A Practical Algorithm for Calculating the Impulse Earthing Resistances of Vertical Earthing Electrodes

Yongzheng Zhang1*, Xiaoqing Zhang2

1Global Energy Interconnection Research Institute Co., Ltd., Beijing, China.

2School of Electrical Engineering, Beijing Jiaotong University, Beijing, China.

*Corresponding author:Yongzheng Zhang, Global Energy Interconnection Research Institute Co., Ltd., Beijing, China. E-mail: zhangyz2@126.com

Citation: Zhang Y, Zhang X. A Practical Algorithm for Calculating the Impulse Earthing Resistances of Vertical Earthing Electrodes. J Electron Adv Electr Eng. 2020;1(2):15-20.

Received Date: October 25, 2020; Accepted Date: December 10, 2020; Published Date: December 15, 2020

Abstract

An algorithm is proposed in this paper for calculating the impulse earthing resistances of vertical earthing electrodes. The proposed algorithm employs the average potential method to derive the formula of the low current earthing resistance. Unlike the previous algorithm, the soil ionization effect under high impulse current is taken into account by introducing a nonlinear characteristic to represent the relationship between the electric field and current density in the ionization zone around the earthing electrode. On the basis of the nonlinear characteristic, the effective radius is evaluated for the equivalent earthing electrode. Then, the impulse earthing resistance can be calculated by substituting the effective radius into the formula of the low current earthing resistance. A comparison is also made between calculated and measured results to confirm the validity of the proposed algorithm.

Keywords: Earthing Resistance; Vertical Earthing Electrode; Average Potential Integral; Soil Ionization; Current Density;

Introduction

The impulse earthing resistance of earthing electrodes is a very important factor which has to be taken into consideration in the lightning protection design of civil buildings and electric substations. An appropriate choice of protection measures against lightning overvoltage depends to a large degree on the knowledge of the values of the impulse earthing resistances of earthing electrodes. As the simplest form of earthing electrodes, vertical erathing electrodes are widely used in practical earthing systems. The impulse resistances of vertical earthing electrodes have been investigated for many years. Although a few algorithms were presented by different authors [1−3], their modeling for soil ionization effect under high impulse current is still a problem. These previous algorithms usually utilized a linear characteristic to represent the relationship between the electric field and current density in the ionization zone around the earthing electrode. In fact, this relationship has pronounced nonlinearity for typical kinds of soils in terms of the experimental investigation [4]. The nonlinear characteristic should therefore betaken into account in order to perform a more accurate calculation of the impulse earthing resistance.

The aim of this paper is to propose an efficient algorithm for calcu-lating the impulse resistances of vertical earthing electrodes. For the sake of engineering application, the inhomogeneity of soil is neglected. In the algorithm, the vertical earthing electrode is divided into a series of segments and the average potential method is employed to calculate the resistance matrix. The formula for the low current earthing resistance is derived from the resistance matrix. The soil ionization effect under high impulse current is further considered by using a nonlinear characteristic to represent the relationship between the electric field and current density in the ionization zone. With a simplified treatment made for the ionization zone, the effective radius of the equivalent earthing electrode is evaluated from the nonlinear characteristic. The impulse earthing resistance is then obtained by substituting the effective radius into the formula of the low current earthing resistance. Validity of the proposed algorithm has been verified by comparing calculated and measured results.

Low Current Earthing resistance

Consider a vertical earthing electrode in a homogeneous soil with the resistivity ρ, as shown in Figure 1. When a low current I diffuses from it into the soil, the soil ionization cannot be caused. According to the electromagnetic field theory, the presence of the interface between

Figure 1: Vertical earthing

air and soil can be taken into account by installing the image electrode[5], as illustrated in Figure 2. The image electrode depicted by the dotted line is installed above the earth surface; meanwhile the actual and image electrodes are symmetrical about the earth surface. In consideration of

Figure 2: Division of actual and image electrodes into N segments.

the diffused current distribution along the electrode length, the actual electrode and its image are divided into N segments, respectively (see Figure 2). At an arbitrary point on the surface of jth actual segment, as shown in Figure 3, the potential generated by the currents of jth actual segment and its image segment (j′ th segment) is calculated as [5]


ϕ j ( z s )= ρ 4π   z 2   z 1   τ dz (z z s ) 2 + r 0 2 +  z 1   z 2 τ j  d z ( z z s ) 2 + r 0 2                  (1) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaabQgaaeqaaOGaaiikaiaadQhadaWgaaWcbaGaam4Caaqa baGccaGGPaGaeyypa0ZaaSaaaeaacaqGbpaabaGaaGinaiaabc8aaa WaamWaaeaanmaapedakeaacaqGGaWaaSaaaeaacqaHepaDdaWgaaWc baGaaeOAaiaabccaaeqaaOGaamizaiaadQhaaeaadaGcaaqaaiaacI cacaWG6bGaeyOeI0IaamOEaKqbaoaaBaaaleaajugabiaabohaaSqa baGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOCamaaDa aaleaacaaIWaaabaGaaGOmaaaaaeqaaaaakiabgUcaR0Waa8qmaOqa amaalaaabaGafqiXdqNbauaadaWgaaWcbaGaaeOAaaqabaGccaqGGa GaamizaiqadQhagaqbaaqaamaakaaabaGaaiikaiqadQhagaqbaiab gkHiTiaadQhajuaGdaWgaaWcbaqcLbqacaqGZbaaleqaaOGaaiykam aaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkhadaqhaaWcbaGaaGim aaqaaiaaikdaaaaabeaaaaaabaqcLbqacaqGGaGaaeOEaKqbaoaaBa aameaajugabiaabgdaaWqabaaaleaajugabiaabccacaWG6bqcfa4a aSbaaWqaaKqzaeGaaGOmaaadbeaaaKqzGkGaey4kIipaaSqaaKqzae GaaeiiaiabgkHiTiaabQhajuaGdaWgaaadbaqcLbqacaqGYaaameqa aaWcbaqcLbqacaqGGaGaeyOeI0IaamOEaKqbaoaaBaaameaajugabi aaigdaaWqabaaajugOciabgUIiYdaakiaawUfacaGLDbaacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bIcacaqGXaGaaeykaaaa@8AAA@

where τj and τj are the linear current densities of jth segment and its image, respectively. As the segment is rather short, the diffused current distribution on the segment is approximately considered to be uniform. Letting Ij and Ij denote the diffused currents of jth segment and its image, respectively,

Figure 3: Sketch for calculating self resistance of jth segment.

τj and τj become τj= Ij/(z2-z1) and τj= Ij/(z2-z1). Owing to the image symmetry, we have Ij= Ij. Thus, (2.1) is rewritten as


ϕ j ( z s )= ρ I j 4 π(z 2 z 1 )   z 2   z 1   dz (z z s ) 2 + r 0 2 +  z 1   z 2 d z ( z z s ) 2 + r 0 2                 (2) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaabQgaaeqaaOGaaiikaiaadQhadaWgaaWcbaGaae4Caaqa baGccaGGPaGaeyypa0ZaaSaaaeaacaqGbpGaamysamaaBaaaleaaca qGQbaabeaaaOqaaiaaisdacaqGapGaaeikaiaabQhadaWgaaWcbaGa aeOmaaqabaGccqGHsislcaWG6bWaaSbaaSqaaiaaigdaaeqaaOGaai ykaaaadaWadaqaa0Waa8qmaOqaaiaabccadaWcaaqaaiaadsgacaWG 6baabaWaaOaaaeaacaGGOaGaamOEaiabgkHiTiaadQhajuaGdaWgaa WcbaqcLbqacaqGZbaaleqaaOGaaiykamaaCaaaleqabaGaaGOmaaaa kiabgUcaRiaadkhadaqhaaWcbaGaaGimaaqaaiaaikdaaaaabeaaaa GccqGHRaWknmaapedakeaadaWcaaqaaiaadsgaceWG6bGbauaaaeaa daGcaaqaaiaacIcaceWG6bGbauaacqGHsislcaWG6bqcfa4aaSbaaS qaaKqzaeGaae4CaaWcbeaakiaacMcadaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcaWGYbWaa0baaSqaaiaaicdaaeaacaaIYaaaaaqabaaaaa qaaKqzaeGaaeiiaiaabQhajuaGdaWgaaadbaqcLbqacaqGXaaameqa aaWcbaqcLbqacaqGGaGaamOEaKqbaoaaBaaameaajugabiaaikdaaW qabaaajugOciabgUIiYdaaleaajugabiaabccajugWaiabgkHiTKqz aeGaaeOEaKqbaoaaBaaameaajugabiaabkdaaWqabaaaleaajugabi aabccajugWaiabgkHiTKqzaeGaamOEaKqbaoaaBaaameaajugabiaa igdaaWqabaaajugOciabgUIiYdaakiaawUfacaGLDbaacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabk dacaqGPaaaaa@8E2C@

By taking the integral average of φj(zs) over the length of jth segment, the average potential is obtained as


ϕ  j a = 1 (z 2 z 1 )   z 2   z 1   ϕ j ( z s )dz       = ρ I j 4 π(z 2 z 1 ) 2   z 2   z 1   z 2   z 1   dzd z s (z z s ) 2 + r 0 2 +   z 2   z 1  z 1   z 2 d z d z s ( z z s ) 2 + r 0 2       = ρ r 0 I j 4 π(z 2 z 1 ) 2 2+2Ψ z 2 z 1 r 0 2Ψ z 1 + z 2 r 0 +Ψ 2 z 1 r 0 +Ψ 2 z 2 r 0             (3) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHvp GzdaqhaaWcbaGaaeiiaiaabQgaaeaajugabiaabggaaaGccqGH9aqp daWcaaqaaiaaigdaaeaacaqGOaGaaeOEamaaBaaaleaacaqGYaaabe aakiabgkHiTiaadQhadaWgaaWcbaGaaGymaaqabaGccaGGPaaaa0Wa a8qmaOqaaiaabccacqaHvpGzdaWgaaWcbaGaaeOAaaqabaGccaGGOa GaamOEamaaBaaaleaacaqGZbaabeaakiaacMcacaWGKbGaamOEaaWc baqcLbqacaqGGaGaeyOeI0IaaeOEaKqbaoaaBaaameaajugabiaabk daaWqabaaaleaajugabiaabccajugWaiabgkHiTKqzaeGaamOEaKqb aoaaBaaameaajugabiaaigdaaWqabaaajugOciabgUIiYdaakeaaju gGbiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiabg2da9OWa aSaaaeaacaqGbpGaamysamaaBaaaleaacaqGQbaabeaaaOqaaiaais dacaqGapGaaeikaiaabQhadaWgaaWcbaGaaeOmaaqabaGccqGHsisl caWG6bWaaSbaaSqaaiaaigdaaeqaaOGaaiykamaaCaaaleqabaGaaG OmaaaaaaGcdaWadaqaa0Waa8qmaOqaa0Waa8qmaOqaaiaabccadaWc aaqaaiaadsgacaWG6bGaamizaiaadQhadaWgaaWcbaGaae4Caaqaba aakeaadaGcaaqaaiaacIcacaWG6bGaeyOeI0IaamOEaKqbaoaaBaaa leaajugabiaabohaaSqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaO Gaey4kaSIaamOCamaaDaaaleaacaaIWaaabaGaaGOmaaaaaeqaaaaa kiabgUcaR0Waa8qmaOqaa0Waa8qmaOqaamaalaaabaGaamizaiqadQ hagaqbaiaadsgacaWG6bWaaSbaaSqaaiaabohaaeqaaaGcbaWaaOaa aeaacaGGOaGabmOEayaafaGaeyOeI0IaamOEaKqbaoaaBaaaleaaju gabiaabohaaSqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamOCamaaDaaaleaacaaIWaaabaGaaGOmaaaaaeqaaaaaaeaaju gabiaabccacaqG6bqcfa4aaSbaaWqaaKqzaeGaaeymaaadbeaaaSqa aKqzaeGaaeiiaiaadQhajuaGdaWgaaadbaqcLbqacaaIYaaameqaaa qcLbQacqGHRiI8aaWcbaqcLbqacaqGGaqcLbmacqGHsisljugabiaa bQhajuaGdaWgaaadbaqcLbqacaqGYaaameqaaaWcbaqcLbqacaqGGa qcLbmacqGHsisljugabiaadQhajuaGdaWgaaadbaqcLbqacaaIXaaa meqaaaqcLbQacqGHRiI8aaWcbaqcLbqacaqGGaqcLbmacqGHsislju gabiaabQhajuaGdaWgaaadbaqcLbqacaqGYaaameqaaaWcbaqcLbqa caqGGaqcLbmacqGHsisljugabiaadQhajuaGdaWgaaadbaqcLbqaca aIXaaameqaaaqcLbQacqGHRiI8aaWcbaqcLbqacaqGGaqcLbmacqGH sisljugabiaabQhajuaGdaWgaaadbaqcLbqacaqGYaaameqaaaWcba qcLbqacaqGGaGaeyOeI0IaamOEaKqbaoaaBaaameaajugabiaaigda aWqabaaajugOciabgUIiYdaakiaawUfacaGLDbaaaeaacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacqGH9aqpdaWcaaqaaiaabg8a caWGYbWaaSbaaSqaaiaabcdaaeqaaOGaamysamaaBaaaleaacaqGQb aabeaaaOqaaiaaisdacaqGapGaaeikaiaabQhadaWgaaWcbaGaaeOm aaqabaGccqGHsislcaWG6bWaaSbaaSqaaiaaigdaaeqaaOGaaiykam aaCaaaleqabaGaaGOmaaaaaaGcdaWadaqaaiaaikdacqGHRaWkcaaI YaGaamiQdmaabmaabaWaaSaaaeaacaWG6bWaaSbaaSqaaiaaikdaae qaaOGaeyOeI0IaamOEamaaBaaaleaacaaIXaaabeaaaOqaaiaadkha daWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaGaeyOeI0IaaG OmaiaadI6adaqadaqaamaalaaabaGaamOEamaaBaaaleaacaaIXaaa beaakiabgUcaRiaadQhadaWgaaWcbaGaaGOmaaqabaaakeaacaWGYb WaaSbaaSqaaiaaicdaaeqaaaaaaOGaayjkaiaawMcaaiabgUcaRiaa dI6adaqadaqaamaalaaabaGaaGOmaiaadQhadaWgaaWcbaGaaGymaa qabaaakeaacaWGYbWaaSbaaSqaaiaaicdaaeqaaaaaaOGaayjkaiaa wMcaaiabgUcaRiaadI6adaqadaqaamaalaaabaGaaGOmaiaadQhada WgaaWcbaGaaGOmaaqabaaakeaacaWGYbWaaSbaaSqaaiaaicdaaeqa aaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabIcacaqGZaGaaeykaaaaaa@114B@

where

Ψ ξ =ξ  sinh -1 ξ 1+ ξ 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiQdmaabm aabaGaeqOVdGhacaGLOaGaayzkaaGaeyypa0JaeqOVdGNaaeiiaiaa bohacaqGPbGaaeOBaiaabIgadaahaaWcbeqaaKqzGeGaamylaSGaae ymaaaakiabe67a4jabgkHiTmaakaaabaGaaGymaiabgUcaRiabe67a 4naaCaaaleqabaGaaGOmaaaaaeqaaaaa@4ADA@

The self resistance of jth segment is given as


R jj = ϕ  j a I j        (4) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaqGQbGaaeOAaaqabaGccqGH9aqpdaWcaaqaaiabew9aMnaa DaaaleaacaqGGaGaaeOAaaqaaKqzaeGaaeyyaaaaaOqaaiaadMeada WgaaWcbaGaaeOAaaqabaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabIcacaqG0aGaaeykaaaa@4747@

In a similar manner, the average potential generated by the currents of kth segment and its image on the surface of jth segment, as shown in Figure 4, is found as


ϕ jk a = ρ I k 4 π(z 2 z 1 ) z 4 z 3   z 2   z 1   z 4   z 3   dzd z s (z z s ) 2 + r 0 2 +   z 2   z 1  z 3   z 4 d z d z s ( z z s ) 2 + r 0 2       = ρ r 0 I k 4 π(z 2 z 1 ) z 4 z 3 Ψ z 3 z 2 r 0 Ψ z 3 z 1 r 0 +Ψ z 4 z 1 r 0 Ψ z 3 z 1 r 0                                            Ψ z 2 + z 3 r 0 Ψ z 1 + z 3 r 0 +Ψ z 1 + z 4 r 0 Ψ z 4 + z 2 r 0             (5) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHvp GzdaqhaaWcbaGaaeOAaiaabUgaaeaajugabiaabggaaaGccqGH9aqp daWcaaqaaiaabg8acaWGjbWaaSbaaSqaaiaabUgaaeqaaaGcbaGaaG inaiaabc8acaqGOaGaaeOEamaaBaaaleaacaqGYaaabeaakiabgkHi TiaadQhadaWgaaWcbaGaaGymaaqabaGccaGGPaWaaeWaaeaacaqG6b WaaSbaaSqaaiaabsdaaeqaaOGaeyOeI0IaamOEamaaBaaaleaacaaI ZaaabeaaaOGaayjkaiaawMcaaaaadaWadaqaa0Waa8qmaOqaa0Waa8 qmaOqaaiaabccadaWcaaqaaiaadsgacaWG6bGaamizaiaadQhadaWg aaWcbaGaae4CaaqabaaakeaadaGcaaqaaiaacIcacaWG6bGaeyOeI0 IaamOEaKqbaoaaBaaaleaajugabiaabohaaSqabaGccaGGPaWaaWba aSqabeaacaaIYaaaaOGaey4kaSIaamOCamaaDaaaleaacaaIWaaaba GaaGOmaaaaaeqaaaaakiabgUcaR0Waa8qmaOqaa0Waa8qmaOqaamaa laaabaGaamizaiqadQhagaqbaiaadsgacaWG6bWaaSbaaSqaaiaabo haaeqaaaGcbaWaaOaaaeaacaGGOaGabmOEayaafaGaeyOeI0IaamOE aKqbaoaaBaaaleaajugabiaabohaaSqabaGccaGGPaWaaWbaaSqabe aacaaIYaaaaOGaey4kaSIaamOCamaaDaaaleaacaaIWaaabaGaaGOm aaaaaeqaaaaaaeaajugabiaabccacaqG6bqcfa4aaSbaaWqaaKqzae Gaae4maaadbeaaaSqaaKqzaeGaaeiiaiaadQhajuaGdaWgaaadbaqc LbqacaaI0aaameqaaaqcLbQacqGHRiI8aaWcbaqcLbqacaqGGaqcLb macqGHsisljugabiaabQhajuaGdaWgaaadbaqcLbqacaqGYaaameqa aaWcbaqcLbqacaqGGaqcLbmacqGHsisljugabiaadQhajuaGdaWgaa adbaqcLbqacaaIXaaameqaaaqcLbQacqGHRiI8aaWcbaqcLbqacaqG GaqcLbmacqGHsisljugabiaabQhajuaGdaWgaaadbaqcLbqacaqG0a aameqaaaWcbaqcLbqacaqGGaqcLbmacqGHsisljugabiaadQhajuaG daWgaaadbaqcLbqacaaIZaaameqaaaqcLbQacqGHRiI8aaWcbaqcLb qacaqGGaqcLbmacqGHsisljugabiaabQhajuaGdaWgaaadbaqcLbqa caqGYaaameqaaaWcbaqcLbqacaqGGaGaeyOeI0IaamOEaKqbaoaaBa aameaajugabiaaigdaaWqabaaajugOciabgUIiYdaakiaawUfacaGL DbaaaeaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacqGH9a qpdaWcaaqaaiaabg8acaWGYbWaaSbaaSqaaiaabcdaaeqaaOGaamys amaaBaaaleaacaqGRbaabeaaaOqaaiaaisdacaqGapGaaeikaiaabQ hadaWgaaWcbaGaaeOmaaqabaGccqGHsislcaWG6bWaaSbaaSqaaiaa igdaaeqaaOGaaiykamaabmaabaGaaeOEamaaBaaaleaacaqG0aaabe aakiabgkHiTiaadQhadaWgaaWcbaGaaG4maaqabaaakiaawIcacaGL PaaaaaWaamqaaeaacaWGOoWaaeWaaeaadaWcaaqaaiaadQhadaWgaa WcbaGaaG4maaqabaGccqGHsislcaWG6bWaaSbaaSqaaiaaikdaaeqa aaGcbaGaamOCamaaBaaaleaacaaIWaaabeaaaaaakiaawIcacaGLPa aacqGHsislcaWGOoWaaeWaaeaadaWcaaqaaiaadQhadaWgaaWcbaGa aG4maaqabaGccqGHsislcaWG6bWaaSbaaSqaaiaaigdaaeqaaaGcba GaamOCamaaBaaaleaacaaIWaaabeaaaaaakiaawIcacaGLPaaacqGH RaWkcaWGOoWaaeWaaeaadaWcaaqaaiaadQhadaWgaaWcbaGaaGinaa qabaGccqGHsislcaWG6bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOC amaaBaaaleaacaaIWaaabeaaaaaakiaawIcacaGLPaaacqGHsislca WGOoWaaeWaaeaadaWcaaqaaiaadQhadaWgaaWcbaGaaG4maaqabaGc cqGHsislcaWG6bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOCamaaBa aaleaacaaIWaaabeaaaaaakiaawIcacaGLPaaaaiaawUfaaaqaaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiamaadiaabaGaamiQdmaabmaabaWaaSaaaeaacaWG6bWaaSba aSqaaiaaikdaaeqaaOGaey4kaSIaamOEamaaBaaaleaacaaIZaaabe aaaOqaaiaadkhadaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzk aaGaeyOeI0IaamiQdmaabmaabaWaaSaaaeaacaWG6bWaaSbaaSqaai aaigdaaeqaaOGaey4kaSIaamOEamaaBaaaleaacaaIZaaabeaaaOqa aiaadkhadaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaGaey 4kaSIaamiQdmaabmaabaWaaSaaaeaacaWG6bWaaSbaaSqaaiaaigda aeqaaOGaey4kaSIaamOEamaaBaaaleaacaaI0aaabeaaaOqaaiaadk hadaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaGaeyOeI0Ia amiQdmaabmaabaWaaSaaaeaacaWG6bWaaSbaaSqaaiaaisdaaeqaaO Gaey4kaSIaamOEamaaBaaaleaacaaIYaaabeaaaOqaaiaadkhadaWg aaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaaacaGLDbaacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGOaGaaeynaiaabMcaaaaa@3C80@

where Ik is the current of kth segment. The mutual resistance between jth and kth segments is given as


R jk = ϕ  jk a I k        (6) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaqGQbGaae4AaaqabaGccqGH9aqpdaWcaaqaaiabew9aMnaa DaaaleaacaqGGaGaaeOAaiaabUgaaeaajugabiaabggaaaaakeaaca WGjbWaaSbaaSqaaiaabUgaaeqaaaaakiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGOaGaaeOnaiaabMcaaaa@4839@

Figure 4: Sketch for calculating mutual resistance between jth and kth segments.

By taking the self and mutual resistances as the matrix elements, the resistance matrix of the vertical earthing electrode is formed as

R= R 11 R 12 R 1N R 21 R 22 R 2N R N1 R N2 R NN                (7) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGaa8Nuai abg2da9maadmaabaqbaeqabqabaaaaaeaacaWGsbWaaSbaaSqaaiaa igdacaaIXaaabeaaaOqaaiaadkfadaWgaaWcbaGaaGymaiaaikdaae qaaaGcbaqcLbKacqGHflY1ieqacaGFGaGaeyyXICTaa4hiaiabgwSi xdGcbaGaamOuamaaBaaaleaacaaIXaGaamOtaaqabaaakeaacaWGsb WaaSbaaSqaaiaaikdacaaIXaaabeaaaOqaaiaadkfadaWgbaWcbaGa aGOmaiaaikdaaeqaaaGcbaqcLbKacqGHflY1caGFGaGaeyyXICTaa4 hiaiabgwSixdGcbaGaamOuamaaBaaaleaacaaIYaGaamOtaaqabaaa keaajugGbiabl6UinbGcbaGaeSO7I0eabaGaeSO7I0eabaGaeSO7I0 eabaGaamOuamaaBaaaleaacaWGobGaaGymaaqabaaakeaacaWGsbWa aSbaaSqaaiaad6eacaaIYaaabeaaaOqaaKqzajGaeyyXICTaa4hiai abgwSixlaa+bcacqGHflY1aOqaaiaadkfadaWgaaWcbaGaamOtaiaa d6eaaeqaaaaaaOGaay5waiaaw2faaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabIcacaqG3aGaaeykaaaa@7FFE@

Using R gives the relationship between voltage and current


U 1 U 2 U N = R 11 R 12 R 1N R 21 R 22 R 2N R N1 R N2 R NN   I 1 I 2 I N              (8) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeabbaaaaeaacaWGvbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyv amaaBaaaleaacaaIYaaabeaaaOqaaiabl6UinbqaaiaadwfadaWgaa WcbaGaamOtaaqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaeaa faqabeabeaaaaaqaaiaadkfadaWgaaWcbaGaaGymaiaaigdaaeqaaa GcbaGaamOuamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaajugqciab gwSixJqabiaa=bcacqGHflY1caWFGaGaeyyXICnakeaacaWGsbWaaS baaSqaaiaaigdacaWGobaabeaaaOqaaiaadkfadaWgaaWcbaGaaGOm aiaaigdaaeqaaaGcbaGaamOuamaaBeaaleaacaaIYaGaaGOmaaqaba aakeaajugqciabgwSixlaa=bcacqGHflY1caWFGaGaeyyXICnakeaa caWGsbWaaSbaaSqaaiaaikdacaWGobaabeaaaOqaaKqzagGaeSO7I0 eakeaacqWIUlstaeaacqWIUlstaeaacqWIUlstaeaacaWGsbWaaSba aSqaaiaad6eacaaIXaaabeaaaOqaaiaadkfadaWgaaWcbaGaamOtai aaikdaaeqaaaGcbaqcLbKacqGHflY1caWFGaGaeyyXICTaa8hiaiab gwSixdGcbaGaamOuamaaBaaaleaacaWGobGaamOtaaqabaaaaaGcca GLBbGaayzxaaGaaeiiamaadmaabaqbaeqabqqaaaaabaGaamysamaa BaaaleaacaaIXaaabeaaaOqaaiaadMeadaWgaaWcbaGaaGOmaaqaba aakeaacqWIUlstaeaacaWGjbWaaSbaaSqaaiaad6eaaeqaaaaaaOGa ay5waiaaw2faaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGa aeioaiaabMcaaaa@9135@

where Uj (j=1, 2, ..., N) is the voltage of jth segment. As the conducing surfaces are equipotential, we have U1=U2= ··· =UN=U [5]. Calculation of the inverse matrix R−1 gives the following expression


I 1 I 2 I N = G 11 G 12 G 1N G 21 G 22 G 2N G N1 G N2 G NN   U U U = k=1 N G 1k k=1 N G 2k k=1 N G Nk U                 (9) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeabbaaaaeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamys amaaBaaaleaacaaIYaaabeaaaOqaaiabl6UinbqaaiaadMeadaWgaa WcbaGaamOtaaqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaeaa faqabeabeaaaaaqaaiaadEeadaWgaaWcbaGaaGymaiaaigdaaeqaaa GcbaGaam4ramaaBaaaleaacaaIXaGaaGOmaaqabaaakeaajugqciab gwSixJqabiaa=bcacqGHflY1caWFGaGaeyyXICnakeaacaWGhbWaaS baaSqaaiaaigdacaWGobaabeaaaOqaaiaadEeadaWgaaWcbaGaaGOm aiaaigdaaeqaaaGcbaGaam4ramaaBeaaleaacaaIYaGaaGOmaaqaba aakeaajugqciabgwSixlaa=bcacqGHflY1caWFGaGaeyyXICnakeaa caWGhbWaaSbaaSqaaiaaikdacaWGobaabeaaaOqaaKqzagGaeSO7I0 eakeaacqWIUlstaeaacqWIUlstaeaacqWIUlstaeaacaWGhbWaaSba aSqaaiaad6eacaaIXaaabeaaaOqaaiaadEeadaWgaaWcbaGaamOtai aaikdaaeqaaaGcbaqcLbKacqGHflY1caWFGaGaeyyXICTaa8hiaiab gwSixdGcbaGaam4ramaaBaaaleaacaWGobGaamOtaaqabaaaaaGcca GLBbGaayzxaaGaaeiiamaadmaabaqbaeqabqqaaaaabaGaamyvaaqa aiaadwfaaeaacqWIUlstaeaacaWGvbaaaaGaay5waiaaw2faaiabg2 da9maadmaabaqbaeqabqqaaaaabaWaaabCaeaacaWGhbWaaSbaaSqa aiaaigdacaWGRbaabeaaaeaacaWGRbGaeyypa0JaaGymaaqaaiaad6 eaa0GaeyyeIuoaaOqaamaaqahabaGaam4ramaaBaaaleaacaaIYaGa am4AaaqabaaabaGaam4Aaiabg2da9iaaigdaaeaacaWGobaaniabgg HiLdaakeaacqWIUlstaeaadaaeWbqaaiaadEeadaWgaaWcbaGaamOt aiaadUgaaeqaaaqaaiaadUgacqGH9aqpcaaIXaaabaGaamOtaaqdcq GHris5aaaaaOGaay5waiaaw2faaiaadwfacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG5aGa aeykaaaa@AF95@

where Gjk (j, k=1, 2, ..., N) is the element of inverse matrix R−1. In terms of (9), the total current is found as


I= j=1 N I j =U j=1 N k=1 N G jk       (10) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2 da9maaqahabaGaamysamaaBaaaleaacaqGQbaabeaaaeaacaWGQbGa eyypa0JaaGymaaqaaiaad6eaa0GaeyyeIuoakiabg2da9iaadwfanm aaqahakeaanmaaqahakeaacaWGhbWaaSbaaSqaaiaabQgacaqGRbaa beaaaeaacaWGRbGaeyypa0JaaGymaaqaaiaad6eaaKqzajGaeyyeIu oaaSqaaiaadQgacqGH9aqpcaaIXaaabaGaamOtaaqcLbKacqGHris5 a0GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabg dacaqGWaGaaeykaaaa@57B7@

As a result, the earthing resistance can be obtained as


R e = U I = 1 j=1 N k=1 N G jk              (11) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaqGLbaabeaakiabg2da9maalaaabaGaamyvaaqaaiaadMea aaGaeyypa0ZaaSaaaeaacaaIXaaabaqddaaeWbGcbaqddaaeWbGcba Gaam4ramaaBaaaleaacaqGQbGaae4AaaqabaaabaGaam4Aaiabg2da 9iaaigdaaeaacaWGobaajugqciabggHiLdaaleaacaWGQbGaeyypa0 JaaGymaaqaaiaad6eaaKqzajGaeyyeIuoaaaGccaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeikaiaabgdacaqGXaGaaeykaaaa@574F@

As shown in Figure 1, a vertical earthing electrode of r0=0.015m is considered here. The earthing electrode is made of plain carbon steel (con-ductivity σ=0.5×107 S•m-1). The soil is kind of clay. Its resistivity is ρ=42Ω·m and water content is about 20%. The earthing resistances calculated from (11) are given in Figure 5. The corresponding measured values are given simultaneously in Figure 5, which was obtained by the fall of potential method [6]. As indicated in Figure 5, the calculated values are close to the measured ones.

Figure 5: Calculated and measured values of low current earthing resistance.

Impulse Earthing Resistance

For a high impulse current with crest value I, representative of lightning, the impulse resistance Rie of the earthing electrode is defined as a ratio of the crest value Um of the impulse potential on the earthing electrode to I, i.e. Rie=Um/I [7]. The impulse current can produces great current density and high electric field intensity near the earthing electrode. When the electric field intensity on the surface of the earthing electrode exceeds the critical value Ec of soil ionization gradient, the breakdown will occur. This process can be illustrated by Figure 6 [8].

Figure 6: Impulse breakdown of soil around a vertical earthing electrode, 1—arc zone, 2—streamer zone, 3—semiconductive zone, 4—constant resistivity zone.

As the current increases, streamers are developed and in turn arcs are generated. Within the streamer and arc zones, the resistivity decreases from its original value to a limit approaching conductor [8]. In addition, there is a semiconductive zone between the streamer zone and the constant resistivity zone. For the purpose of simplifying calculation, this process can be described by an equivalent model shown in Figure 7. In the equivalent model, the semiconductive zone is neglected since it is small.

Figure 7: Ionization zone, (a) non-uniform along electrode length, (b) uniform along electrode length.

The streamer and arc zones are equivalently modeled as an ionization zone [8,9]. As the vertical earthing electrodes used in the actual cases usually have a shorter length (less than 5m), the non-uniformity of the ionization zone along the electrode length, as shown in Figure 7(a), is not pronounced and may be neglected. Therefore, the ionization zone is approximately considered to be uniform along the electrode length, as shown in Figure 7(b) [8−10]. The border of theionizationzoneis delimitedbythecriticalelectric field value Ec. A nonlinear characteris-tic is introduced to represent the relationship between the electric field andcurrent density in theionizationzone. The nonlinear characteristic is given as [4]


E=a J b      (12) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9iaadggacaWGkbWaaWbaaSqabeaacaWGIbaaaOGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabkdacaqGPaaaaa@4086@

where a and b are constants and J is the current density


J= I 2π r l+2π r 2 = I 2πr r+l          (13) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9maalaaabaGaamysaaqaaiaaikdacaqGapGaamOCamaaBaaaleaa aeqaaOGaamiBaiabgUcaRiaaikdacaqGapGaamOCamaaDaaaleaaae aacaaIYaaaaaaakiabg2da9maalaaabaGaamysaaqaaiaaikdacaqG apGaamOCamaabmaabaGaamOCaiabgUcaRiaadYgaaiaawIcacaGLPa aaaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeikaiaabgdacaqGZaGaaeykaaaa@5341@

Substituting (12) into (13), the electric field intensity is rewritten as

E=a I b 2π b r b r+l b         (14) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9iaadggadaWcaaqaaiaadMeadaahaaWcbeqaaiaadkgaaaaakeaa daqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaai aadkgaaaGccaWGYbWaaWbaaSqabeaacaWGIbaaaOWaaeWaaeaacaWG YbGaey4kaSIaamiBaaGaayjkaiaawMcaamaaCaaaleqabaGaamOyaa aaaaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabIcacaqGXaGaaeinaiaabMcaaaa@4F26@

On the border of the ionization zone, E should satisfy the boundary condition.


E r= r i = E c       (15) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqGaaeaaca WGfbaacaGLiWoadaWgaaWcbaqcLbqacaWGYbGaeyypa0JaamOCaKqb aoaaBaaameaajugabiaadMgaaWqabaaaleqaaOGaeyypa0Jaamyram aaBaaaleaacaqGJbaabeaakiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabIcacaqGXaGaaeynaiaabMcaaaa@479D@

The values of a, b and Ec can be found in [4] for typical kinds of soils. Thus, the boundary radius of the ionization zone can be derived from (14) and (15)


r i = l+ l 2 +4Q 2        (16) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiabg2da9maalaaabaGaeyOeI0IaamiBaiab gUcaRmaakaaabaGaamiBamaaCaaaleqabaGaaGOmaaaakiabgUcaRi aaisdacaWGrbaaleqaaaGcbaGaaGOmaaaacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG2aGaaeykaa aa@4859@

where


Q= I 2π a E c 1 b         (17) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuaiabg2 da9maalaaabaGaamysaaqaaiaaikdacaqGapaaamaabmaabaWaaSaa aeaacaWGHbaabaGaamyramaaBaaaleaacaqGJbaabeaaaaaakiaawI cacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaadkgaaaaa aOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGOaGaaeymaiaabEdacaqGPaaaaa@48DB@

The soil ionization is basically equivalent to an increase in the dimension of the earthing electrode, which can be taken into account by the effective radius re of the equivalent earthing electrode in constant resistivity zone, asshowninFigure 8. The voltage between the earthing electrode and the border of the ionization zone, as shown in Figure 8(a), is expressed as


U i =   r 0   r i a I b 2π b 1 r b r+1 b dr= a I b 2π b   W          (18) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGPbaabeaakiabg2da90Waa8qmaOqaamaalaaabaGaamyy aiaadMeadaahaaWcbeqaaiaadkgaaaaakeaadaqadaqaaiaaikdaca qGapaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGIbaaaaaakmaalaaa baGaaGymaaqaaiaadkhadaahaaWcbeqaaiaadkgaaaGcdaqadaqaai aadkhacqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWG IbaaaaaakiaadsgacaWGYbGaeyypa0ZaaSaaaeaacaWGHbGaamysam aaCaaaleqabaGaamOyaaaaaOqaamaabmaabaGaaGOmaiaabc8aaiaa wIcacaGLPaaadaahaaWcbeqaaiaadkgaaaaaaOGaaeiiaaWcbaqcLb qacaqGGaGaamOCaKqbaoaaBaaameaajugWaiaaicdaaWqabaaaleaa jugabiaabccacaWGYbWcdaWgaaadbaqcLboacaWGPbaameqaaaqcLb QacqGHRiI8aKqzagGaam4vaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabI dacaqGPaaaaa@6B30@

where

W=   r 0   r i 1 r b r+1 b dr         (19) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da90Waa8qmaOqaamaalaaabaGaaGymaaqaaiaadkhadaahaaWcbeqa aiaadkgaaaGcdaqadaqaaiaadkhacqGHRaWkcaaIXaaacaGLOaGaay zkaaWaaWbaaSqabeaacaWGIbaaaaaakiaadsgacaWGYbGaaeiiaaWc baqcLbqacaqGGaGaamOCaKqbaoaaBaaameaajugWaiaaicdaaWqaba aaleaajugabiaabccacaWGYbWcdaWgaaadbaqcLboacaWGPbaameqa aaqcLbQacqGHRiI8a0GaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGXaGaaeyoaiaabMcaaaa@564C@

On the other hand, the voltage between the equivalent earthing electrode and the border of the ionization zone, as shown in Figure 8(b), is expressed as


U e =  re   r i ρI 2πr r+l   dr= ρI 2πl ln r i r e +l r e r i +l              (20) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaqGLbaabeaakiabg2da90Waa8qmaOqaamaalaaabaGaeqyW diNaamysaaqaaiaaikdacaqGapGaamOCamaabmaabaGaamOCaiabgU caRiaadYgaaiaawIcacaGLPaaaaaGaaeiiaaWcbaqcLbqacaqGGaGa amOCaSGaaeyzaaqaaKqzaeGaaeiiaiaadkhalmaaBaaameaajug4ai aadMgaaWqabaaajugOciabgUIiYdqcLbyacaWGKbGaamOCaiabg2da 9OWaaSaaaKqzagqaaiabeg8aYjaadMeaaeaacaaIYaGaaeiWdiaadY gaaaGaciiBaiaac6gakmaalaaajugGbeaacaWGYbWcdaWgaaqaaiaa dMgaaeqaaOWaaeWaaeaacaWGYbWaaSbaaSqaaiaabwgaaeqaaOGaey 4kaSIaamiBaaGaayjkaiaawMcaaaqcLbyabaGaamOCaSWaaSbaaeaa caqGLbaabeaakmaabmaabaGaamOCamaaBaaaleaacaWGPbaabeaaki abgUcaRiaadYgaaiaawIcacaGLPaaaaaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGYaGaaeimaiaabMcaaaa@76A0@

Figure 8: Sketch for evaluation of the effective radius, 1—ionization zone, 2—constant resistivity zone, (a) ionization zone around the actual earthing electrode, (b) constant resistivity zone around the equivalent earthing electrode.

Because Ui and Ue must be equal, the effective radius of the equivalent eathing electrode can be determined from (18) and (20)


r e = r i l r i λ1 +λl           (21) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaqGLbaabeaakiabg2da9maalaaabaGaamOCamaaBaaaleaa caWGPbaabeaakiaadYgaaeaacaWGYbWaaSbaaSqaaiaadMgaaeqaaO WaaeWaaeaacqaH7oaBcqGHsislcaaIXaaacaGLOaGaayzkaaGaey4k aSIaeq4UdWMaamiBaaaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqGXaGa aeykaaaa@4FD0@

where λ=exp(χ) and


χ= al I b1 ρ (2π) b1 W         (22) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey ypa0ZaaSaaaeaacaWGHbGaamiBaiaadMeadaahaaWcbeqaaiaadkga cqGHsislcaaIXaaaaaGcbaGaeqyWdiNaaiikaiaaikdacaqGapGaai ykamaaCaaaleqabaGaamOyaiabgkHiTiaaigdaaaaaaOGaam4vaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabIcacaqGYaGaaeOmaiaabMcaaaa@4F65@

As the integration for (19) is difficult to be evaluated analytically, a numerical solution is given to obtain W


W Δr 2 n=1 N 1 r n1 b r n1 +l b + 1 r n b r n +l b           (23) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabgI Ki7oaalaaabaGaeuiLdqKaamOCaaqaaiaaikdaaaqddaaeWbGcbaWa amWaaeaadaWcaaqaaiaaigdaaeaacaWGYbWaa0baaSqaaiaab6gacq GHsislcaaIXaaabaGaamOyaaaakmaabmaabaGaamOCamaaBaaaleaa caqGUbGaeyOeI0IaaeymaaqabaGccqGHRaWkcaWGSbaacaGLOaGaay zkaaWaaWbaaSqabeaacaWGIbaaaaaakiabgUcaRmaalaaabaGaaGym aaqaaiaadkhadaqhaaWcbaGaamOBaaqaaiaadkgaaaGcdaqadaqaai aadkhadaWgaaWcbaGaamOBaaqabaGccqGHRaWkcaWGSbaacaGLOaGa ayzkaaWaaWbaaSqabeaacaWGIbaaaaaaaOGaay5waiaaw2faaaWcba GaamOBaiabg2da9iaaigdaaeaacaWGobaajugOciabggHiLdqdcaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeikaiaabkdacaqGZaGaaeykaaaa@66AD@

where Δr is the radial step and N=(ri-r0)/Δr.

By replacing r0 with re in (11), the impulse earthing resistance can be obtained. To check the validity of the proposed algorithm, two numerical examples are given here. The data for the first example are: r0=0.0127m, l=3.05m, ρ=87.2Ω·m, Ec=127kV/m, a=3094.6 and b=0.51 [4]. At different current crest values,the calculated impulse earthing resistances are shown in Figure 9.

Figure 9: Calculated and measured impulse earthing resistances.

The data for the second example are: r0=0.025m, l=1m, ρ=43.5Ω·m, Ec=350kV/m, a=219.05 and b=0.82 [4]. The calculated crest potentials on the earthing electrode (Um=RieI) are shown in Figure 10. Furthermore, the corresponding measured results [11,12] are given in Figures 9 and 10, respectively, for comparison. It can be seen from Figures 9 and 10 that a better agreement appears between calculated and measured results.

Figure 10: Calculated and measured crest potentials.

Conclusions

An algorithm for calculating the impulse earthing resistances of vertical earthing electrodes has been proposed in this paper. It uses the average potential method to derive the formula of the low current earthing resistances. In the ionization zone, a nonlinear characteristic is intro-duced in representingtherelationship between electric field and current density and has the capability of giving an appropriate consideration for the soil ionization effect under high impulse current. On the basis of the nonlinear characteristic, a simplified procedure has been developed to evaluate the effective radius of the equivalent earthing electrode. The impulse earthing resistance can then be obtained by substituting the effective radius into the formula of the low current earthing resistance. The calculated results are compared whith the measured ones and an agreement appears between them, which shows the applicability of the proposed algorithm in practical lightning protection design.

References

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