric field is essentially determined by the characteristics of the conductive layer and it can be computed considering a homogeneous ground characterized by the conductive layer conductivity as long as the depth of the upper layer remains below 10 m or so. In general, the fields penetrating into the ground are markedly affected by the soil stratification. The electromagnetic field components inside the stratified soil are generally characterized by faster rise times compared to those of the field components in the case of a homogeneous ground with the upper layer characteristics. The peak value of the horizontal electric field is found to be very sensitive to the ground stratification. The horizontal electric field peak decreases considerably in the presence of a lower layer of higher conductivity. On the other hand, the presence of a lower layer with lower conductivity results in an increase of the peak value of the underground horizontal electric field. Index Terms—Electromagnetic fields, finite-difference time domain (FDTD), lightning, stratified media.
I. INTRODUCTION
O
NE of the first studies on the propagation of electromagnetic waves along a stratified medium is due Wait who
Manuscript received March 11, 2013; revised July 28, 2013; accepted September 7, 2013. Date of publication October 9, 2013; date of current version March 24, 2014. A. Mimouni is with the Laboratoire de G´enie Electrique et des Plasmas, Universit´e Ibn Khaldoun, Tiaret 14000, Algeria (e-mail: abdenbi.mimouni@ gmail.com). F. Rachidi is with the Electromagnetic Compatibility Laboratory, Swiss Federal Institute of Technology, Lausanne 1015, Switzerland (e-mail: Farhad.
[email protected]). M. Rubinstein is with the Institute for Information and communication Technologies, University of Applied Sciences of Western Switzerland, Yverdon-lesBains 1400, Switzerland (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2013.2282995
showed that the concept of attenuation function and surface impedance can be used to represent the effect of a multilayered soil [1]. Wait [2], [3], and Hill and Wait [4] derived the attenuation function for the vertical electric field propagating over a horizontally stratified ground. The theory was then used by Ming and Cooray [5], and Cooray and Cummins [6] to evaluate such effects on lightning return stroke electromagnetic fields. Recently, Shoory et al. [7], [8] examined the accuracy of Wait’s formulations for a horizontally stratified ground taking as a reference full-wave simulations obtained using the finite-difference time-domain (FDTD) technique. They found that Wait’s formula is able to reproduce the vertical electric field peak and waveform with a reasonably good accuracy at far distances (typically 10–100 km) from the lightning channel. Other studies have considered the effect of a mixed propagation path or verticallystratified ground (e.g., [9], [10]). Delfino et al. [11] proposed an efficient algorithm for the evaluation of the exact expressions for the fields generated by a lightning discharge above the horizontally stratified ground. The theory in [11] along with its time-domain implementation was used by Shoory et al. [12] for the assessment of the validity of simplified approaches for the evaluation of electromagnetic fields above a two-layer ground. They also proposed a new formula for the evaluation of the horizontal electric field at a given height above the air–ground interface. The formula can be viewed as the generalization of the Cooray–Rubinstein formula for the case of a two-layer ground. In this paper, we present a comprehensive characterization of the electromagnetic fields, both above and within a twolayer ground, in the immediate vicinity of a lightning channel. For the analysis, Maxwell’s equations are solved using the FDTD technique, and the return stroke channel is modeled using the modified transmission line with exponential decay (MTLE) engineering model. This paper is organized as follows: In Section II, the computational model is briefly described. In Section III, we present the numerical simulations along with the relevant discussion. Finally, conclusions are given in Section IV.
II. ANALYSIS METHOD AND COMPUTATIONAL MODELS A. Electromagnetic Field Computation The electromagnetic fields generated by lightning are computed using the FDTD method, implemented in a FORTRAN code developed by the first author. The details of the technique can be found in [13]–[15]. This technique solves Maxwell’s time-dependent curl equations directly in the time domain by converting them into
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TABLE I ELECTRIC PARAMETERS OF THE TWO-LAYER GROUND
Fig. 1.
Side view of the simulation domain of the FDTD technique.
finite-difference equations that are then solved in a time matching sequence by alternately calculating the electric and magnetic fields in an interlaced spatial grid. For the analysis, the 2-D cylindrical coordinates are adopted (e.g., [14], [16], [17]), and the first-order Mur absorbing boundary conditions [18] are used to truncate the computational domain. The FDTD algorithm requires specific considerations. The grid size should be a fraction of the wavelength. In addition, to avoid numerical instabilities, the time increment should be determined satisfying the Courant stability criterion [16], namely Δt ≤ min(Δr, Δz)/2 c. The function min(Δr, Δz) gives the minimum value between Δr and Δz. In the FDTD simulations, a value of 0.5 ns was selected for the time step. The overall computational time interval was set to Tm ax = 10 μs. This corresponds to 20 000 time steps. The spatial discretization interval was chosen to be 0.5 m. The simulation domain, illustrated in Fig. 1, was truncated at rm ax = 1000 m, zm ax = 1500 m, and zm in = −100 m, using, as previously mentioned, the first-order Mur absorbing boundary conditions (see Fig. 1). Making use of the axial symmetry of the problem (left most boundary coinciding with the z-axis), this corresponds to 2000 × 3200 spatial cells. The simulations were carried out on a 64-bit computational platform having 32 GB of available memory. B. Return Stroke Model The MTLE model [19], [20] was adopted for the analysis and included in the FDTD algorithm. The current decay constant along the channel was assumed to be λ = 2 km and a return stroke speed of v = 1.5 × 108 m/s was adopted. The used channel–base current, represented using Heilder’s functions [21], is typical of subsequent return strokes with a peak value of 12 kA and a maximum steepness of 40 kA/μs (see [22] for the parameters of the function). C. Ground Parameters The adopted values for the electrical parameters of the ground are given in Table I. The depth of the upper soil layer was set to h1 = 5 m. Four cases were considered in the simulations: Cases 1 and 2 are base cases corresponding to a homogeneous soil,
Fig. 2. Vertical electric field 50 m from the lightning channel and 10 m above the ground surface.
while cases 3 and 4 represent two configurations of two-layer soils. III. SIMULATION RESULTS A. Above-Ground Electromagnetic Fields Figs. 2, 3 and 4 present simulation results of the vertical electric field, azimuthal magnetic field, and horizontal electric field for an observation point located at a horizontal distance of 50 m from the lightning channel and at a height of 10 m above the ground surface. It can be seen, in agreement with [23], that the vertical electric field and the azimuthal magnetic field are virtually unaltered by the ground stratification. Indeed, at the considered distance range, these field components are not significantly affected by the ground finite conductivity and they can be evaluated using the approximation of a perfectly conducting plane (e.g., [24]). It is also interesting to observe that the horizontal electric field (see Fig. 4) is nearly identical for cases 2, 3, and 4. In other words, for the considered configuration, this field component is essentially determined by the characteristics of the conductive layer and it can be computed considering a homogeneous ground characterized by the conductive layer conductivity (see also the discussion in [25]). These results are especially useful in the context of lightning induced voltages on overhead transmission lines, since the horizontal electric field component at
MIMOUNI et al.: ELECTROMAGNETIC FIELDS OF A LIGHTNING RETURN STROKE IN PRESENCE OF A STRATIFIED GROUND
Fig. 3. Azimuthal magnetic field 50 m from the lightning channel and 10 m above the ground surface.
Fig. 4. Horizontal electric field 50 m from the lightning channel and 10 m above the ground surface.
the height of the line (typically of the order of 10 m) is the source component used in the coupling model of Agrawal et al. [26]. In order to verify if such observations are still valid for different depths of the upper layer, we recalculated the horizontal electric field at a distance of 50 m from the lightning channel and at a height of 10 m above ground, setting the depth of the upper soil layer to h1 = 10 m and 20 m.The results are illustrated in Figs. 5 and 6, respectively. It can be seen that changing the depth of the upper layer from h1 = 5 m to h1 = 10 m, does not significantly affect the horizontal electric field (compare Figs. 4 and 5). On the other hand, for h1 = 20 m (see Fig. 6), one can see that when the upper layer is less conductive (case 3), the early-time response of the horizontal electric field is essentially determined by the upper layer, while its late-time response is governed by the lower, more conductive layer. It is important to note that the fact that the effect of the ground’s finite conductivity and/or stratification on the electromagnetic field components is not significant is essentially due to the fact that we are considering observation points at very close distances to the lightning channel. At farther distances, the electromagnetic fields will be affected by the ground losses/stratification in a more significant way (see e.g. [23]).
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Fig. 5. Horizontal electric field at a distance of 50 m from the lightning channel and at a height of 10 m above the ground surface (h 1 = 10 m).
Fig. 6. Horizontal electric field at a distance of 50 m from the lightning channel and at a height of 10 m above the ground surface (h 1 = 20 m).
B. Underground Electromagnetic Fields Computational simulations were performed to calculate the vertical electric field, azimuthal magnetic field, and horizontal electric field for an observation point located at a horizontal distance of 50 m from the lightning channel and at a depth of 3 m. The results for case 3 (upper layer with a lower conductivity, see Table I) are presented in Figs. 7–9. On the same plots, we have shown the results corresponding to the case of a homogeneous soil with the upper layer electrical characteristics (case 1). It can be seen from Figs. 7–9 that among the three field components, the horizontal electric field is the one that is more markedly affected by the soil stratification. In general, the electromagnetic field components inside the stratified soil (upper layer with lower conductivity) are characterized by faster rise times compared to the rise times of the field components in the case of a homogeneous ground (with the same properties of the upper layer). The ground stratification results in a slight increase (in the order of 10–15%) of the peak values of the vertical electric field and the azimuthal magnetic field. On the other hand, the horizontal electric field decreases by a factor of 3 or so in the case of a stratified ground with the considered parameters.
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Fig. 7. Vertical electric field at a depth of 3 m inside (i) a homogeneous ground (case 1) and (ii) a stratified ground (case 3).
Fig. 8. Azimuthal magnetic field at a depth of 3 m inside (i) a homogeneous ground (case 1) and (ii) a stratified ground (case 3).
In Figs. 10, 11 and 12, we present the electromagnetic field waveforms at a depth of 3 m for case 4 (upper layer with higher conductivity, see Table I). Again, on the same plot, we have included the results corresponding to the case of a homogeneous soil with the upper layer electrical characteristics (case 2). It can be seen, again, that the stratification affects all the field components penetrating into the ground. In this case, the vertical electric field is characterized by a shorter width and an inversion of polarity at about 2 μs. The azimuthal magnetic field peak is reduced by about 30% due to the presence of the second layer. Finally, the presence of the second ground layer results in an increase of the horizontal electric field peak of about 20%. Table II summarizes the underground electromagnetic field peaks and rise times for the considered cases. Furthermore, one can see in Fig. 11 that the magnetic field for the two-layer case is lower than for the case of a single layer. In Fig. 12, however, one can see that the horizontal electric field exhibits the opposite behavior. A possible explanation for this fact is that the reflection coefficients for the electric and magnetic field at the boundary between the higher-conductivity upper layer and the lower-conductivity lower layer have oppo-
Fig. 9. Horizontal electric field at a depth of 3 m inside (i) a homogeneous ground (case 1) and (ii) a stratified ground (case 3).
Fig. 10. Vertical electric field at a depth of 3 m inside (i) a homogeneous ground (case 2) and (ii) a stratified ground (case 4).
Fig. 11. Azimuthal magnetic field at a depth of 3 m inside (i) a homogeneous ground (case 2) and (ii) a stratified ground (case 4).
site signs. The horizontal electric field sees a positive reflection coefficient and the magnetic field encounters a negative reflection coefficient. The reported conclusions remain valid for distances ranging from 10 m to 100 m for which simulations were carried out but not shown here.
MIMOUNI et al.: ELECTROMAGNETIC FIELDS OF A LIGHTNING RETURN STROKE IN PRESENCE OF A STRATIFIED GROUND
Fig. 12. Horizontal electric field at a depth of 3 m inside (i) a homogeneous ground (case 2) and (ii) a stratified ground (case 4). TABLE II UNDERGROUND ELECTROMAGNETIC FIELD PEAKS AND RISE TIMES
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2) The above-ground vertical electric field and the azimuthal component of the magnetic field can be calculated assuming the ground as a perfectly conducting plane. 3) The above-ground horizontal electric field is essentially determined by the characteristics of the conductive layer and it can be computed considering a homogeneous ground characterized by the conductive layer conductivity as long as the depth of the upper layer remains below 10 m or so. 4) In general, the fields penetrating into the ground are markedly affected by the soil stratification. 5) The electromagnetic field components inside the stratified soil are generally characterized by faster rise times compared to those of the field components in the case of a homogeneous ground with the upper layer characteristics. 6) The peak value of the horizontal electric field is found to be very sensitive to the ground stratification. The horizontal electric field peak decreases considerably in the presence of a lower layer of higher conductivity. On the other hand, the presence of a lower layer with lower conductivity results in an increase of the peak value of the underground horizontal electric field.
REFERENCES
IV. CONCLUSION In this paper, we presented an analysis of the nearby electromagnetic fields generated by lightning discharges in the presence of a stratified, two-layer ground. To the best of our knowledge, this is the first time the effect of ground stratification on underground fields generated by lightning is analyzed. The analysis was performed by solving Maxwell’s equations using the FDTD technique. The return stroke channel was modeled using the MTLE model with current parameters typical of subsequent return strokes. The effect of the soil stratification on both, the above-ground fields and the fields penetrating into the ground were illustrated and discussed for two different cases characterized, respectively, by an upper layer more conductive than the lower level, and vice versa. The main conclusions from the analysis presented in this paper and which are applicable to close distance ranges (within 100 m or so) are as follows. 1) The ground stratification does not appreciably affect the above-ground electromagnetic fields.
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[13] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp. 302–307, May 1966. [14] A. Mimouni, F. Rachidi, and Z. Azzouz, “Electromagnetic environment in the immediate vicinity of a lightning return stroke,” J. Lightning Res., vol. 2, pp. 64–75, 2007. [15] A. Mimouni, F. Rachidi, and Z. Azzouz, “A finite-difference timedomain approach for the evaluation of electromagnetic fields radiated by lightning to tall structures,” J. Electrostatics, vol. 866, pp. 504–513, 2008. [16] 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, Feb. 2004. [17] A. Mimouni, F. Delfino, R. Procopio, and F. Rachidi, “On the computation of underground electromagnetic fields generated by lightning: A comparison between different approaches,” presented at the IEEE PES PowerTech, Lausanne, Switzerland, Jul. 1–5, 2007. [18] G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 4, pp. 377–382, Nov. 1981. [19] C. A. Nucci, C. Mazzetti, F. Rachidi, and M. Ianoz, “On lightning return stroke models for LEMP calculations,” presented at the 19th Int. Conf. Lightning Protection, Graz, Austria, Apr. 1988. [20] F. Rachidi and C. A. Nucci, “On the Master, Uman, Lin, Standler and the modified transmission line lightning return stroke current models,” J. Geophysical Res., vol. 95, pp. 20389–20394, 1990. [21] F. Heidler, “AnalytischeBlitzstrom function zur LEMP-Berechnung,” in Proc. 18th Int. Conf. Lightning Protection, Munich, Germany, 1985, pp. 63–66. [22] 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. [23] A. Shoory, F. Rachidi, and V. Cooray, “Propagation effects on electromagnetic fields generated by lightning return strokes: Review of simplified formulas and their validity assessment,” Lightning Electromagnetics, Stevenage, U.K.: IET, 2012, Ch. 12, pp. 485–513. [24] F. Rachidi, C. A. Nucci, M. Ianoz, and C. Mazzetti, “Influence of a lossy ground on lightning-induced voltages on overhead lines,” IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 250–264, Aug. 1996. [25] F. Delfino, R. Procopio, M. Rossi, A. Shoory, and F. Rachidi, “The effect of a horizontally stratified ground on lightning electromagnetic fields,” presented at the IEEE Int. Symp. Electromagn. Compat., Fort Lauderdale, FL, USA, Jul. 2010. [26] A. K. Agrawal, H. J. Price, and S. H. Gurbaxani, “Transient response of multiconductor transmission lines excited by a nonuniform electromagnetic field,” IEEE Trans. Electromagn. Compat., vol. EMC-22, no. 2, pp. 119–129, May 1980.
Abdenbi Mimouni was born in Algeria, on October 16, 1970. He received the Engineer’s and Magister’s degrees from the University of Tiaret, Tiaret, Algeria, and the University of Sciences and Technology of Oran, Oran, Algeria, both in electrical engineering, in 1994 and 2000, respectively, and the Ph.D. degree from the University of Sciences and Technology of Oran, Oran, Algeria, in December 2007. From 2000 to 2006, he was with the Department of Electrical Engineering, University of Tiaret, Algeria, as an Assistant and the LGP Laboratory, at the same university, as a Researcher. From April 2006 to August 2007, he was with the EMC group in the Power Systems Laboratory of Swiss Federal Institute of Technology, Lausanne, Switzerland as an Invited-Assistant. He is currently a Lecturer and Researcher in the Department of Electrical Engineering, University of Tiaret. His research interests include electromagnetic field theory, numerical techniques applied to electromagnetic compatibility and lightning electromagnetics. He is author or coauthor of more than 40 scientific papers published in reviewed journals or presented at national and international conferences.
Farhad Rachidi (M’93–SM’02–F’10) received the M.S. degree in electrical engineering and the Ph.D. degree from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1986 and 1991, respectively. He worked at the Power Systems Laboratory of the same institute until 1996. In 1997, he joined the Lightning Research Laboratory of the University of Toronto in Canada and from April 1998 to September 1999, he was with Montena EMC in Switzerland. He is currently a Titular Professor and the Head of the electromagnetic compatibility Laboratory at the Swiss Federal Institute of Technology, Lausanne, Switzerland. He is the author or coauthor of over 300 scientific papers published in reviewed journals and presented at international conferences. Dr. Rachidi served as the Vice-Chair of the European COST Action on the Physics of Lightning Flash and its Effects (2005–2009) and the Chairman of the 2008 European Electromagnetics International Symposium. He is currently the Editor-in-Chief of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, the President of the International Conference on Lightning Protection, and the President of the Swiss National Committee of the International Union of Radio Science. He received the IEEE Technical Achievement Award and the CIGRE Technical Committee Award in 2005. In 2006, he received the Blondel Medal from the French Association of Electrical Engineering, Electronics, Information Technology and Communication.
Marcos Rubinstein (SM’11) received the Bachelor’s degree in electronics from the Universidad Simon Bolivar, Caracas, Venezuela, in 1982, and the Master’s and Ph.D. degrees in electrical engineering from the University of Florida, Gainesville, FL, USA, in 1986 and 1991, respectively. In 1992, he joined the Swiss Federal Institute of Technology in Lausanne, where he was active in the fields of electromagnetic compatibility and lightning in close cooperation with the former Swiss PTT. In 1995, he took a position at Swisscom, where he was involved in numerical electromagnetics and electromagnetic compatibility (EMC) in telecommunications and led a number of coordinated projects covering the fields of EMC and biological effects of electromagnetic radiation. In 2001, he moved to the University of Applied Sciences of Western Switzerland HES-SO, Yverdon-les-bains, Switzerland, where he is currently a Professor in telecommunications and a Member of the IICT institute team. His current research interests include lightning, EMC in telecommunication systems, PLC, wireless technologies, and layer-2 network security. He is the author or coauthor of more than 100 scientific publications in reviewed journals and international conferences. Prof. Rubinstein received the best Master’s Thesis award from the University of Florida. He received the IEEE achievement award and is a corecipient of NASA’s recognition for innovative technological work. He is a member of the Swiss Academy of Sciences and of the International Union of Radio Science.
