Fdtd calculation model for the transient analyses of grounding systems

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 56, NO. 5, OCTOBER 2014 1155

FDTD Calculation Model for the Transient Analysesof Grounding Systems

Run Xiong, Bin Chen, Member, IEEE, Cheng Gao, Member, IEEE, Yun Yi, and Wen Yang

Abstract—Transient performance of grounding systems is sim-ulated using the finite-difference time-domain (FDTD) method forsolving Maxwell’s equations. The derivation of the transient cur-rent is first adjusted to obtain accurate impedance. Then, the FDTDcalculation model is optimized to predict the grounding systemimpedance accurately without resulting in huge computational re-sources. From evaluation, the direction parallel to the connectingline is supposed to integrate the transient voltage. The transientvoltage should be integrated 20 m in length and the reference elec-trode to be 30 m from the grounding system, and these lengthsshould be further enlarged for a large dimension grounding sys-tem. The transient impedance of the five lightning current compo-nents is very close to each other except the multiple burst, and thecurrent should be injected close to the ground.

Index Terms—Calculation model, finite-difference time-domain (FDTD) method, grounding system, transient groundingimpedance.

I. INTRODUCTION

GROUNDING systems are often parts of the lightning pro-tection systems, which provide a channel for current to

flow into ground. These systems should have sufficiently lowimpedance and current-carrying capacity to prevent voltage risethat may result in undue hazard to connected equipments and topersons [1], [2]. Grounding electrode performance under normaland fault conditions is well understood [3]. However, the dy-namic behavior of grounding electrodes might be quite differentduring lightning discharge.

Transient response of grounding systems has been inves-tigated by experimental work [4], [5], simplified computa-tional model [6]–[15], and numerical analyses based on themethod of moments, finite element method, and finite-differencetime-domain (FDTD) method for solving Maxwell’s equations[16]–[22].

The FDTD method [23], which has been widely applied inanalyzing many types of electromagnetic problems, is very suit-

Manuscript received October 8, 2013; revised December 27, 2013 and Febru-ary 17, 2014; accepted March 25, 2014. Date of publication April 18, 2014; dateof current version September 26, 2014. This work was supported in part by theChinese National Science Foundation under Grant 51277182, Grant 41305017,and Grant 61271106, and in part by the School Foundation under Grant KYGY-ZLYY1306.

R. Xiong is with the National Key laboratory on Electromagnetic Environmentand Electro-optical Engineering, PLA University of Science and Technology,Nanjing 210007, China and also with Command Institute of Engineering Corps,Xuzhou 221004, China (e-mail: [email protected]).

B. Chen, C. Gao, and Y. Yi are with the National Keylaboratory on Electromagnetic Environment and Electro-optical Engineering,PLA University of Science and Technology, Nanjing 210007, China (e-mail:[email protected]; [email protected]; [email protected]).

W. Yang is with the Engineering and Design Institute, Chengdu Military Areaof PLA, Yunnan 650222, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2014.2313918

able for the transient analyses of grounding systems. In [19]pioneer work, the FDTD method is used to evaluate the hori-zontal grounding electrode performance, as it can simulate thetransient and steady-state characteristics of grounding systems(low-frequency performance).

The FDTD calculation model is widely used to investigatethe grounding system [19]–[22] and thin wire performances[24]–[26]. The transient voltage is integrated 8–50 m verticaland parallel to the current lead wire in [19], while integrated 50 mvertical to the current lead wire in [20] and [22]. The distancebetween the reference electrode and the grounding system is setto be 8–15 m in [19], 20–50 m in [20], and 30 m in [24]–[26].The connecting line height is 1 m in [24]–[26] and 4 m in [19].

The transient current is derived from magnetic field integralalong the FDTD cell edge in [20] and [22], but the magnetic fieldis varied along the edge in an FDTD cell. In [20], parallel im-plementation is introduced to analyze a large grounding system,but the FDTD calculation model has not been evaluated. Whilein [19], transient responses of a horizontal grounding electrodeis analyzed to evaluate the voltage reference direction effect, butthe computational domain and the voltage reference wire lengtheffect are not considered.

In this paper, the derivation of the transient current is first ad-justed to obtain accurate impedance. Then, the transient ground-ing impedance calculation model is evaluated to find the opti-mized parameters to predict the grounding impedance accu-rately without resulting in huge computational resources. Fromevaluation of three grounding system impedance under variedvoltage integrating path length, reference electrode length anddistance and current type, and injected height, the optimizedFDTD calculation model parameters are derived to predict tran-sient impedance accurately without resulting in huge computa-tional resources.

II. TRANSIENT IMPEDANCE CALCULATION MODEL

The computational model as shown in Fig. 1 is adopted[19]–[22]. A remote electrode, which is used to provide a pathfor current flowing into ground, is placed parallel to and awayfrom the grounding electrode. The grounding electrode and thereference electrode are connected by vertical lifting lines andconnected by an overhead horizontal thin wire.

In Fig. 1, Lc is the connecting line length, hc is the connectingline height from the ground, Li is the transient voltage integrat-ing path length, lr is the reference electrode length, and hs isthe injected source height.

A homogenous ground is involved in this paper, and it isassumed that the ground has a constant constitutive parameter.

0018-9375 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

1156 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 56, NO. 5, OCTOBER 2014

Fig. 1. Transient impedance calculation model.

The conductivity is set at σg = 0.005 S/m, and the relativepermittivity of the ground is εr g = 10.

Lightning restrike is selected as the current source, which canbe represented by

I(t) = I0(e−αt − e−βt) (1)

where I0 = 109 405 V/m, α = 22 708 s−1 , β = 1 294 530 s−1 .The transient impedance is defined as a ratio of the transient

voltage to the transient current [10], [27]

Z(t) =V (t)I(t)

. (2)

A. Transient Voltage

The transient voltage V (t) is the transient potential from thelifting line to an infinite distance, which can be derived from

V (t) =∫ ∞

Nl

E · dl (3)

where Nl is FDTD mesh index of the point of the lifting lineentering ground in Fig. 1.

In the FDTD analysis, the voltage between the two sides of acell can be defined as [20]–[23]

Vj = Ej · Δsj . (4)

By integrating the electric field along the air–ground interfacefrom the lifting line to the computational domain boundary (seepoint K of Fig. 1), the transient voltage V (t) can be obtained

V (t) = −NK∑

j=Nl

Vj = −NK∑

j=Nl

EjΔsj (5)

where Ej is the electric field component in the air–ground inter-face, Δsj is the grid dimension, and NK is FDTD mesh indexesof the point K of Fig. 1.

B. Transient Current

To derive the current I(t) injected to the grounding system,Ampere’s circuit law is applied to the FDTD cells containingthe grounding system lifting line [20] as shown in Fig. 2, and the

Fig. 2. Ampere’s loop to derive the transient current.

Fig. 3. Three used grounding systems, where system S is a single verticalelectrode, system T is a line-set system composed of three vertical electrodes,and system F is a cross-set system composed of five vertical electrodes.

calculation would refer to the space distribution of the currentin the ground. In [20] and [22], I(t) is derived from magneticfield integral along the FDTD cell edge. However, as pointedout in [24] and [28], the magnetic field is varied as 1/r near aline, where r is the distance from the metal line. The distancefrom the integral path to the lifting line is varied from Δx /2 at(i0 , j0 − 1

2 , k0 + 12 ) to

√Δ2

x + Δ2z /2 at (i0 − 1

2 , j0 − 12 , k0 +

12 ) for the Hx component, which means the magnetic field Hx at(i0 , j0 − 1

2 , k0 + 12 ) is

√2 times larger than that at (i0 − 1

2 , j0 −12 , k0 + 1

2 ) for the case that Δ = Δx = Δz .In this paper, the current I(t) is derived from integration of

the magnetic field in the Ampere’s circuit whose radius is Δ/2,as shown in Fig. 3, and I(t) can be obtained from

I(t) = πΔHΔ/2 . (6)

A round Ampere circuit is used here instead of square cir-cuit [20], [22]. The magnetic field component along the roundintegral path, whose radius is Δ/2, is constant. Thus, the I(t)derived from (6) is an more accurate simulation of the groundingsystem injected current.

The magnetic field component HΔ/2 in (6) is approximatedfrom the weighted average of the four magnetic field compo-nents adjacent to the lifting line (7), as shown at the bottom ofthe next page, where (i0 , j0 , k0) is the point of the lifting lineentering ground. Substituting (7) into (6) derives

I(t) =

[Hz

(i0− 1

2 , j0− 12 , k0

)−Hz

(i0 + 1

2 , j0− 12 , k0

)]πΔz

4

+

[Hx

(i0 , j0− 1

2 , k0 + 12

)−Hx

(i0 , j0− 1

2 , k0− 12

)]πΔx

4.

(8)

XIONG et al.: FDTD CALCULATION MODEL FOR THE TRANSIENT ANALYSES OF GROUNDING SYSTEMS 1157

Fig. 4. Four typical transient voltage integrating paths.

The three selected grounding systems in this paper are shownin Fig. 3, where the vertical electrode radius is 1 cm. The radiusof the lifting line, connecting line and reference electrode is1 mm. Cubic FDTD cells with the grid dimension Δ = Δx =Δy = Δz = 0.15 m is used and the time step is Δt = Δ/2 c,where c is the speed of light in the free space. The computationaldomain is terminated by a 15-layer CPML [29].

III. EVALUATION OF THE CALCULATION MODEL

In this section, transient voltage integrating path direction andlength, reference electrode length and distance, current typeand injected height are evaluated, respectively, to derive theoptimized transient impedance calculation model parameters.

A. Transient Voltage Integrating Direction

The transient voltage integrating direction effect on the tran-sient impedance of the grounding system is analyzed in this part.There are four typical paths for integrating the transient voltage,as shown in Fig. 4. Here, path K1 is below the connecting line inthe −z direction; K2 is parallel to the connecting line and to thecomputational edge in the z-direction; K3 and K4 are verticalto the connecting line and to the computational edge in x and−x direction, respectively.

First of all, the field distribution in the air–ground interfaceis evaluated. Considering that the integrated field component isEx in path K1 ,K2 and Ez in K3 ,K4 , the |Ex | and |Ez | fielddistribution in the air–ground interface is observed, respectively,as shown in Fig. 5, where Lc = 10 m and Li = 6 m.

It can be seen from Fig. 5 that the electric field componentdecreases rapidly from the lifting line and the reference elec-trode to the surrounding areas. Both the |Ex | and |Ez | fields aresymmetrically distributed in the x-direction in the air–groundinterface.

Fig. 5. Electric field distribution in the air–ground interface when system S isused as the grounding system and L = 10 m: (a) the |Ex | distribution and (b)the |Ez | distribution.

From Fig. 5(a), one can see that field strength singularitiesoccur in the z-direction near the lifting line and the referenceelectrode and under the connecting line for the Ex field compo-nent. Additionally, the Ex field is much larger in the x-directionthan that the z-direction at the same distance from the liftingline and the reference electrode.

From Fig. 5(b), we can see that singularities occur in the xdirection near the lifting line and the reference electrode for theEz field component. Additionally, the Ez field is much smallerin the x-direction than that the z-direction at the same distancefrom the lifting line and the reference electrode in the air–groundinterface. What’s more, the Ez field in path K1 is larger thanthat in path K2 at the same distance from the lifting line.

The electric field in K2 is absolutely determined by the currentdissipating through the grounding system, while the electricfield in K1 is mainly determined by the grounding system andstrongly enhanced by the reference electrode. Additionally, thefield in K3 and K4 is mainly determined by the groundingsystem and affected by the reference electrode to some extend.Thus, the voltage derived from K1 is much larger than that fromthe other paths, while the voltage derived from K2 is a littlelarger than that from K3 and K4 .

HΔ/2 =1

4Δ

⎧⎪⎪⎨⎪⎪⎩

[Hz

(i0 −

12, j0 −

12, k0

)− Hz

(i0 +

12, j0 −

12, k0

)]Δz

+[Hx

(i0 , j0 −

12, k0 +

12

)− Hx

(i0 , j0 −

12, k0 −

12

)]Δx

⎫⎪⎪⎬⎪⎪⎭

(7)


Fig. 6. Transient impedance calculated from the three transient voltage inte-grating paths.

Second, the transient impedance of grounding system S andF calculated from different transient voltage integrating pathsis plotted in Fig. 6, where the path K4 impedance is not graphedbecause of its symmetry to path K3 .

It can be seen from Fig. 6 that the transient voltage integratingpath K1 gives much larger transient impedance than the otherpaths, which enhances the conclusion drawn from Fig. 5. It canalso be seen from Fig. 6 that the impedance derived from pathK2 and K3 is very close to each other, and both of them canbe occupied. However, the involved computational domain isquite different when the two paths are occupied to integrate thetransient voltage.

It worth to note that the calculated resistance for a singleelectrode at late time as shown in Fig. 6(a) is 64.1 Ω, while thecalculated quasistatic resistance from [30] is 64.8 Ω. The tworesults show good agreement with each other.

Third, the computational domain usage is compared whenpath K2 and K3 are involved. It is demonstrated in Part B thatthe transient voltage integrating length path should be 20 mand Part C points out that the distance between the lifting lineand the reference electrode should be L = 30 m. The distancebetween the computational edge and the grounding system is6 m here. Thus, the computational domain is 26 m × 42 m inthe xoz plane when path K3 is occupied, compared with 12 m ×56 m when path K2 is occupied. This means the computationaldomain when path K2 is occupied is only 65% of that whenpath K3 is occupied.

The initial impedance peak is a manifestation of the groundplane surge response, and it is affected by the lifting wire heightand the ground parameter [31]–[34].

Therefore, it can be concluded that the transient voltage inte-grating path K2 , which is parallel to the connecting line in thez-direction, is the optimized path to integrate the transient volt-age. In the following analyses, path K2 is used to integrate thetransient voltage. The conclusion drawn here show good agree-ment with the Zed-Meter calculations result [35], [36] usingfrequency-domain methods (NEC 4).

B. Transient Voltage Integrating Path Length

To get the transient voltage accurately, it is needed to integratethe electric field from the lifting line to a point where the electric

Fig. 7. Transient voltage integrating path length effect on the grounding sys-tem impedance. (a) Transient impedance of various integrating path lengths,where the impedance at infinite distance from the grounding system derivedfrom linear regression is also graphed. (b) Steady impedance of groundingsystems versus the integrating path length in different soil conductivity.

field vanishes. However, in the numerical calculation it is im-possible to simulate so large a domain. Thus, it is necessary tofind the length convergence of the transient voltage integratingpath with respect to acceptable accurate.

To observe the transient voltage integrating path length effecton the grounding system impedance, the transient impedanceof the grounding system at varying transient voltage integratingpath lengths is calculated as shown in Fig. 7(a). To providea benchmark, linear regression of impedance versus inversedistance is used to derive the impedance at infinite distance [36].By taking linear regression of the resistance when Li = 2, 4, 6,and 8 m against the inverse of distance, the impedance at infinitedistance are obtained, which are also graphed in Fig. 7(a).

It can be seen that the transient voltage integrating path lengthdoes not affect the transient performance of the grounding sys-tem at the first 0.1 μs, because the current has only dissipatedto a limited area near the grounding system. However, the pathlength effect the transient impedance appears as time goes on.The steady impedance increases as the length of transient volt-age integrating path increases but the slope decreases.

To determine the transient voltage integrating path length,the path length is varied from 0.15 to 200 m and the transientimpedance is calculated. Fig. 7(b) plots the steady impedancevariation versus integrating path length when the three


TABLE IRELATIVE STEADY IMPEDANCE ERROR Rerror COMPARED WITH THAT WHEN

INTEGRATING TO AN INFINITE DISTANCE AWAY (%)

grounding systems are considered, where soil conductivity ef-fect can also be evaluated. It can be seen that steady impedanceincreases significantly as integrating path length increaseswhen it is shorter than 10 m, and the steady impedance does notincrease obviously when the integrating path length is longerthan 20 m. From comparison of the impedance of groundingsystem F at different soil conductivity, it can also be seen thatthe steady impedance is much lower for better conductivity soilat the same grounding system and integrating path length.

To show the steady impedance convergence as integratinglength increases, the relative steady impedance error

Rerror =Zf − Zl

Zf(9)

is chosen, where Zf is impedance extrapolated to infinite dis-tance from the grounding system derived from linear regression,and Zl is the steady impedance when the integrating length is Li .Table I gives the relative steady impedance error Rerror whenthe integrating path length is increased from 10 to 50 m.

It can be seen from Fig. 7(b) that better ground conductivityand large dimension grounding system result in lower steadyimpedance. However, from comparing the relative error whenthe grounding conductivity is 0.005 and 0.01 S/m in Table I,it can be seen that the ground conductivity affects the lengthconvergence of the transient voltage integrating path slightly.

From comparison of the relative impedance error of ground-ing system S, T , and F , it can be seen a longer integrating pathlength is needed to obtain the impedance convergence for a largedimension grounding system.

As can be seen from Fig. 7(a), the transient impedance ofthe peak impedance is hardly affected by the transient voltageintegrating path length. While the transient impedance varia-tion versus time after 0.2 μs is in accordance with the steadyimpedance variation. So the integrating path length conclusionsdrawn from the analysis of the steady impedance can be appliedto the transient voltage integrating path length.

Therefore, the transient voltage should be integrated at least20 m for a small dimension grounding system and the voltageintegrating path should be further enlarged when a much largerdimension grounding system is involved.

C. Reference Electrode

To derive the grounding system impedance accurately, thereference electrode should be far enough from the groundingsystem to ensure that the return reflection from the referenceelectrode will not arrive before the FDTD simulation is termi-

Fig. 8. Electric field |Ez | distribution in the air–ground interface at differentreference electrode distances when system S is used as the grounding system:(a) Lc = 10 m and (b) Lc = 15 m.

nated. However, that would result in huge computational re-sources or even make it impossible to be simulated. Thus, it isneeded to find a reasonable reference electrode program for theFDTD simulation.

To analyze the reference electrode effect on the transientimpedance, the Ez component distribution in the air–groundinterface is first monitored. Then, the transient grounding sys-tem impedance at different reference electrode distances is alsocompared with each other to derive the optimized referenceelectrode position. Third, the length of the reference electrodeis chose.

First, electric field component |Ez | distribution in the air–ground interface when Lc = 10 m as shown in Fig. 8(a) iscompared with that when Lc = 15 m, as shown in Fig. 8(b).The reference electrode is located at the point (6 m, 6 m) andthe grounding electrode is located at the point (6 m, 16 m) forFig. 8(a) and (6 m, 21 m) for Fig. 8(b).

As graphed in Fig. 8, the electric field Ez decreases from thelifting line and the reference electrode to the surrounding areas.Additionally, the region area of the same field strength at theinner side is larger than that at the outer side.

From Fig. 8(a), it can be seen that the electric field compo-nent near the grounding electrode is seriously affected by thereference electrode field when Lc = 10 m. However, when theconnecting line length is enlarged to Lc = 15 m, the effect ofthe reference electrode on the field near the grounding electrodeis greatly eased. It can also be seen that the inner side high field


Fig. 9. Transient impedance of varied reference electrode distances from thegrounding system lifting line.

TABLE IIREFLECTION REACHES TIME AS THE REFERENCE ELECTRODE

DISTANCE VARIES

strength region area when Lc = 10 m is larger than that whenLc = 15 m.

From comparing Fig. 8(a) and (b), it is easy to see that thecalculated impedance when Lc = 10 m is more seriously af-fected by the mutual impedance of the reference electrode thanthat when Lc = 15 m. Thus, the reference electrode should beset to a certain distance away from the grounding electrode toreduce the mutual resistance.

Second, to observe the reference electrode distance effecton the system impedance, the transient impedance of the threegrounding systems as the reference electrode distance Lc variedfrom 6 to 250 m is calculated and graphed in Fig. 9. The transientimpedance when Lc = 250 m was selected as a reference valuehere, because the return reflection introduced by the referenceelectrode has not reached for only a 1.5 μs simulation is involvedhere.

It can be seen from Fig. 9 that the transient impedance of 0–0.2 μs when Lc = 30 m is the same as that when Lc = 250 m, butsingularity occurs at 0.2 μs. Similar conclusions can be drawnfor Lc = 50 m and Lc = 80 m, but the time when the impedancesingularity occurs is delayed as the reference electrode distanceenlarges.

To evaluate the relationship between the transient impedancesingularity occur time and the reference electrode distance Lc ,Table II gives the singularity occurs time as Lc varies from 30to 120 m, where the electromagnetic wave propagating distanceat these times is also given.

From Table II, it can be seen that the singularity occur time isvery close to the time that the electromagnetic wave propagatesto the reference electrode and back. Thus, it can be concludedthat the reflections in Fig. 9 are mainly brought about by the mis-match between connecting line surge impedance and referenceelectrode resistance. An efficient way of reducing the mismatchis selecting a matching resistor [37]. It can be demonstrated that

Fig. 10. Transient impedance at varied reference electrode lengths.

TABLE IIILIGHTNING INDIRECT EFFECT WAVEFORM PARAMETERS

the matching resistor can diminish the reflection by a factor ofthree or more.

To avoid singularity brought about by reference electrode,it should be set to be far enough from the grounding system tomake sure that the reflection will not reach before the simulationis terminated. However, it is difficult to simulate so large a do-main in the numerical calculation, and we suggest the referenceelectrode to be 30 m from the lifting line.

Third, the length of the reference electrode is varied from 1to 50 m when Lc = 30 m, and the transient impedance of thegrounding system is calculated and graphed in Fig. 10. It canbe seen that the reference electrode length has limited effect onthe grounding system impedance and the impedance when thelr = 3 m is very close to that when lr = 50 m. Thus, a 3-m-longreference electrode can be efficient.

Therefore, the reference electrode should be located at least30 m for a small dimension grounding system and the dis-tance should be further enlarged when a much larger dimensiongrounding system is involved. The reference electrode shouldbe lr = 3 m in length.

D. Injected Source

In this part, both the injected source type and height effecton the transient impedance are considered. First, the five cur-rent components [38] of different lightning steps, as shown inTable III, are occupied as the injected source, and the transientimpedance is calculated and graphed in Fig. 11(a).

Fig. 11(a) shows that the transient impedance is very closeto each other when the current components A,B, and D areused as the injected source. However, the transient impedance


Fig. 11. Injected current component effect on the transient impedance.(a) Transient impedance of different current components. (b) Transientimpedance at different injected positions.

of the current component H is quite different from the transientimpedance of the other current components before the time1.0 μs, because the frequency spectra of A,B, and D are veryclose to each other, while H is quite different from the others.

Second, the current component D is selected as the sourceand the injected position effect on the transient impedance isanalyzed. Here, the injected height is varied from hc = 0.225to 2.025 m. The transient impedance is calculated as shown inFig. 11(b), where the 0–0.06 μs transient impedance of systemS is zoomed. The current component is introduced through theAmpere’s law, which means the injected source is located at halfan FDTD cell position. The transient current is integrated halfan FDTD cell above the ground (j0-0.5)Δ, thus the source caninjected 1.5Δ (0.225 m) or higher above the ground.

As can be seen from Fig. 11(b) that the source injected po-sition can hardly affect the transient impedance after 0.1 μs,but the peak impedance increases as the injected current heightincreases. The peak impedance is 83.1 Ω, when hc = 0.225 m,and increase to 86.2, 89.9, 96.2, and 100.0 Ω as hc increases to0.525, 0.975, 1.575, and 2.025 m, respectively. Additionally, thehigh injected source position results in a late peak impedancevalue time.

Therefore, the multiple burst transient impedance is quitedifferent from the other four lightning steps and the currentshould be injected at a low position above the ground to get thereal peak impedance of the grounding system.

IV. CONCLUSION

In this study, the derivation of the transient current is firstadjusted to obtain accurate impedance. Then, the FDTD cal-culation model is evaluated in order to predict the transientgrounding system impedance accurately without resulting inhuge computational resources. From evaluation, it can be con-cluded that:

1) The transient voltage should be integrated along the pathparallel to the connecting line.

2) The transient voltage should be integrated at least 20 mfor a small dimension grounding system and longer for alarge dimension grounding system.

3) The reference electrode should be located at least 30 mfrom the grounding system for a small dimension systemand the distance should be further enlarged when a largedimension grounding system is involved. The referenceelectrode should be 3 m in length.

4) Among the current components of different lightningsteps, the transient impedance of the current componentmultiple burst is different from the other steps. The currentshould be injected at the point close to the ground.

The adjusted and optimized FDTD calculation model wouldbe useful in impedance calculation of the grounding systems.

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Run Xiong was born in Sichuan, China, in 1983.He received the B.S. and M.S. degrees in electricsystems and automation in 2005 and 2010, respec-tively, from the Engineering Institute of Corps of En-gineers, PLA University of Science and Technology,Nanjing, China, where he is currently working towardthe Ph.D. degree.

He is currently with the National Key Laboratoryon Electromagnetic Environment and Electro-opticalEngineering, PLA University of Science and Tech-nology. His current research interests include com-

putational electromagnetics and EMC.

Bin Chen (S’02–M’03) was born in Jiangsu, China,in 1957. He received the B.S. and M.S. degrees inelectrical engineering from Beijing Institute of Tech-nology, Beijing, China, in 1982 and 1987, respec-tively, and the Ph.D. degree in electrical engineeringfrom Nanjing University of Science and Technology,Nanjing, China, in 1997.

He is currently a Professor at the National KeyLaboratory on Electromagnetic Environment andElectro-optical Engineering, PLA University of Sci-ence and Technology, Nanjing, China. His current

research interests include computational electromagnetics, EMC, and EMP.

Cheng Gao (M’98) was born in Jiangsu, China, in1964. He received the B.S. degree from the NavalAeronautical Engineering Institute, Yantai, China,the M.S. degree from the Nanjing University of Aero-nautics and Astronautics, Nanjing, China, and thePh.D. degree from the Nanjing Engineering Institute,Nanjing, in 1985, 1994, and 2003, all in electricalengineering.

He is currently a Professor at the National KeyLaboratory on Electromagnetic Environment andElectro-optical Engineering, PLA University of Sci-

ence and Technology, Nanjing. His current research interests include electro-magnetic compatibility and electromagnetic pulse protection.

Yun Yi was born in Changsha, China, in 1978. Shereceived both the B.S. and M.S. degrees in electricsystem and its automation in 2000 and 2003, respec-tively, from Nanjing Engineering Institute, Nanjing,China, where she is currently working toward thePh.D. degree in disaster prevention and reduction en-gineering and protective engineering.

Her current research interest includes computa-tional electromagnetics.

Wen Yang was born in Tongren, China, in 1982.He received the B.S. degree in electric system andits automation from Nanjing Engineering Institute,Nanjing, China, in 2005.

He is currently a Pioneer Engineer with the Engi-neering and Design Institute, Chengdu Military Areaof PLA, Yunnan, China. His current research inter-ests include computational electromagnetics and theelectric system design.