# Evaluation of Horizontal Electric Field from the Lightning Channel by Electromagnetic Field Equations of Moving Charges

Authors: Yunfeng Zhang; Erchun Zhang; Jialiang Gu
Journal: International Journal of Antennas and Propagation
 (2019)

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## Abstract

The horizontal electric field from the lightning return-stroke channel is evaluated by the electromagnetic field equations of moving charges in this paper. When a lightning flash strikes the ground, the charges move upward the lightning channel at the return-stroke speed, thereby producing the electromagnetic fields. According to the electromagnetic field equations of moving charges, the detained charges, uniformly moving charges, and decelerating (or accelerating) charges in each segment of the channel generate electrostatic fields, velocity fields, and radiation fields, respectively. The horizontal component of the sum is the horizontal electric field over the perfectly conducting ground. For the real soil with finite conductivity, the Wait formula is used here for the evaluation of the horizontal electric field over the realistic soil. The proposed method can avoid the oscillation of the fields in the long distance by the FDTD method and the singularity problem of the integral equation by the Sommerfeld integral method. The influences of the return-stroke speed, distance, and soil conductivity on the horizontal electric field are also analyzed by the proposed method. The conclusions can be drawn that the horizontal electric field decreases with the increasing of the return-stroke speed; the negative offset increases with the increasing of horizontal distance and with the decreasing of the soil conductivity, thereby forming the bipolar waveform. These conclusions will be practically valuable for the protection of lightning-induced overvoltage on the transmission lines.


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## Body

## 1. Introduction

How to protect the power distribution lines against lightning damage efficiently has become an important research topic in power system design. It is necessary to obtain the accurate lightning-induced voltage of the power distribution lines. One of the keys to calculating the lightning-induced overvoltage is to obtain the horizontal electric field. It is difficult to measure the horizontal electric field over the ground. Therefore, the numerical calculation of the horizontal electric field from the lightning channel is very important.Many scholars have proposed different methods to calculate the horizontal electric field from the lightning channel. These methods are mainly classified into two categories: analytical algorithms and numerical algorithms. Sommerfeld [1, 2] first solved the electromagnetic fields over the finite conducting ground by using the dipole antenna theory, but the Sommerfeld integral converges very slowly. There are also singular points in the Sommerfeld integral and the oscillating Bessel function, which make the Sommerfeld algorithm limited. In order to overcome some shortcomings of the Sommerfeld algorithm and apply it to the actual electromagnetic problems, Norton [3] first simplified the Sommerfeld algorithm to get the right results. Zeddam and Degauque [4] compared the Norton approximation algorithm with the Sommerfeld algorithm. They found that the Norton approximation algorithm is very high precision. In order to apply the Sommerfeld algorithm into the lightning electromagnetic fields, in 1992, Cooray [5] used the surface impedance theory to calculate the horizontal electric field on the ground surface, but the surface impedance method can only be applied to calculate the electromagnetic fields on the ground surface. After that, based on the Ampere loop law and Faraday’s law of electromagnetic induction, Rubin [6] generalized the surface impedance method and the generalized method is called the Cooray–Rubinstein (C-R) algorithm [7], which can be used to evaluate the horizontal electric field at different heights from the lightning channel. The finite difference time domain (FDTD) method was first introduced into the electromagnetic fields by Yee [8] and later evolved into an important tool in computational electromagnetics through the improvement of Berenger et al. [9]. Yang and Zhou et al. [10] applied FDTD to calculate the electromagnetic fields within 1 km from the lightning channel on the finite conducting ground. Baba and Rakov et al. [11] applied FDTD to calculate the electromagnetic fields of different return-stroke models at different distances from the lightning channel on the finite conducting ground. Zhang et al. [12] calculated the horizontal electric field of vertical stratification of soil at different distances and tested the C-R formula. Li et al. [13] used three-dimensional FDTD to analyze the influence of the rough surface on the horizontal electric field. Though FDTD is widely used and has high calculation accuracy, the waveforms often appear oscillating in the long distance.The theory of electromagnetic fields generated by moving charges is described in any standard book on electromagnetic theory, and it suffices to quote the results directly. Cooray and Cooray [14] demonstrated that the electromagnetic field equations of moving charges can be utilized to evaluate the vertical fields on the ground from the lightning channel. Chen et al. [15] applied the electromagnetic field equations of moving charges to calculate the vertical electric field from the lightning channel. Zhang et al. [16] used the electromagnetic field equations of moving charges to evaluate the radiation field from the horizontal conductor struck by a lightning flash. Zhang et al. [17] used the electromagnetic field equations of moving charges to calculate the vertical electric field from a tower struck by a lightning flash. Although the vertical electric field by the electromagnetic field equations of moving charges was calculated, it is not real for the soil with an infinite conductivity. In this paper, the finite conductivity of the soil is considered when the horizontal electric field is calculated by the electromagnetic field equations of moving charges. We will derive the expression of the horizontal electric field and azimuth magnetic field of the modified transmission line with linear decay (MTLL) model over the perfectly conducting ground. Then the horizontal electric field and the azimuth magnetic field calculated are substituted into the Wait formula [18] to obtain the right horizontal field over the ground with a finite conductivity.The organization of the paper is as follows: in Section2, the computational methodology for horizontal electric fields both over a perfectly ground and over a finite conductive ground is described. In Section 3, comparison between the proposed method and the FDTD method is used to test effectiveness of the proposed method. The influences of the return-stroke speed, distance from the channel, and soil conductivity on the horizontal electric field are analyzed by the proposed method in Section 4. Finally, conclusions are presented in Section 5.


## 2. Theory and Model

### 2.1. Electromagnetic Field Equations of Moving Charges

In classical electrodynamic theory, the static charges and uniform linear moving charges produce no radiation fields, while accelerating or decelerating charges generate radiation electromagnetic fields outward.An infinite small particle with the chargeq moves in a velocity of ν and an acceleration of ν˙, as it is shown in Figure 1. It is assumed that the direction of ν does not change with time; both ν and ν˙ have the same direction. P is the field point, and A is the source point. The electric field Ε and the magnetic induction Β generated at time t are given, respectively, by(1)Εt=q4πε0r211−β⋅αr3αr−β1−β2+q4πε0cr11−β⋅αr3αr×αr×β˙,(2)Βt=q4πε0cr211−β⋅αr3β×αr1−β2+q4πε0c2r11−β⋅αr3αr×αr×αr×β˙,where β points to the direction of the moving charge (with β=ν/c), αr represents the unit vector from the source point to the field point (with αr=r/r), β˙ points to the direction of the acceleration(with β˙=∂β/∂t), and r is the distance from the source point pointing to the field point. It can be found from equations (1) and (2) that both the electric field and the magnetic induction are composed of two terms. The first term is the velocity field, which is related to the velocity of the moving charge and varies with 1/r2. The second term is the radiation field, which is related to the acceleration and varies with 1/r.Figure 1
Schematic diagram of radiational principle from the moving charge.


### 2.2. Horizontal Electric Field over a Perfectly Conducting Ground

The sketch for the base current traveling along the lightning return-stroke channel is shown in Figure2. In Figure 2, R stands for the horizontal distance, h stands for the vertical height, H stands for the height of the lightning channel, αz stands for the vertical direction unit vector, αρ stands for the horizontal direction unit vector, and the ground is assumed to be the perfect ground. It is assumed that the x-axis is the positive direction of the horizontal electric field and the base current is ibt. The horizontal electric field at the point P can be obtained by using equation (1). The horizontal electric field at the time t and with a height h over the perfectly conducting ground, Eρph,t, comprises of four components: (1) the horizontal component of the radiation field generated as the charge accelerates at the bottom of the channel, Eρp,rad1h,t; (2) the horizontal component of the velocity field generated as the current pulse moves along the lightning channel, Eρp,velh,t; (3) the horizontal component of the radiation field generated by the decelerating charge as the current decays along the lightning channel, Eρp,rad2h,t; and (4) the horizontal component of the electrostatic field generated by the decelerating charge as the charge is deposed in the lightning channel, Eρp,stath,t. The expressions are given, respectively, by(3)Eρp,rad1h,t=ibt−r/cvsinθ4πε0c2r11−vcosθ/ccosθ+ibt−r/cvsinθ4πε0c2r11+vcosθ/ccosθ,(4)Eρp,velh,t=∫0HPzibt−z/v−r1/c1−v2/c24πε0r121−v/ccosθ12sinθ1vdz−∫0HPzibt−z/v−r2/c1−v2/c24πε0r221−v/ccosθ22sinθ2vdz,(5)Eρp,rad2h,t=−∫0H∂Pz/∂zibt−z/v−r1/cvsinθ14πε0c2r11−vcosθ1/ccosθ1dz+∫0H∂Pz/∂zibt−z/v−r2/cvsinθ24πε0c2r21−vcosθ2/ccosθ2dz,(6)Eρp,stath,t=14πε0∫Ht∂Pz∂z1r12dz∫z/v+r1/cHibτdτsinθ1−14πε0∫0H∂Pz∂z1r22dz∫z/v+r2/ctibτdτsinθ2,where Pz is the current attenuation coefficient in the channel; v is the return-stroke speed; and c is the light speed, and c=3×108m/s.Figure 2
Geometry model of electromagnetic fields from the lightning channel.The total horizontal electric fieldEρph,t is the sum of equations (3)–(6).


### 2.3. Azimuth Magnetic Field over a Perfectly Conducting Ground

The direction of the magnetic induction generated by the accelerating charge is assumed to be positive. The magnetic induction of point P can be obtained by using equation (2). The three components of magnetic induction at point P are as follows: (1) the magnetic radiation induction generated as the charge accelerates at the bottom of the channel, Bφp,rad1h,t; (2) the magnetic velocity induction generated by the current pulse moves along the lightning channel, Bφp,velh,t; and (3) the magnetic radiation induction generated by the decelerating charge as the current decays along the lightning channel, Bφp,rad2h,t. The expressions are given, respectively, by(7)Bφp,rad1h,t=ibt−rcvsinθ14πε0c3r11−v/ccosθ+ibt−rcvsinθ14πε0c3r11+v/ccosθ,(8)Bφp,velh,t=∫0HPzibt−r1/c−z/v1−v2/c2sinθ14πε0c2r121−v/ccosθ12dz+∫0HPzibt−r2/c−z/v1−v2/c2sinθ24πε0c2r221−v/ccosθ22dz,(9)Bφp,rad2h,t=−∫0H∂Pz/∂zibt−r1/c−z/vsinθ1v4πε0r1c31−vcosθ1/cdz−∫0H∂Pz/∂zibt−r2/c−z/vsinθ2v4πε0r2c31−vcosθ2/cdz.The total magnetic inductionBφph,t is the sum of equations (7)–(9).ForB=μH, the azimuth magnetic field at the time t and with a height h over the perfectly conducting ground, Hφph,t, is given by(10)Hφph,t=Bφph,tμ.


### 2.4. Horizontal Electric Field over a Finite Conducting Ground

In Sections2.2 and 2.3, it has been discussed how to apply the proposed method to acquire the horizontal and azimuthal magnetic fields at the height h over the perfectly conductive ground. Then we use the above results to calculate the horizontal electric field over the ground with a finite soil conductivity. Here the Wait formula [18] will be applied to get the results over a finite conducting ground. Since Eρph,t and Hφph,t are expressions in the time domain, the Fourier transforms in each frequency point [19] of them are Eρph,jω and Hφph,jω. The expression of the horizontal electric field over the finite conducting ground in the frequency domain is given by(11)Eρh,jω=Eρph,jω−Hφp0,jωFjωZjω,where ω stands for the angular frequency, Eρh,jω represents the horizontal electric field over the finite conducting ground in the frequency domain, Eρph,jω stands for the horizontal electric field over the perfectly conducting ground in the frequency domain, Hφp0,jω represents the azimuthal magnetic field on the perfectly conducting ground in the frequency domain. Fjω is defined as(12)Fjω=1−jπpe−perfcjp,where erfc is the error correction function and p is the numerical distance. The expression of p is given by(13)p=−0.5γ0ρΔ2,where γ0 stands for the free space wave number and Δ stands for the normalized surface impedance of the soil.γ0 and Δs are defined as(14)γ0=jωμ0ε0,(15)Δ=Zjωμ0/ε0,and the expression of Zjω in equation (15) is given by(16)Zjω=jωμ0σ+jωε0εr.To obtain the expression of the horizontal electric field over the finite conducting ground in the time domain, inverse Fourier transforms can be used [19]. The expression of the horizontal electric field over the finite conducting ground in the time domain is given by(17)Eρh,t=F−1Eρh,jω.


### 2.5. Model of the Lightning Channel

The lightning return-stoke model used in this paper is the MTLL model [20]. When the return-stroke current pulse moves along the lightning channel, it decays linearly with height. The longitudinal channel current at any height z and any time t of the MTLL model is(18)Iz,t=Pzib0,t−zv,where Pz=1−z/H represents the current attenuation coefficient in the return-stroke channel, H is assumed to be 7500 m, and v stands for the return-stroke speed. ib0,t is the base current with the double Heidler function [21], which is commonly used in the literature and has been adopted for the lightning study. The base current ib is depicted as follows:(19)ib0,t=I01η1t/τ11n1t/τ11n1+1e−t/τ12+I02η2t/τ12n2t/τ12n2+1e−t/τ22,where I01 and I02 are the first stroke current and the subsequent stroke current peak in the lightning channel, respectively; τ11 and τ21 are the rising edge time; τ12 and τ22 are the falling edge time; and n1 and n2 are the current steepness factor. η1 and η2 in equation (20) are the current peak correction factor, where(20)η1=exp−τ11τ12n1τ12τ111/n1,(21)η2=exp−τ21τ22n2τ22τ211/n2.The numerical values of two channel-base current waveforms corresponding, respectively, to typical first and subsequent return stroke are given in Table1 [22]. These values are commonly used in the lightning literature.Table 1
The base current parameters of the lightning.
I
01
kA
τ
11
μ
s
τ
12
μ
s
n
1
I
02
kA
τ
21
μ
s
τ
22
μ
s
n
2
10.7
0.25
2.5
2
6.5
2
230
2


## 2.1. Electromagnetic Field Equations of Moving Charges

In classical electrodynamic theory, the static charges and uniform linear moving charges produce no radiation fields, while accelerating or decelerating charges generate radiation electromagnetic fields outward.An infinite small particle with the chargeq moves in a velocity of ν and an acceleration of ν˙, as it is shown in Figure 1. It is assumed that the direction of ν does not change with time; both ν and ν˙ have the same direction. P is the field point, and A is the source point. The electric field Ε and the magnetic induction Β generated at time t are given, respectively, by(1)Εt=q4πε0r211−β⋅αr3αr−β1−β2+q4πε0cr11−β⋅αr3αr×αr×β˙,(2)Βt=q4πε0cr211−β⋅αr3β×αr1−β2+q4πε0c2r11−β⋅αr3αr×αr×αr×β˙,where β points to the direction of the moving charge (with β=ν/c), αr represents the unit vector from the source point to the field point (with αr=r/r), β˙ points to the direction of the acceleration(with β˙=∂β/∂t), and r is the distance from the source point pointing to the field point. It can be found from equations (1) and (2) that both the electric field and the magnetic induction are composed of two terms. The first term is the velocity field, which is related to the velocity of the moving charge and varies with 1/r2. The second term is the radiation field, which is related to the acceleration and varies with 1/r.Figure 1
Schematic diagram of radiational principle from the moving charge.


## 2.2. Horizontal Electric Field over a Perfectly Conducting Ground

The sketch for the base current traveling along the lightning return-stroke channel is shown in Figure2. In Figure 2, R stands for the horizontal distance, h stands for the vertical height, H stands for the height of the lightning channel, αz stands for the vertical direction unit vector, αρ stands for the horizontal direction unit vector, and the ground is assumed to be the perfect ground. It is assumed that the x-axis is the positive direction of the horizontal electric field and the base current is ibt. The horizontal electric field at the point P can be obtained by using equation (1). The horizontal electric field at the time t and with a height h over the perfectly conducting ground, Eρph,t, comprises of four components: (1) the horizontal component of the radiation field generated as the charge accelerates at the bottom of the channel, Eρp,rad1h,t; (2) the horizontal component of the velocity field generated as the current pulse moves along the lightning channel, Eρp,velh,t; (3) the horizontal component of the radiation field generated by the decelerating charge as the current decays along the lightning channel, Eρp,rad2h,t; and (4) the horizontal component of the electrostatic field generated by the decelerating charge as the charge is deposed in the lightning channel, Eρp,stath,t. The expressions are given, respectively, by(3)Eρp,rad1h,t=ibt−r/cvsinθ4πε0c2r11−vcosθ/ccosθ+ibt−r/cvsinθ4πε0c2r11+vcosθ/ccosθ,(4)Eρp,velh,t=∫0HPzibt−z/v−r1/c1−v2/c24πε0r121−v/ccosθ12sinθ1vdz−∫0HPzibt−z/v−r2/c1−v2/c24πε0r221−v/ccosθ22sinθ2vdz,(5)Eρp,rad2h,t=−∫0H∂Pz/∂zibt−z/v−r1/cvsinθ14πε0c2r11−vcosθ1/ccosθ1dz+∫0H∂Pz/∂zibt−z/v−r2/cvsinθ24πε0c2r21−vcosθ2/ccosθ2dz,(6)Eρp,stath,t=14πε0∫Ht∂Pz∂z1r12dz∫z/v+r1/cHibτdτsinθ1−14πε0∫0H∂Pz∂z1r22dz∫z/v+r2/ctibτdτsinθ2,where Pz is the current attenuation coefficient in the channel; v is the return-stroke speed; and c is the light speed, and c=3×108m/s.Figure 2
Geometry model of electromagnetic fields from the lightning channel.The total horizontal electric fieldEρph,t is the sum of equations (3)–(6).


## 2.3. Azimuth Magnetic Field over a Perfectly Conducting Ground

The direction of the magnetic induction generated by the accelerating charge is assumed to be positive. The magnetic induction of point P can be obtained by using equation (2). The three components of magnetic induction at point P are as follows: (1) the magnetic radiation induction generated as the charge accelerates at the bottom of the channel, Bφp,rad1h,t; (2) the magnetic velocity induction generated by the current pulse moves along the lightning channel, Bφp,velh,t; and (3) the magnetic radiation induction generated by the decelerating charge as the current decays along the lightning channel, Bφp,rad2h,t. The expressions are given, respectively, by(7)Bφp,rad1h,t=ibt−rcvsinθ14πε0c3r11−v/ccosθ+ibt−rcvsinθ14πε0c3r11+v/ccosθ,(8)Bφp,velh,t=∫0HPzibt−r1/c−z/v1−v2/c2sinθ14πε0c2r121−v/ccosθ12dz+∫0HPzibt−r2/c−z/v1−v2/c2sinθ24πε0c2r221−v/ccosθ22dz,(9)Bφp,rad2h,t=−∫0H∂Pz/∂zibt−r1/c−z/vsinθ1v4πε0r1c31−vcosθ1/cdz−∫0H∂Pz/∂zibt−r2/c−z/vsinθ2v4πε0r2c31−vcosθ2/cdz.The total magnetic inductionBφph,t is the sum of equations (7)–(9).ForB=μH, the azimuth magnetic field at the time t and with a height h over the perfectly conducting ground, Hφph,t, is given by(10)Hφph,t=Bφph,tμ.


## 2.4. Horizontal Electric Field over a Finite Conducting Ground

In Sections2.2 and 2.3, it has been discussed how to apply the proposed method to acquire the horizontal and azimuthal magnetic fields at the height h over the perfectly conductive ground. Then we use the above results to calculate the horizontal electric field over the ground with a finite soil conductivity. Here the Wait formula [18] will be applied to get the results over a finite conducting ground. Since Eρph,t and Hφph,t are expressions in the time domain, the Fourier transforms in each frequency point [19] of them are Eρph,jω and Hφph,jω. The expression of the horizontal electric field over the finite conducting ground in the frequency domain is given by(11)Eρh,jω=Eρph,jω−Hφp0,jωFjωZjω,where ω stands for the angular frequency, Eρh,jω represents the horizontal electric field over the finite conducting ground in the frequency domain, Eρph,jω stands for the horizontal electric field over the perfectly conducting ground in the frequency domain, Hφp0,jω represents the azimuthal magnetic field on the perfectly conducting ground in the frequency domain. Fjω is defined as(12)Fjω=1−jπpe−perfcjp,where erfc is the error correction function and p is the numerical distance. The expression of p is given by(13)p=−0.5γ0ρΔ2,where γ0 stands for the free space wave number and Δ stands for the normalized surface impedance of the soil.γ0 and Δs are defined as(14)γ0=jωμ0ε0,(15)Δ=Zjωμ0/ε0,and the expression of Zjω in equation (15) is given by(16)Zjω=jωμ0σ+jωε0εr.To obtain the expression of the horizontal electric field over the finite conducting ground in the time domain, inverse Fourier transforms can be used [19]. The expression of the horizontal electric field over the finite conducting ground in the time domain is given by(17)Eρh,t=F−1Eρh,jω.


## 2.5. Model of the Lightning Channel

The lightning return-stoke model used in this paper is the MTLL model [20]. When the return-stroke current pulse moves along the lightning channel, it decays linearly with height. The longitudinal channel current at any height z and any time t of the MTLL model is(18)Iz,t=Pzib0,t−zv,where Pz=1−z/H represents the current attenuation coefficient in the return-stroke channel, H is assumed to be 7500 m, and v stands for the return-stroke speed. ib0,t is the base current with the double Heidler function [21], which is commonly used in the literature and has been adopted for the lightning study. The base current ib is depicted as follows:(19)ib0,t=I01η1t/τ11n1t/τ11n1+1e−t/τ12+I02η2t/τ12n2t/τ12n2+1e−t/τ22,where I01 and I02 are the first stroke current and the subsequent stroke current peak in the lightning channel, respectively; τ11 and τ21 are the rising edge time; τ12 and τ22 are the falling edge time; and n1 and n2 are the current steepness factor. η1 and η2 in equation (20) are the current peak correction factor, where(20)η1=exp−τ11τ12n1τ12τ111/n1,(21)η2=exp−τ21τ22n2τ22τ211/n2.The numerical values of two channel-base current waveforms corresponding, respectively, to typical first and subsequent return stroke are given in Table1 [22]. These values are commonly used in the lightning literature.Table 1
The base current parameters of the lightning.
I
01
kA
τ
11
μ
s
τ
12
μ
s
n
1
I
02
kA
τ
21
μ
s
τ
22
μ
s
n
2
10.7
0.25
2.5
2
6.5
2
230
2


## 3. Validity of Algorithm

To validate the proposed algorithm, the FDTD method with cylindrical coordinate is used to compare with the proposed method in this paper. The range of the analog domain is set to be 1000 m × 8000 m, and the size of the space step is set to be 1 m × 1 m. The time step ∆t is set to 1.67 ns to meet the Courant stability condition. The first-order Mur absorption boundary is used in this paper [23].It is assumed that the height of the field point is 10 m, the horizontal distance from the lightning channel is 200 m, the return-stroke speed is 0.5c, and soil conductivity is ∞. The comparison chart of the proposed method and FDTD is shown in Figure 3. It is assumed that the height is 10 m, return-stroke speed is 0.5c, distance is 200 m, and soil conductivity is 0.01 S/m. The comparison chart of the proposed method and the FDTD method is shown in Figure 4.Figure 3
The horizontal electric field over the perfectly ground with 10 m height and 200 m distance.Figure 4
The horizontal electric field over the finite conductive ground with 10 m height and 200 m distance.From Figures3 and 4, it can be found that the waveform and amplitude calculated by the FDTD method and the proposed method in this paper agree with each other well. This means that the proposed method in this paper is suitable for calculating the horizontal electric field over the finite conducting ground.Due to the problem of computer capacity, calculating long-distance electromagnetic fields by FDTD must make the grid larger, which will cause the waveform to oscillate. The proposed method does not experience such a problem.


## 4. Results and Analysis

### 4.1. Influence of the Return-Stroke Speed

To analyze the influence of the return-stroke speed on the horizontal electric field, the distance is assumed to be 200 m, the vertical height is assumed to be 10 m, and the soil conductivity is assumed to be 0.1 S/m. The return-stroke speeds are set to be 0.5c, 0.7c, 0.9c, and c, respectively. The horizontal electric field over the finite conducting ground is obtained as shown in Figure 5. In order to explain the result from Figure 5, we need obtain the horizontal electric field and the azimuth magnetic field over the perfectly conducting ground under the same parameters, as shown in Figures 6 and 7.Figure 5
Horizontal electric field of different return-stroke speeds.Figure 6
Horizontal electric field of different return-stroke speeds over the perfectly conducting ground.Figure 7
Azimuth magnetic field of different return-stroke speeds on the perfectly conducting ground.As seen in Figure5, the horizontal electric field decreases as the return-stroke speed increases. From Figures 6 and 7, it can be found that the horizontal electric field over the perfectly conducting ground decreases as the return-stroke speed increases and the azimuth magnetic field on the perfectly conducting ground increases as the return-stroke speed increases. The horizontal electric field and the azimuth magnetic field over the perfectly conducting ground are substituted into the Wait formula. The first term of the Wait formula decreases as the return-stroke speed increases, and the second term decreases as the return-stroke speed increases, which result that the horizontal electric field over the finite conducting ground decreases as the return-stroke speed increases.


### 4.2. Influence of Distance

To analyze the influence of distance on the horizontal electric field, the height is assumed to be 10 m, soil conductivity is assumed to be 0.01 S/m, and return-stroke speed is assumed to be 0.5c. It sets the distances to be 100 m, 200 m, 800 m, and 1000 m, respectively. The horizontal electric field of different distances is shown in Figures 8 and 9.Figure 8
Horizontal electric field of 100 m and 200 m.Figure 9
Horizontal electric field of 800 m and 1000 m.Figures8 and 9 depict that the peak value of the horizontal electric field decreases and the rising edge time of the wave head increases with the increase of the distance. This situation is caused that the component of high frequency attenuates and the energy reduces with the increase of distance. The waveform of horizontal electric field presents bipolar features. The horizontal distance is the farther, and the negative offset is larger, which can be explained from the Wait formula. The horizontal electric field in the Wait formula consists of two parts, in which one part is the ideal field term and the other part is the surface impedance term. The horizontal distance is farther, and the contribution of the coefficient in the surface impedance term to the total horizontal electric field is larger.


### 4.3. Influence of Soil Conductivity

To analyze the influence of soil conductivity on the horizontal electric field, the horizontal distance is assumed to be 200 m, the height is assumed to be 10 m, the return-stroke speed is set to be 0.5c, and the soil conductivities are set to be 0.1 S/m, 0.01 S/m, and 0.001 S/m. The horizontal electric field of different soil conductivities is shown in Figure 10.Figure 10
Horizontal electric field of different soil conductivities.It can be seen from Figure10 that the peak value of the horizontal electric field increases with the increase of conductivity. This situation is caused that the attenuation of horizontal electric field decreases with the increase of conductivity. The waveform of horizontal electric field is bipolar. The conductivity is the smaller and the negative offset is larger, which can be explained from the Wait formula. The horizontal electric field in the Wait formula consists of two parts, in which one part is the ideal field term and the other part is the surface impedance term. The conductivity is the smaller, and Zjω in the surface impedance term is the lager, which makes the negative offset larger at the head of the wave.


## 4.1. Influence of the Return-Stroke Speed

To analyze the influence of the return-stroke speed on the horizontal electric field, the distance is assumed to be 200 m, the vertical height is assumed to be 10 m, and the soil conductivity is assumed to be 0.1 S/m. The return-stroke speeds are set to be 0.5c, 0.7c, 0.9c, and c, respectively. The horizontal electric field over the finite conducting ground is obtained as shown in Figure 5. In order to explain the result from Figure 5, we need obtain the horizontal electric field and the azimuth magnetic field over the perfectly conducting ground under the same parameters, as shown in Figures 6 and 7.Figure 5
Horizontal electric field of different return-stroke speeds.Figure 6
Horizontal electric field of different return-stroke speeds over the perfectly conducting ground.Figure 7
Azimuth magnetic field of different return-stroke speeds on the perfectly conducting ground.As seen in Figure5, the horizontal electric field decreases as the return-stroke speed increases. From Figures 6 and 7, it can be found that the horizontal electric field over the perfectly conducting ground decreases as the return-stroke speed increases and the azimuth magnetic field on the perfectly conducting ground increases as the return-stroke speed increases. The horizontal electric field and the azimuth magnetic field over the perfectly conducting ground are substituted into the Wait formula. The first term of the Wait formula decreases as the return-stroke speed increases, and the second term decreases as the return-stroke speed increases, which result that the horizontal electric field over the finite conducting ground decreases as the return-stroke speed increases.


## 4.2. Influence of Distance

To analyze the influence of distance on the horizontal electric field, the height is assumed to be 10 m, soil conductivity is assumed to be 0.01 S/m, and return-stroke speed is assumed to be 0.5c. It sets the distances to be 100 m, 200 m, 800 m, and 1000 m, respectively. The horizontal electric field of different distances is shown in Figures 8 and 9.Figure 8
Horizontal electric field of 100 m and 200 m.Figure 9
Horizontal electric field of 800 m and 1000 m.Figures8 and 9 depict that the peak value of the horizontal electric field decreases and the rising edge time of the wave head increases with the increase of the distance. This situation is caused that the component of high frequency attenuates and the energy reduces with the increase of distance. The waveform of horizontal electric field presents bipolar features. The horizontal distance is the farther, and the negative offset is larger, which can be explained from the Wait formula. The horizontal electric field in the Wait formula consists of two parts, in which one part is the ideal field term and the other part is the surface impedance term. The horizontal distance is farther, and the contribution of the coefficient in the surface impedance term to the total horizontal electric field is larger.


## 4.3. Influence of Soil Conductivity

To analyze the influence of soil conductivity on the horizontal electric field, the horizontal distance is assumed to be 200 m, the height is assumed to be 10 m, the return-stroke speed is set to be 0.5c, and the soil conductivities are set to be 0.1 S/m, 0.01 S/m, and 0.001 S/m. The horizontal electric field of different soil conductivities is shown in Figure 10.Figure 10
Horizontal electric field of different soil conductivities.It can be seen from Figure10 that the peak value of the horizontal electric field increases with the increase of conductivity. This situation is caused that the attenuation of horizontal electric field decreases with the increase of conductivity. The waveform of horizontal electric field is bipolar. The conductivity is the smaller and the negative offset is larger, which can be explained from the Wait formula. The horizontal electric field in the Wait formula consists of two parts, in which one part is the ideal field term and the other part is the surface impedance term. The conductivity is the smaller, and Zjω in the surface impedance term is the lager, which makes the negative offset larger at the head of the wave.


## 5. Conclusion

In this paper, the electromagnetic field equations of moving charges are applied to obtain the horizontal electric field over the ground with both infinite and finite conductivity. For the MTLL model of lightning return stroke considered in this paper, in the lightning channel over the perfectly conducting ground, the electrostatic fields given by the detained charges, the velocity fields generated by the uniformly moving charges, and the radiation fields produced by the decelerating (or accelerating) charges are obtained by the electromagnetic field equations of moving charges. The horizontal component of the sum is the horizontal electric field above the perfectly conducting ground. Substituting the expression of horizontal electric field and azimuth magnetic field in the frequency domain into the Wait formula, the horizontal electric field over the finite conductive ground in the frequency domain can be obtained. The expression in the time domain can be obtained by the inverse Fourier transforms. In order to prove the proposed method in this paper, the FDTD method with cylindrical coordinate is used to compare with it. It has been found that results obtained by the FDTD method have good consistency with those by the proposed method.The effects of the return-stroke speed, distance, and soil conductivity on the horizontal electric field over the finite conducting ground are also analyzed, and the following conclusions are obtained: (1) The horizontal electric field over the finite conducting ground decreases as the return-stroke speed increases. (2) The peak value of the horizontal electric field becomes smaller and the rising edge time becomes larger with increase of distance. (3) The waveform of horizontal electric field presents bipolar features. (4) The negative offset is larger with increase of distance (or with decrease of the conductivity). (5) The peak value of the horizontal electric field is larger with increase of the conductivity. These conclusions have practical value for the protection of lightning-induced overvoltage on the transmission lines.



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*Source: 4091586-2019-12-14.xml*