# Analytical Solution for the Transient Current Induced along the Vertical Grounding Electrode

Authors: Ante Soldo; Silvestar Šesnić
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The paper deals with the analysis of the Pocklington integro-differential equation in the time domain and its analytical solution for the transient current induced along the vertical grounding electrode and, as such, represents a continuation of the research pertaining to the horizontal grounding electrode previously reported by one of the authors. The electrode is excited by a current source impressed at one end of the electrode, representing the model of the lightning strike current. A novel analytical solution is derived using the Laplace transform. Results obtained with this time domain solution are compared with the results obtained via analytical solution of the corresponding Pocklington equation in the frequency domain in conjunction with inverse fast Fourier transform (IFFT). The results obtained via different methods are shown to be in a very good agreement. The results of the time domain analytical solution for the vertical electrode are also compared with the results of the time domain analytical solution for the horizontal electrode, where greater influence of earth-air interface on an induced current is observed. These procedures can be implemented as benchmarking techniques as well as tools for rapid estimation of the observed phenomena.


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## Body

## 1. Introduction

Tall structures are, principally, very vulnerable to lightning strikes (e.g., telecommunication towers, wind turbines…) which facilitates the necessity to design and model an appropriate grounding and lightning protection systems (LPS) making this an important area in current and future scientific research [1–7]. Vertical grounding electrode is considered to be one of the essential components of realistic grounding systems, particularly for the protection of wind turbines [8]. Current induced along the vertical grounding electrode represents a quantity that can be subsequently used for determining other parameters of interest.Time domain current induced along a thin wire structure is governed by integro-differential equation, either of Hallen or Pocklington type. These equations are usually solved using various numerical methods representing an established practice when dealing with highly complex systems [9, 10]. Analytical solutions, on the other hand, can be obtained by implementing appropriate approximations when dealing with canonical geometries [11–14]. The advantages of analytical solutions are: can be used for fast engineering assessment of the observed phenomenon; can be applied for reference purposes; can be used within hybrid approaches, where computational costs can be reduced [9]. The main drawback of any analytical model is its suitability for simplified canonical geometries.In this paper, time domain analytical solution for the induced current governed with Pocklington integro-differential equation for vertical grounding electrode excited by a lightning strike is derived. In [15], the author has presented the analytical solution for the horizontal grounding electrode. However, the solution derived in this paper significantly differs from the one previously published. Taking into account vertical orientation of the grounding electrode, more complex Green’s function is used which subsequently results in a differential equation with variable coefficients to be solved analytically. This approach ensures that the novel closed-form solution for the induced current along the vertical electrode is obtained.Section2 defines Pocklington integro-differential formulation for a current flowing along the vertical electrode buried into a lossy half-space and excited with a transient current source representing lightning current strike. Thin wire approximation is applied, while the influence of the ground-air interface is taken into account using a simplified reflection coefficient derived from the Modified Image Theory (MIT) [10]. In Section 3, approximation of the current distribution, Laplace transform and Cauchy residue theorem are used to derive the solution to the corresponding Pocklington equation. Furthermore, a model of an equivalent current source is defined. The obtained results are compared with the results for the analytical solution of the corresponding Pocklington equation in the frequency domain and the subsequent IFFT in Section 4. Additionally, the results obtained via analytical time domain solution for the vertical and horizontal grounding electrode, respectively, are compared. In Section 5, concluding remarks with the reference for future work are discussed.


## 2. Transient Model of the Vertical Grounding Electrode

Vertical grounding electrode of lengthL is buried in a lossy medium. Medium is characterized with electrical permittivity ε and electrical conductivity σ. The electrode is excited at one end with an equivalent current source. The geometry of the posed problem is shown in Figure 1.Figure 1
Vertical grounding electrode buried in a lossy half-space.Tangential component of the total field, in the frequency domain, is equal to zero for perfectly conducting (PEC) material [15].(1)e⟶z⋅E⟶inc+E⟶sct=0.Since grounding electrodes are made of materials with very high conductivity, they can be considered as perfectly conducting as it is shown in [16].Governing equation for the current induced along the electrode is derived as time-domain Pocklington integro-differential equation:(2)∂2∂z2−μσ∂∂t−με∂2∂t2⋅μ4π∫−d−d−LIz′,t−Rve−1/τgR/vRdz′−μ4π∫0t∫−d−d−LΓrefMITτIz′,t−R∗v−τe−1/τgR∗/vR∗dτdz′=0.where Iz′,t−R/v represents the unknown current. Detailed derivation of (2) can be found in [17].As can be seen in Figure2, the distance from the source point in the wire axis to the observation point on the wire surface is defined as(3)R=z−z′2+a2.Figure 2
Vertical electrode and its image obtained by applying image theory.However, the distance from the source point in the image wire to the observation point on the wire surface, according to the image theory is(4)R∗=z+z′2+a2.The time constant and propagation velocity in a lossy medium depend on the properties of the medium and are given as [17](5)τg=2εσ,v=1με.For the sake of the simplicity and considering the objective of obtaining a simple analytical solution, reflection coefficient arising from the modified image theory is used in this paper and is given as [18](6)ΓrefMITt=−τ1τ2δt+1τ21−τ1τ2e−tτ2,where(7)τ1=ε0εr−1σ,τ2=ε0εr+1σ.Reflection coefficient (6) represents the simplest characterization of the earth-air interface, taking into account only medium properties. However, an extensive validation of the application of this coefficient to thin wire problems has been carried out in [10].


## 3. Analytical Solution

### 3.1. Current Approximation

The solution for the unknown current in (2) can be obtained either by using some numerical method as in [19] or analytically using certain approximations. As it has been reported in [20, 21] and somewhat later by the authors in [17], an approximation suitable for simplifying (2) is taken as(8)∫−d−d−Lz′,t−Rve−1/τgR/vRdz′I=Iz,t−av∫−d−d−Le−1/τgR/vRdz′.Main advantage of this approximation is simplification of the governing equation. The disadvantage is that it assumes a time retardation asa/v, which can cause some discrepancies investigated in [21].If (2) is rewritten, it yields(9)∂2∂z2−μσ∂∂t−με∂2∂t2⋅μ4πIz,t−av∫−d−d−Le−1/τgR/vRdz′−μ4π∫0tΓrefMITτIz,t−av−τ∫−d−d−Le−1/τgR∗/vR∗dz′dτ=0.


### 3.2. Laplace Transform and Differential Equation Solution

The transformation of the second order differential (9) into the Laplace domain is performed under the following physically justified assumptions [17].(10)Iz,t−a/v=0,t≤0,∂Iz,t−a/v∂t=0,t<0.In the Laplace domain, (9) becomes(11)μ4π∂2∂z2−μσs−μεs2Iz,se−a/vs⋅∫−d−d−Le−1/τgR/vRdz′−ΓrefMITs∫−d−d−Le−1/τgR∗/vR∗dz′=0.where s=jω is the Laplace variable.If the ground conductivity is equal to or less than 100 mS/m, condition [17],(12)σ≪2aεμ0,is satisfied and integrals in (11) are given with [13](13)Ψz,s=∫−d−d−Le−1/τgR/vRdz′−ΓrefMITs∫−d−d−Le−1/τgR∗/vR∗dz′=lnz+d+z+d2+a2z+d+L+z+d+L2+a2−ΓrefMITslnz−d−L+z−d−L2+a2z−d+z−d2+a2.Taking into account (13), (11) becomes(14)μ4π∂2∂z2−μσs−μεs2Ψz,sIz,se−a/vs=0.Propagation constant is(15)γ=μεs2+σεs.Enabling the differential equation (14) to be written as follows:(16)∂2∂z2Ψz,sIz,s−γ2Ψz,sIz,s=0,which, subsequently, leads to the second order differential equation of the form:(17)Ψz,s∂2Iz,s∂z2+2∂Ψz,s∂z∂Iz,s∂z+∂2Ψz,s∂z2−γ2Ψz,sIz,s=0.Boundary conditions at the electrode ends are(18)I−d,s=Igs,I−d−L,s=0.And, the solution of (17) is obtained(19)Iz,s=IgsΨ−d,sΨz,ssinhγz+d+LsinhγL.(19) represents current distribution in the frequency domain [22].


### 3.3. Cauchy Residue theorem

In order to derive the solution for the space-time distribution of the induced current, it is necessary to perform an inverse Laplace transform applying Cauchy's residue theorem [23]:(20)ft=limy⟶∞1j2π∫x−jyx+jyetsFsds=∑k=1nRessk,where(21)Ressk=lims⟶sks−sketsFs,while sk represents the poles of the function Fs.Current source in the frequency domain is taken as(22)Igs=1A.Corresponding to the impulse excitation in the time domain.Undertaking long and tedious mathematical manipulations, the residues of current function (19) are determined according to (21) and the time domain counterpart of (19) is given as(23)Iz,t=I1zetsΨ+I2z⋅ets1,2n.Corresponding functions in (23) are given with(24)I1z=Ψ−d,sΨklnz−d+z−d2+a2/z−d−L+z−d−L2+a2sinhγΨz+d+LsinhγΨL,I2z=2πμεL2∑n=1∞−1n−1n±b2−4cnsinnπz+d+LLΨ−d,s1,2nΨz,s1,2n.And, coefficients can be defined as(25)Ψ−d,sΨ=lnaL+L2+a2+sΨτ1+1sΨτ2+1⋅ln−2d−L+2d+L2+a2−2d+4d2+a2,sΨ=h−1τ1−hτ2,h=lnz+d+z+d2+a2/z+d+L+z+d+L2+a2lnz−d+z−d2+a2/z−d−L+z−d−L2+a2,k=1sΨτ2+1τ2sΨτ1+1sΨτ2+1−τ1,γΨ=μεsΨ2+σεsΨ,s1,2n=12−b±b2−4cn,b=σε,cn=n2π2μεL2,n=1,2,3,...,Ψ−d,s1,2n=lnaL+L2+a2+s1,2nτ1+1s1,2nτ2+1,⋅ln−2d−L+2d+L2+a2−2d+4d2+a2,Ψz,s1,2n=lnz+d+z+d2+a2z+d+L+z+d+L2+a2+s1,2nτ1+1s1,2nτ2+1,⋅lnz−d−L+z−d−L2+a2z−d+z−d2+a2.(23) defines an analytical solution for the space-time distribution of the current induced along the vertical grounding electrode excited by an equivalent current source in the form of a Dirac pulse.


### 3.4. Equivalent Current Source

One of the most commonly used functions for representing lightning strike is a double exponential pulse [24].(26)Igt=I0e−αt−e−βt.An analytical convolution of (23) and (26) is performed to obtain the current flowing along the vertical electrode which is given with(27)Iz,t=I0I1zesΨt−e−αtsΨ+α−esΨt−e−βtsΨ+β+I2zes1,2nt−e−αts1,2n+α−es1,2nt−e−βts1,2n+β.(27) is an expression for the space-time distribution of current flowing along the vertical grounding electrode due to a lightning strike represented with double exponential function.


## 3.1. Current Approximation

The solution for the unknown current in (2) can be obtained either by using some numerical method as in [19] or analytically using certain approximations. As it has been reported in [20, 21] and somewhat later by the authors in [17], an approximation suitable for simplifying (2) is taken as(8)∫−d−d−Lz′,t−Rve−1/τgR/vRdz′I=Iz,t−av∫−d−d−Le−1/τgR/vRdz′.Main advantage of this approximation is simplification of the governing equation. The disadvantage is that it assumes a time retardation asa/v, which can cause some discrepancies investigated in [21].If (2) is rewritten, it yields(9)∂2∂z2−μσ∂∂t−με∂2∂t2⋅μ4πIz,t−av∫−d−d−Le−1/τgR/vRdz′−μ4π∫0tΓrefMITτIz,t−av−τ∫−d−d−Le−1/τgR∗/vR∗dz′dτ=0.


## 3.2. Laplace Transform and Differential Equation Solution

The transformation of the second order differential (9) into the Laplace domain is performed under the following physically justified assumptions [17].(10)Iz,t−a/v=0,t≤0,∂Iz,t−a/v∂t=0,t<0.In the Laplace domain, (9) becomes(11)μ4π∂2∂z2−μσs−μεs2Iz,se−a/vs⋅∫−d−d−Le−1/τgR/vRdz′−ΓrefMITs∫−d−d−Le−1/τgR∗/vR∗dz′=0.where s=jω is the Laplace variable.If the ground conductivity is equal to or less than 100 mS/m, condition [17],(12)σ≪2aεμ0,is satisfied and integrals in (11) are given with [13](13)Ψz,s=∫−d−d−Le−1/τgR/vRdz′−ΓrefMITs∫−d−d−Le−1/τgR∗/vR∗dz′=lnz+d+z+d2+a2z+d+L+z+d+L2+a2−ΓrefMITslnz−d−L+z−d−L2+a2z−d+z−d2+a2.Taking into account (13), (11) becomes(14)μ4π∂2∂z2−μσs−μεs2Ψz,sIz,se−a/vs=0.Propagation constant is(15)γ=μεs2+σεs.Enabling the differential equation (14) to be written as follows:(16)∂2∂z2Ψz,sIz,s−γ2Ψz,sIz,s=0,which, subsequently, leads to the second order differential equation of the form:(17)Ψz,s∂2Iz,s∂z2+2∂Ψz,s∂z∂Iz,s∂z+∂2Ψz,s∂z2−γ2Ψz,sIz,s=0.Boundary conditions at the electrode ends are(18)I−d,s=Igs,I−d−L,s=0.And, the solution of (17) is obtained(19)Iz,s=IgsΨ−d,sΨz,ssinhγz+d+LsinhγL.(19) represents current distribution in the frequency domain [22].


## 3.3. Cauchy Residue theorem

In order to derive the solution for the space-time distribution of the induced current, it is necessary to perform an inverse Laplace transform applying Cauchy's residue theorem [23]:(20)ft=limy⟶∞1j2π∫x−jyx+jyetsFsds=∑k=1nRessk,where(21)Ressk=lims⟶sks−sketsFs,while sk represents the poles of the function Fs.Current source in the frequency domain is taken as(22)Igs=1A.Corresponding to the impulse excitation in the time domain.Undertaking long and tedious mathematical manipulations, the residues of current function (19) are determined according to (21) and the time domain counterpart of (19) is given as(23)Iz,t=I1zetsΨ+I2z⋅ets1,2n.Corresponding functions in (23) are given with(24)I1z=Ψ−d,sΨklnz−d+z−d2+a2/z−d−L+z−d−L2+a2sinhγΨz+d+LsinhγΨL,I2z=2πμεL2∑n=1∞−1n−1n±b2−4cnsinnπz+d+LLΨ−d,s1,2nΨz,s1,2n.And, coefficients can be defined as(25)Ψ−d,sΨ=lnaL+L2+a2+sΨτ1+1sΨτ2+1⋅ln−2d−L+2d+L2+a2−2d+4d2+a2,sΨ=h−1τ1−hτ2,h=lnz+d+z+d2+a2/z+d+L+z+d+L2+a2lnz−d+z−d2+a2/z−d−L+z−d−L2+a2,k=1sΨτ2+1τ2sΨτ1+1sΨτ2+1−τ1,γΨ=μεsΨ2+σεsΨ,s1,2n=12−b±b2−4cn,b=σε,cn=n2π2μεL2,n=1,2,3,...,Ψ−d,s1,2n=lnaL+L2+a2+s1,2nτ1+1s1,2nτ2+1,⋅ln−2d−L+2d+L2+a2−2d+4d2+a2,Ψz,s1,2n=lnz+d+z+d2+a2z+d+L+z+d+L2+a2+s1,2nτ1+1s1,2nτ2+1,⋅lnz−d−L+z−d−L2+a2z−d+z−d2+a2.(23) defines an analytical solution for the space-time distribution of the current induced along the vertical grounding electrode excited by an equivalent current source in the form of a Dirac pulse.


## 3.4. Equivalent Current Source

One of the most commonly used functions for representing lightning strike is a double exponential pulse [24].(26)Igt=I0e−αt−e−βt.An analytical convolution of (23) and (26) is performed to obtain the current flowing along the vertical electrode which is given with(27)Iz,t=I0I1zesΨt−e−αtsΨ+α−esΨt−e−βtsΨ+β+I2zes1,2nt−e−αts1,2n+α−es1,2nt−e−βts1,2n+β.(27) is an expression for the space-time distribution of current flowing along the vertical grounding electrode due to a lightning strike represented with double exponential function.


## 4. Numerical Results

In this chapter, the results for transient response of the vertical grounding electrode excited by a lightning strike obtained directly in the time domain via (27) are presented. For the purpose of the model verification, the results are first compared with the results obtained in the frequency domain and transformed via IFFT as presented in [25]. Subsequently, the influence of different parameters on the current distribution along the vertical electrode is examined. Additionally, the comparison with the response for the horizontal electrode for the same input parameters and the same excitation pulse are presented.


### 4.1. Verification of the Analytical Time Domain Solution

Transient currents for the vertical electrode with radiusa = 5 mm, buried at depth d = 0.5 m are calculated. Two current source pulses with different parameters are considered as excitation. The parameters of 0.1/1 μs pulse are defined as (26)(28)I0=1.1043A,α=0.07924⋅1071s,β=4.0011⋅1071s.And, for the 1/10μs pulse, it is given with(29)I0=1.1043A,α=0.07924⋅1061s,β=4.0011⋅1061s.The 0.1/1μs pulse has higher frequency content than a 1/10 μs pulse, so it can be used to determine the effectiveness of the proposed model for stimuli with higher frequency content [24].Figure3 shows the results for the transient current induced in the center of the electrode calculated with (27) for 0.1/1 μs pulse (28) and compared with the analytical result obtained in the frequency domain for the same input parameters.Figure 3
Current at the center of the electrode,L = 10 (m); σ = 1 mS/m.It can be seen that the agreement between the results is very satisfactory, which is additionally confirmed with Figure456 where different electrode lengths and different ground conductivities are used.Figure 4
Current at the center of the electrode,L = 10 (m); σ = 10 mS/m.Figure 5
Current at the center of the electrode,L = 1 (m); σ = 1 mS/m.Figure 6
Current at the center of the electrode,L = 1 (m); σ = 0.1 mS/m.Since only limited number of results for model verification is presented, it is important to note that the authors extensively examined and determined the validity of the model for various combinations of input parameters.


### 4.2. Influence of Different Physical Parameters

Examining the input parameters that affect the results, the influence of the electrode length is first shown in Figure7. Electric properties of the ground are taken as σ = 1 mS/m and εr = 10. The lightning pulse (28) is used as an excitation function in all presented results.Figure 7
Current at the center of the electrode for different electrode lengths.From the results shown in Figure7 it can be seen that the current in the center of the electrode is equal to zero for the early time period, which is proportional to the length of the electrode and represents the time for the propagation of the current pulse from the end of the electrode where the current source is applied to its center. In the case of longer electrodes, due to the longer propagation period and the corresponding increased energy loss, the maximum value is lower.Figure8 shows the grounding electrode transient response for different ground conductivities. The length of the electrode is L = 10 m, while electrical permittivity is εr = 10. It can be seen that the decrease in the ground conductivity increases the maximum value of the current at the early time period. Additionally, the magnitude of oscillations is very slowly damped for low conductivity of 0.1 mS/m. Since the length of the electrode is constant, the response delay to the center of the electrode is approximately 50 ns, which shows that the conductivity of the Earth has no effect on the propagation time period.Figure 8
Current at the center of the electrode for different ground conductivities.The influence of ground permittivity is shown in Figure9. The delay of induced current in the early time period is due to the the propagation speed in a lossy medium defined by (6) which is inversely proportional to the electrical permittivity, i.e., increasing permittivity decreases the velocity. Additionally, increasing electrical permittivity, the amplitude of the oscillations also increases, which can be explained by increasing the ratio of Ψ−d,s1,2n and Ψz,s1,2n in (24).Figure 9
Current at the center of the electrode for different ground permittivity.The results shown in Figure10 represent transient current in the center of the vertical grounding electrode for two different electrode lengths (1 and 10 m) and for the two types of lightning current pulses (0.1/1 and 1/10 μs). For a 1/10 μs pulse, it can be seen that the current initially rises much slower than for a 0.1/1 μs pulse and that the oscillatory nature of the induced current is virtually nonexistent, which is consistent with lower frequency spectrum of the given pulse.Figure 10
Current at the center of the electrode for different electrode lengths and lightning pulses.Figure11 shows the space-time distribution of the current induced along the vertical grounding electrode with length L = 10 m, buried in the ground with electrical conductivity σ = 10 mS/m. It can be seen that the current decreases with time, as well as along the electrode.Figure 11
Current along the grounding electrode,L = 10 (m); σ = 10 mS/m.


### 4.3. Comparison between Vertical and Horizontal Electrode

The final set of results (Figure12131415) show a comparison between the transient response for the induced current for vertical and the horizontal electrode [21]. The excitation pulse of 0.1/1 μs is used for both electrodes and relative dielectric permeability of the ground is εr = 10. The results are shown for various values of electrode length and Earth conductivity.Figure 12
Comparison of transient currents for vertical and horizontal electrode,L = 10 (m); σ = 1 mS/m.Figure 13
Comparison of transient currents for vertical and horizontal electrode,L = 1 (m); σ = 1 mS/m.Figure 14
Comparison of transient currents for vertical and horizontal electrode,L = 1 (m); σ = 10 mS/m.Figure 15
Comparison of transient currents for vertical and horizontal electrode,L = 20 (m); σ = 1 mS/m.Comparing the results for the transient current between the vertical and horizontal grounding electrodes, it can be seen that the waveform of the induced current is very similar for all calculated parameters, but the maximum amplitude of the induced current for the horizontal electrode is greater by approximately 35–40%. This phenomenon can be explained by the greater influence of the ground-air interface on the induced current along the horizontal electrode as oppose to the vertically buried electrode. Additionally, for the vertical electrode energy dissipation is more emphasized due to medium conductivity.


## 4.1. Verification of the Analytical Time Domain Solution

Transient currents for the vertical electrode with radiusa = 5 mm, buried at depth d = 0.5 m are calculated. Two current source pulses with different parameters are considered as excitation. The parameters of 0.1/1 μs pulse are defined as (26)(28)I0=1.1043A,α=0.07924⋅1071s,β=4.0011⋅1071s.And, for the 1/10μs pulse, it is given with(29)I0=1.1043A,α=0.07924⋅1061s,β=4.0011⋅1061s.The 0.1/1μs pulse has higher frequency content than a 1/10 μs pulse, so it can be used to determine the effectiveness of the proposed model for stimuli with higher frequency content [24].Figure3 shows the results for the transient current induced in the center of the electrode calculated with (27) for 0.1/1 μs pulse (28) and compared with the analytical result obtained in the frequency domain for the same input parameters.Figure 3
Current at the center of the electrode,L = 10 (m); σ = 1 mS/m.It can be seen that the agreement between the results is very satisfactory, which is additionally confirmed with Figure456 where different electrode lengths and different ground conductivities are used.Figure 4
Current at the center of the electrode,L = 10 (m); σ = 10 mS/m.Figure 5
Current at the center of the electrode,L = 1 (m); σ = 1 mS/m.Figure 6
Current at the center of the electrode,L = 1 (m); σ = 0.1 mS/m.Since only limited number of results for model verification is presented, it is important to note that the authors extensively examined and determined the validity of the model for various combinations of input parameters.


## 4.2. Influence of Different Physical Parameters

Examining the input parameters that affect the results, the influence of the electrode length is first shown in Figure7. Electric properties of the ground are taken as σ = 1 mS/m and εr = 10. The lightning pulse (28) is used as an excitation function in all presented results.Figure 7
Current at the center of the electrode for different electrode lengths.From the results shown in Figure7 it can be seen that the current in the center of the electrode is equal to zero for the early time period, which is proportional to the length of the electrode and represents the time for the propagation of the current pulse from the end of the electrode where the current source is applied to its center. In the case of longer electrodes, due to the longer propagation period and the corresponding increased energy loss, the maximum value is lower.Figure8 shows the grounding electrode transient response for different ground conductivities. The length of the electrode is L = 10 m, while electrical permittivity is εr = 10. It can be seen that the decrease in the ground conductivity increases the maximum value of the current at the early time period. Additionally, the magnitude of oscillations is very slowly damped for low conductivity of 0.1 mS/m. Since the length of the electrode is constant, the response delay to the center of the electrode is approximately 50 ns, which shows that the conductivity of the Earth has no effect on the propagation time period.Figure 8
Current at the center of the electrode for different ground conductivities.The influence of ground permittivity is shown in Figure9. The delay of induced current in the early time period is due to the the propagation speed in a lossy medium defined by (6) which is inversely proportional to the electrical permittivity, i.e., increasing permittivity decreases the velocity. Additionally, increasing electrical permittivity, the amplitude of the oscillations also increases, which can be explained by increasing the ratio of Ψ−d,s1,2n and Ψz,s1,2n in (24).Figure 9
Current at the center of the electrode for different ground permittivity.The results shown in Figure10 represent transient current in the center of the vertical grounding electrode for two different electrode lengths (1 and 10 m) and for the two types of lightning current pulses (0.1/1 and 1/10 μs). For a 1/10 μs pulse, it can be seen that the current initially rises much slower than for a 0.1/1 μs pulse and that the oscillatory nature of the induced current is virtually nonexistent, which is consistent with lower frequency spectrum of the given pulse.Figure 10
Current at the center of the electrode for different electrode lengths and lightning pulses.Figure11 shows the space-time distribution of the current induced along the vertical grounding electrode with length L = 10 m, buried in the ground with electrical conductivity σ = 10 mS/m. It can be seen that the current decreases with time, as well as along the electrode.Figure 11
Current along the grounding electrode,L = 10 (m); σ = 10 mS/m.


## 4.3. Comparison between Vertical and Horizontal Electrode

The final set of results (Figure12131415) show a comparison between the transient response for the induced current for vertical and the horizontal electrode [21]. The excitation pulse of 0.1/1 μs is used for both electrodes and relative dielectric permeability of the ground is εr = 10. The results are shown for various values of electrode length and Earth conductivity.Figure 12
Comparison of transient currents for vertical and horizontal electrode,L = 10 (m); σ = 1 mS/m.Figure 13
Comparison of transient currents for vertical and horizontal electrode,L = 1 (m); σ = 1 mS/m.Figure 14
Comparison of transient currents for vertical and horizontal electrode,L = 1 (m); σ = 10 mS/m.Figure 15
Comparison of transient currents for vertical and horizontal electrode,L = 20 (m); σ = 1 mS/m.Comparing the results for the transient current between the vertical and horizontal grounding electrodes, it can be seen that the waveform of the induced current is very similar for all calculated parameters, but the maximum amplitude of the induced current for the horizontal electrode is greater by approximately 35–40%. This phenomenon can be explained by the greater influence of the ground-air interface on the induced current along the horizontal electrode as oppose to the vertically buried electrode. Additionally, for the vertical electrode energy dissipation is more emphasized due to medium conductivity.


## 5. Concluding Remarks

This paper deals with the analytical solution of the Pocklington integro-differential equation in the time domain for the vertical grounding electrode. The solution is derived using an approximation of the unknown current distribution, the Laplace transform and the Cauchy residue theorem, respectively. The obtained solution represents a closed-form expression for the distribution of the induced current along the vertical electrode. The equivalent current source is used as a model of a lightning strike current excitation.1.
Results obtained in the paper are compared with the results for the analytical solution of the Pocklington equation in the frequency domain2.
The obtained results agree satisfactorily for all calculated input parameters3.
Comparison with the results for the analytical solution of the horizontal electrode obtained in the time domain is performedPotential future work includes derivation of analytical expression for the transient scattered voltage that allows calculation of transient impedance for the vertical grounding electrode, as well as detailed Sensitivity Analysis to determine the influence of all parameters of interest.



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*Source: 8501420-2022-03-17.xml*