Research Article On the Lightning Electromagnetic Fields due ...downloads.hindawi.com/journals/mpe/2015/150756.pdfResearch Article On the Lightning Electromagnetic Fields due to Channel

Research ArticleOn the Lightning Electromagnetic Fields due to Channel withVariable Return Stroke Velocity

M Izadi12 M Z A Ab Kadir1 and M Hajikhani1

1Centre for Electromagnetic and Lightning Protection Research (CELP) Faculty of Engineering University Putra Malaysia(UPM) 43400 Serdang Selangor Malaysia2Department of Electrical Engineering Islamic Azad University Firoozkooh Branch 3981838381 Firoozkooh Iran

Correspondence should be addressed to M Izadi aryaphaseyahoocom

Received 1 October 2014 Accepted 30 January 2015

Academic Editor Xiao-Qiao He

Copyright copy 2015 M Izadi et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Numerical field expressions are proposed to evaluate the electromagnetic fields due to the lightning channel with variable values ofreturn stroke velocity Previous calculation methods generally use an average value for the return stroke velocity along a lightningchannel The proposed method can support different velocity profiles along a lightning channel in addition to the widely usedchannel-base current functions and also the general form of the engineering current models directly in the time domain withoutthe need to apply any extra conversions Moreover a sample of the measured lightning current is used to validate the proposedmethod while the velocity profile is simulated by the general velocity functionThe simulated fields based on constant and variablevalues of velocity are compared to the corresponding measured fields The results show that the simulated fields based on theproposed method are in good agreement with the corresponding measured fields

1 Introduction

Lightning is an important natural phenomenon that canaffect power lines Induced voltages are a major effect oflightning on distribution lines that can be created by couplingbetween the electromagnetic fields of the lightning and thepower line [1ndash5]Therefore the evaluation of electromagneticfields associated with lightning is an important objectivewhen considering lightning induced voltages and settingan appropriate protection level for power systems Severalstudies have been undertaken to estimate the electromagneticfields due to a lightning channel [6ndash11] Such studies dependon the lightning channel parameters the geometrical param-eters channel shape channel condition and the ground con-ductivity parameters Among the different channel param-eters the return stroke velocity is an important variablefor the evaluation of lightning electromagnetic fields [1012 13] and this is usually entered into field calculations asan average value of velocities at different heights along alightning channel with a typical value between 1198883 and 21198883(119888 is speed of light in free space) [14]

On the other hand some experimental work has beencarried out to measure the return stroke velocity at differentheights along a channel where the velocity is measured as aheight-dependent variable [14ndash17] Therefore the variationof velocity along a lightning channel can have an effect onthe values of the lightning electromagnetic fields In thisstudy numerical field expressions are proposed to evaluatelightning electromagnetic fields based on a channel withvariable values of return stroke velocity directly in the timedomain Likewise the proposed method is applied to atypical current sample whereby the corresponding returnstroke velocity function is used for considering the velocityprofile along a lightning channel Further the simulatedfields are compared to the corresponding fields based ona constant value of velocity and the results are discussedaccordingly The proposed method can support a widerange of velocity profiles along the channel along withvarious current functions and current models directly in thetime domain without the need to apply any extra conver-sions The basic assumptions in this study are expressed asfollows

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 150756 9 pageshttpdxdoiorg1011552015150756

2 Mathematical Problems in Engineering

Table 1 119875(1199111015840) and V for six engineering current models based on (3)

Model 119875(1199111015840) V

BG (Bruce-Golde model) [18] 1 infin

TCS (traveling current source model) [19] 1 minus119888

TL (transmission line model) [9] 1 V119891

MTLL (modified transmission line linear model with linear decaywith height) [9] (1 minus

1199111015840

119867

) V119891

MTLE (modified transmission line model with exponential decaywith height) [9] exp(minus119911

1015840

120582

) V119891

MTLD (modified transmission line model with current distortion)[20 21] [1 minus exp(minus

119905 minus (1199111015840V)

120591

times

120582119875

1199111015840)](1 minus

1199111015840

119867

) V119891

(1) The lightning channel is assumed to be vertical with-out any branches

(2) The ground conductivity is assumed to be infinite

2 Return Stroke Current

The return stroke currents at the channel base (groundsurface) and at different heights along a lightning channelcan be simulated using current functions and currentmodelsrespectively In this study the sum of two Heidler currentfunctions [22 23] is used to simulate the channel-base currentas expressed by the following equation

119894 (0 119905) = [

11989401

1205781

(119905Γ11)1198991

1 + (119905Γ11)1198991

exp( minus119905

Γ12

)

+

11989402

1205782

(119905Γ21)1198992

1 + (119905Γ21)1198992

exp( minus119905

Γ22

)]

(1)

where 1198940111989402

is the current amplitude of firstsecond Hei-dler function in (1) Γ

11Γ12

is the front time constant offirstsecond Heidler function in (1) Γ

21Γ22is the decay-time

constant in firstsecond Heidler function in (1) and 1198991 1198992are

the exponents (2sim10)

1205781= exp[minus(Γ11

Γ12

)(1198991times

Γ12

Γ11

)

11198991

]


Γ22

)(1198992times

Γ22

Γ21

)

11198992

]

(2)

The general form of the engineering current models isconsidered in this study to cover a wide range of currentmodels as given by (3) Table 1 shows the constant factors ofsome common engineering current models where 119867 is thecloud height 120582 is the decay factor and 120582

119875is the attenuation

factor of the peak [9 20 21] It should be mentioned thatthe MTLE (Modified Transmission Line current model withExponential decay factor) current model was used for thesimulation of electromagnetic fields however the proposedmethod can support a wide range of current models based on(3) [20 21 24]

119868 (1199111015840 119905) = 119868(0 119905 minus

1199111015840

V) times 119875 (119911

1015840) times 119906(119905 minus

1199111015840

V119891

) (3)

where 1199111015840 is temporary charge height along lightning channel119868(1199111015840 119905) is current distribution along lightning channel at any

height 1199111015840 and any time 119905 119868(0 119905) is the channel-base current119875(1199111015840) is attenuation height dependent factor V is the current-

wave propagation velocity V119891is upward propagating front

velocity and 119906 is Heaviside function defined as

119906(119905 minus

1199111015840

V119891

) =

1 for 119905 ge 1199111015840

V119891

0 for 119905 lt 1199111015840

V119891

(4)

3 The Return Stroke Velocity

Several studies have measured subsequent return strokevelocities and they indicate that the return stroke velocity is aheight-dependent variable as expressed by a typical functionof velocity profile given in (5) [17] Table 2 illustrates theunknown variables in (5) for a number of current peaks inthe range 3ndash30 kA

V (1199111015840)

=

V1+ (

V2

2

)2 minus exp(minus (1199111015840minus 1)

1205821

) minus exp(minus (1199111015840minus 1)

1205822

)

1 le 1199111015840le 50

V3exp(minus119911

1015840

1205823

) minus V4exp(minus119911

1015840

1205824

) 1199111015840ge 50

(5)

Furthermore the velocity profiles along a lightning channelfor a number of current peaks are illustrated in Figure 1 wherethe initial parameters are obtained from Table 2

Figure 2 illustrates the average and maximum valuesof the velocities of different lightning channels based onthe velocity profiles from Figure 1 Figure 2 shows that byincreasing the current peak the maximum value of thevelocity is increased while the average velocity along thechannel has an increasing trend against an increasing currentpeak up to 21 kA This increasing trend can be seen againin the current peak ranges from 24 kA to 30 kA It shouldbe mentioned that the average values were calculated infirst 500m of channel as an effective part for the peak ofelectromagnetic field components

Mathematical Problems in Engineering 3

Table 2 The parameters of return stroke velocity profile [17]

119868119901(kA) V

1(times108) V

2(times108) V

3(times108) V

4(times108) 120582

11205822

1205823

1205824

3 072 118 019 171 18 62 400 9004 078 123 0201 181 16 66 120 12006 086 129 086 129 16 68 370 22009 095 134 0687 160 14 74 320 200012 102 135 0711 166 14 74 400 210015 107 137 0488 195 12 80 330 200018 112 136 0496 198 12 78 400 210021 116 136 0504 202 12 80 400 220024 12 134 0254 129 12 76 340 210027 123 134 0257 131 12 78 400 210030 126 133 0259 133 12 78 400 2200

0 50 100 150 200 250 300 350 400 450 5000608

112141618

2222426

Channel height (m)

Retu

rn st

roke

velo

city

(ms

)

times108

Ip = 3kAIp = 4kAIp = 6kAIp = 9kA

Ip = 30kAIp = 12kAIp = 15kA


Figure 1 The velocity profiles for different lightning channels

4 The Proposed ElectromagneticField Expressions

In order to evaluate the electromagnetic fields due to alightning channel with variable velocities along the channelMaxwellrsquos equations are presented in (6) to (9) as follows[8 13 25]

nabla times = minus

120597

120597119905

(6)

nabla times = 119869 +

120597

120597119905

(7)

nabla sdot = 120588V (8)

nabla sdot = 0 (9)

5 10 15 20 25 3012

14

16

18

2

22

24

26

Retu

rn st

roke

velo

city

(ms

)

times108

Ip (kA)

avemax

Figure 2The behavior of Vave and Vmax versus current peak changes

where is the magnetic flux density is the magnetic field is the electric field 119869 is the current density is the electricflux density and 120588V is the free charge density

Themagnetic flux density can be expressed in terms of thevector potential () as given by the following equation [10]

= nabla times (10)

The relation between and can be expressedrespectively by the following equations

= 120583 (11)

= 120576 (12)

Also (7) can be converted to (13) when (11) and (12) aresubstituted into (7) By substituting (10) into (6) (6) becomes(14)

nabla times = 120583 119869 + 120583120576

120597

120597119905

(13)

nabla times = minus

120597

120597119905

= minus

120597 (nabla times )

120597119905

(14)


Therefore (14) can be expressed by (15)

nabla times ( +

120597

120597119905

) = 0

+

120597

120597119905

= minusnablaV119890

(15)

where V119890is the electric scalar potential

Also by substituting (10) into (13) the vector potentialcan be given by the following equations

nabla times nabla times = nabla (nabla sdot ) minus nabla2 = 120583 119869 + 120583120576

120597

120597119905

(16)

nabla (nabla sdot ) minus nabla2 = 120583 119869 minus 120583120576

120597 (nablaV119890)

120597119905

minus 120583120576

1205972

1205971199052 (17)

Therefore (17) can be converted to the following equation

nabla2 minus 120583120576

1205972

1205971199052= minus120583 119869 + nabla(nabla sdot + 120583120576

120597V119890

120597119905

) (18)

According to Lorentzrsquos gauge the relation between potentialvector and scalar potential can be obtained from the followingequation [26]

nabla sdot = minus120583120576

120597V119890

120597119905

(19)

By substituting (19) into (18) and using free space conditions(120583 = 120583

0 120576 = 120576

0) the vector potential can be expressed by the

following equation

nabla2 minus 120583

01205760

1205972

1205971199052= minus1205830119869 (20)

The solution of (20) can be obtained by considering Figure 3where an infinitesimal current source is located in space by

position997888rarr

1199031015840 from the origin Also an observation point (the

point at which is to be evaluated) is located in space at point119875 and 119903 is a vector from the origin to the observation point

Hence = 119903 minus

997888rarr

1199031015840

The infinitesimal current source can be divided into threecurrent elements as follows

(1) volume current element (997888rarr119869119894119889V1015840)

(2) surface current element (1198891198781015840)

(3) line current element ( 1198681198891198711015840)

where 997888rarr

119869119894 and 119868 are the volume current the surface

current and the line current densities respectively Hencethe solution of (20) will produce the following equation

119889 =

1205830[

997888rarr

119869119894] 119889V1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[] 119889119878

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[ 119868] 119889119871

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

(21)

Moreover the geometry of a lightning channel with variablevelocity values along the channel is illustrated in Figure 4

Therefore the derivative of the potential vector can beexpressed by (22) assuming that the lightning channel isperpendicular to the ground surface along the 119911-axis

997888997888rarr

119889119860 =

1205830119894 (1199111015840 119905119899minus 119877 (119911

1015840) 119888) 119889119911

1015840

4120587119877 (1199111015840)

(22)

whereΔℎ is the channel height step 1199111015840 = 119899Δℎ 119899 is the numberof height steps along lightning channel (1 2 119899max) V119899is the return stroke velocity in each per unit of lightningchannel

119905119899=

119899

sum

119894=1

Δℎ

V119894

+

radic(119899Δℎ)2+ 1199032

119888

119877 (1199111015840= 119899Δℎ) = radic(119899Δℎ)

2+ 1199032

119894 (1199111015840 119905119899minus

119877 (1199111015840)

119888

)

= 119875 (1199111015840= 119899Δℎ) times 119894(0 119905

119899minus

119899

sum

119894=1

Δℎ

V119894

minus

radic(119899Δℎ)2+ 1199032

119888

)

(23)

and 119888 is speed of light in free spaceBy applying (22) to (10) and by using the trapezoid

method themagnetic flux density can be evaluated as follows[27 28]

119861119899

120593=

minus10minus7times Δℎ

119899

sum

119895=1

1198651(119895) 119899 gt 0

0 119899 = 0

(24)


where

1198651(119895)

=

119896 times [

119875 (Δℎ)

radicΔℎ2+ 1199032

times

119889 119894 (0 119905119899minus ΔℎV

1minusradicΔℎ2+ 1199032119888)

119889119903

minus 119903 times 119875 (Δℎ)

times 119894(0 119905119899minus

Δℎ

V1

minus


119888

)

times [Δℎ2+ 1199032]

minus32

]

+[

1

119903

times

119889 119894 (0 119905119899minus 119903119888)

119889119903

minus

119894 (0 119905119899minus 119903119888)

1199032

]

if 119895 = 1

119896 times[

[

[

119875 (119895Δℎ)

radic(119895Δℎ)2

+ 1199032

times119889

119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times (119889119903)minus1

minus119903 times 119875 (119895Δℎ)

times119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times [(119895Δℎ)2

+ 1199032]

minus32]

]

]

if 119895 gt 1

119896 =

1 119895 = 119899

2 119895 = 119899

(25)

Furthermore the derivative of the vertical electric field withrespect to time and the vertical electric field expressionsare proposed by (26) and (27) respectively where (24) issubstituted into (13)

119889119864119899

119911

119889119905

=

minusΔℎ

41205871199031205760

119899

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(26)

119864119899

119911=

minusΔℎ

41205871199031205760

119899

sum

119904=1

1198961015840

119904

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(27)

y

z

x

rarr

R998400

P

drarrA

rarrr1

rarrr2

Figure 3 Geometry of an observation point with respect to currentsource

rP

Real channel

Image channel

The perfect ground

Δh

Δh

Δh

Δh

R1

R2

1

1

2

2

Figure 4 The geometry of problem

where

1198961015840=

1

2

[

Δℎ

V1

+


119888

] 119904 = 1

1

2

[

[

[

(

119904

sum

119894=1

Δℎ

V119894

minus

119904minus1

sum

119894=1

Δℎ

V119894

)

+

radic(119904Δℎ)2+ 1199032minus radic([119904 minus 1] Δℎ)

2+ 1199032

119888

]

]

]

119904 = 1

(28)

In order to simulate the electromagnetic fields associatedwitha lightning channel a sample of a measured channel-basecurrent is simulated using the sumof twoHeidler functions asshown in Figure 5 whereas the measured current is obtainedfrom a triggered lightning experiment Table 3 illustrates theevaluated current parameters based on (1) [29 30]


Table 3 The current parameters based on the sum of two Heidlerfunctions

11989401

(kA)11989402

(kA)12059111

(120583s)12059112

(120583s)12059121

(120583s)12059122

(120583s) 1198991

1198992

17793 10753 0434 1775 2611 5764 2 2

0 2 4 6 8 10 12 14 16 18 200

02040608

112141618

2

Time (120583s)

Curr

ent (

A)

Simulated channel base currentMeasured channel base current

times104

Figure 5 Comparison between simulated and measured channel-base current

Figure 5 shows that the simulated current is in goodagreement with the corresponding measured current There-fore themagnetic flux densities and the vertical electric fieldsbased on the variable and constant values of the velocity at 119903 =15m are evaluated based on the proposed method and thesimulated fields are compared to the correspondingmeasuredfields as illustrated in Figures 6 and 7 respectively It shouldbe noted that the constant velocity is set at the average valueof the velocity along the lightning channel and the MTLEmodel is used for the current model Likewise the velocityprofile for the case of variable velocity is based on (5) withthe corresponding parameters that can be obtained from the7th row of Table 2

Figure 6 shows that the simulated magnetic flux densitybased on the variable values of the velocity is in betteragreement with the corresponding measured field comparedto the simulated field that is obtained from using a constantvalue for the velocity However this difference is not greatMoreover the simulated vertical electric field due to thevariable values of the velocity is in good agreement with thecorresponding measured field as shown in Figure 7 whilethe simulated vertical electric field based on a constantvelocity is not closer to the corresponding measured fieldIn previous studies the appropriate average velocity is setas a basic assumption so as to obtain a good agreementbetween the simulated field and themeasured fieldThis valueis usually selected based on trial and error because in orderto determine the average velocity the values of the velocity atdifferent heights along a lightning channel are required andthese are based on recording the velocity values at just a few

0 05 1 15 2 25 3 35 4 45 50

1

2

Time (120583s)

times10minus4

B120601

(Wb

m2)

Measured field

(vave = 1426 lowast 108 ms)Simulated field based on constant velocitySimulated field based on variable

Figure 6 Comparison between simulated and measured magneticflux density at 119903 = 15m

0 05 1 15 2 25 3 35 4 45 50

5

10

15times104

Time (120583s)

Ez

(Vm

)

Measured field


Figure 7 Comparison between simulated and measured verticalelectric field at 119903 = 15m

points along the channel In the present study the averagevalue of the velocity is obtained from the velocity function(5)

The simulated 119889119864119911119889119905 for both the constant and variable

cases are demonstrated in Figure 8 which shows the peak of119889119864119911119889119905 due to the variable values of the velocity are lower

than the similar values based on a constant velocity Thiscould be due to changes in the charge heights at differenttimes along the lightning channel while the charge heightvalues are more effective for the integration of currentcomponents along a lightning channel Moreover the effectof velocity changes on the values of horizontal electric fieldwas considered as shown in Figure 9 Figure 9 illustrates thatthe effect of velocity changes on the values of horizontalelectric fields at closed distances with respect to lightning isnot considerable


0 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

minus050 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

Time (120583s)

minus05

dEzdt

(Vm

s)


Figure 8 Comparison between simulated and measured 119889119864119911119889119905 at

119903 = 15m

0 05 1 15 2 25 3 35 4 45 50123456789

10times104

Time (120583s)

Based on vave = 1426 lowast 108 msBased on vvariable

Er

(Vm

)

Figure 9 Comparison between simulated horizontal electric fieldsat 119903 = 15m 119911 =10m

Figures 10 11 and 12 show the behavior of peak valuesof magnetic flux density vertical electric field and horizontalelectric fields versus height changes (observation point)respectively

The proposed method can consider the different behav-iours of the velocity along a lightning channel directly in thetime domainwithout the need to apply any extra conversionsLikewise the method can support different current functionsand the general form of the engineering current modelsMoreover the results show that the simulated fields basedon the general function of the velocity are closer to thecorresponding measured fields compared to the simulatedfields based on the average values of velocity especially fora vertical electric field

5 Conclusion

In this paper general electromagnetic field expressionsare proposed to consider the variation of velocity along

1 2 3 4 5 6 7 8 9 10250725082509

25125112512251325142515

Height with respect to ground surface (m)

times10minus4

B120601

(Wb

m2)

Figure 10 Behaviour of magnetic flux density (peak) versus heightchanges (observation point) at 119903 = 15m

1 2 3 4 5 6 7 8 9 10100

105

110

115

120

125

130


Ez

(kV

m)

Figure 11 Behaviour of vertical electric field (peak) versus heightchanges (observation point) at 119903 = 15m

1 2 3 4 5 6 7 8 9 10102030405060708090

100


Er

(kV

m)

Figure 12 Behaviour of horizontal electric field (peak) versus heightchanges (observation point) at 119903 = 15m

a lightning channel directly in the time domain while themeasured values of the velocity show that the velocity is aheight-dependent variable which is usually entered into fieldcalculations as a constant value By the simulation of thevelocity behaviour along a lightning channel based on thegeneral velocity function the proposed method is applied toa sample of measured channel-base current from a triggeredlightning experiment and the simulated fields are validated


using the corresponding measured fields The results showthat the simulated fields are in good agreement with themeasured fields Likewise the simulated electromagneticfields based on variable values of the velocity are compared tothe corresponding simulated fields based on a constant valuefor the velocity and the results are discussed accordinglyTheproposed method can support different velocity behaviourscurrent functions and the general form of the engineeringcurrent model directly in the time domain without the needto apply any extra conversions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Rachidi M Rubinstein S Guerrieri and C A Nucci ldquoVolt-ages induced on overhead lines by dart leaders and subsequentreturn strokes in natural and rocket-triggered lightningrdquo IEEETransactions on Electromagnetic Compatibility vol 39 no 2 pp160ndash166 1997

[2] F Rachidi C A Nucci M Ianoz and CMazzetti ldquoResponse ofmulticonductor power lines to nearby lightning return strokeelectromagnetic fieldsrdquo in Proceedings of the 14th IEEE Trans-mission and Distribution Conference pp 294ndash301 September1996

[3] F Rachidi C A Nucci M Ianoz and C Mazzetti ldquoInfluenceof a lossy ground on lightning-induced voltages on overheadlinesrdquo IEEE Transactions on Electromagnetic Compatibility vol38 no 3 pp 250ndash264 1996

[4] F Rachidi ldquoFormulation of the field-to-transmission line cou-pling equations in terms of magnetic excitation fieldrdquo IEEETransactions on Electromagnetic Compatibility vol 35 no 3 pp404ndash407 1993

[5] M Paolone CANucci E Petrache andF Rachidi ldquoMitigationof lightning-induced overvoltages in medium voltage distribu-tion lines by means of periodical grounding of shielding wiresand of surge arresters modeling and experimental validationrdquoIEEE Transactions on Power Delivery vol 19 no 1 pp 423ndash4312004

[6] M Izadi M Z A Ab Kadir C Gomes and W F Wan AhmadldquoAn analytical second-fdtd method for evaluation of electricand magnetic fields at intermediate distances from lightningchannelrdquo Progress in Electromagnetics Research vol 110 pp329ndash352 2010

[7] M Izadi M Z A A Ab Kadir C Gomes and W F WAhmad ldquoNumerical expressions in time domain for electro-magnetic fields due to lightning channelsrdquo International Journalof Applied Electromagnetics and Mechanics vol 37 no 4 pp275ndash289 2011

[8] M Izadi M Z Ab Kadir C Gomes and W F H Wan AhmadldquoAnalytical expressions for electromagnetic fields associatedwith the inclined lightning channels in the time domainrdquoElectric Power Components and Systems vol 40 no 4 pp 414ndash438 2012

[9] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part I return stroke current models with specifiedchannel-base current for the evaluation of the return strokeelectromagnetic fieldsrdquo Electra vol 161 pp 75ndash102 1995

[10] R Thottappillil and V Rakov ldquoReview of three equivalentapproaches for computing electromagnetic fields from anextending lightning dischargerdquo Journal of Lightning Researchvol 1 pp 90ndash110 2007

[11] RThottappillil V A Rakov and NTheethayi ldquoExpressions forfar electric fields produced at an arbitrary altitude by lightningreturn strokesrdquo Journal of Geophysical Research D Atmospheresvol 112 no 16 Article ID D16102 2007

[12] M Rubinstein and M A Uman ldquoMethods for calculatingthe electromagnetic fields from a known source distributionapplication to lightningrdquo IEEE Transactions on ElectromagneticCompatibility vol 31 no 2 pp 183ndash189 1989

[13] M Izadi M Z A Ab Kadir C Gomes and W F Wan AhmadldquoEvaluation of the electromagnetic fields due to lightningchannel with respect to the striking anglerdquo International Reviewof Electrical Engineering vol 6 no 2 pp 1013ndash1023 2011

[14] V Rakov ldquolightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[15] DWang N Takagi TWatanabe V A Rakov andM A UmanldquoObserved leader and return-stroke propagation characteristicsin the bottom 400 m of a rocket-triggered lightning channelrdquoJournal of Geophysical Research D Atmospheres vol 104 no 12pp 14369ndash14376 1999

[16] R C Olsen III D M Jordan V A Rakov M A Uman and NGrimes ldquoObserved one-dimensional return stroke propagationspeeds in the bottom 170m of a rocket-triggered lightningchannelrdquo Geophysical Research Letters vol 31 no 16 2004

[17] V CoorayThe lightning Flash IET Press 2003[18] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo The

Journal of the Institute of Electrical Engineers Part 2 vol 88 1941[19] F Heidler ldquoTravelling current source model for LEMP cal-

culationrdquo in Proceedings of the 6th Symposium and TechnicalExhibition on Electromagnetic Compability Zurich Switzerland1985

[20] Y Baba and M Ishii ldquoLightning return-stroke model incorpo-rating current distortionrdquo IEEETransactions on ElectromagneticCompatibility vol 44 no 3 pp 476ndash478 2002

[21] Y Baba S Miyazaki and M Ishii ldquoReproduction of light-ning electromagnetic field waveforms by engineering model ofreturn strokerdquo IEEE Transactions on Electromagnetic Compati-bility vol 46 no 1 pp 130ndash133 2004

[22] F Heidler ldquoAnalytische Blitzstromfunktion zur LEMP- Berech-nungrdquo in Conference Proceedings ICLP rsquo85 18th InternationalConference on Lightning Protection Hotel Hilton Munich Fed-eral Republic of Germany September 16ndash20 1985 1985

[23] M Izadi M Z A Ab Kadir C Gomes and V CoorayldquoEvaluation of lightning return stroke current using measuredelectromagnetic fieldsrdquo Progress in Electromagnetics Researchvol 130 pp 581ndash600 2012

[24] V Rakov and A Dulzon ldquoA modified transmission line modelfor lightning return stroke field calculationsrdquo in Proceedingsof the 9th International Zurich Symposium on ElectromagneticCompatibility pp 229ndash235 Zurich Switzerland 1991

[25] M Izadi M Z A A Kadir and C Gomes ldquoEvaluationof electromagnetic fields associated with inclined lightningchannel using second order FDTD-hybrid methodsrdquo Progressin Electromagnetics Research vol 117 pp 209ndash236 2011

[26] R Nevels andC-S Shin ldquoLorenz Lorentz and the gaugerdquo IEEEAntennas amp Propagation Magazine vol 43 no 3 pp 70ndash722001


[27] E Kreyszig Advanced Engineering Mathematics Wiley-IndiaNew Delhi India 2007

[28] M N O SadikuNumerical Technique in Electromagnetics CRCPress New York NY USA 2001

[29] D Lovric S Vujevic and T Modric ldquoOn the estimationof Heidler function parameters for reproduction of variousstandardized and recorded lightning currentwaveshapesrdquo Inter-national Transactions on Electrical Energy Systems vol 23 no 2pp 290ndash300 2013

[30] S Vujevic D Lovric and I Juric-Grgic ldquoLeast squares estima-tion of Heidler function parametersrdquo European Transactions onElectrical Power vol 21 no 1 pp 329ndash344 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 119875(1199111015840) and V for six engineering current models based on (3)

Model 119875(1199111015840) V

BG (Bruce-Golde model) [18] 1 infin

TCS (traveling current source model) [19] 1 minus119888

TL (transmission line model) [9] 1 V119891

MTLL (modified transmission line linear model with linear decaywith height) [9] (1 minus

1199111015840

119867

) V119891

MTLE (modified transmission line model with exponential decaywith height) [9] exp(minus119911

1015840

120582

) V119891

MTLD (modified transmission line model with current distortion)[20 21] [1 minus exp(minus

119905 minus (1199111015840V)

120591

times

120582119875

1199111015840)](1 minus

1199111015840

119867

) V119891

(1) The lightning channel is assumed to be vertical with-out any branches

(2) The ground conductivity is assumed to be infinite

2 Return Stroke Current

The return stroke currents at the channel base (groundsurface) and at different heights along a lightning channelcan be simulated using current functions and currentmodelsrespectively In this study the sum of two Heidler currentfunctions [22 23] is used to simulate the channel-base currentas expressed by the following equation

119894 (0 119905) = [

11989401

1205781

(119905Γ11)1198991

1 + (119905Γ11)1198991

exp( minus119905

Γ12

)

+

11989402

1205782

(119905Γ21)1198992

1 + (119905Γ21)1198992

exp( minus119905

Γ22

)]

(1)

where 1198940111989402

is the current amplitude of firstsecond Hei-dler function in (1) Γ

11Γ12

is the front time constant offirstsecond Heidler function in (1) Γ

21Γ22is the decay-time

constant in firstsecond Heidler function in (1) and 1198991 1198992are

the exponents (2sim10)


Γ12

)(1198991times

Γ12

Γ11

)

11198991

]


Γ22

)(1198992times

Γ22

Γ21

)

11198992

]

(2)

The general form of the engineering current models isconsidered in this study to cover a wide range of currentmodels as given by (3) Table 1 shows the constant factors ofsome common engineering current models where 119867 is thecloud height 120582 is the decay factor and 120582

119875is the attenuation

factor of the peak [9 20 21] It should be mentioned thatthe MTLE (Modified Transmission Line current model withExponential decay factor) current model was used for thesimulation of electromagnetic fields however the proposedmethod can support a wide range of current models based on(3) [20 21 24]

119868 (1199111015840 119905) = 119868(0 119905 minus

1199111015840

V) times 119875 (119911

1015840) times 119906(119905 minus

1199111015840

V119891

) (3)

where 1199111015840 is temporary charge height along lightning channel119868(1199111015840 119905) is current distribution along lightning channel at any

height 1199111015840 and any time 119905 119868(0 119905) is the channel-base current119875(1199111015840) is attenuation height dependent factor V is the current-

wave propagation velocity V119891is upward propagating front

velocity and 119906 is Heaviside function defined as

119906(119905 minus

1199111015840

V119891

) =

1 for 119905 ge 1199111015840

V119891

0 for 119905 lt 1199111015840

V119891

(4)

3 The Return Stroke Velocity

Several studies have measured subsequent return strokevelocities and they indicate that the return stroke velocity is aheight-dependent variable as expressed by a typical functionof velocity profile given in (5) [17] Table 2 illustrates theunknown variables in (5) for a number of current peaks inthe range 3ndash30 kA

V (1199111015840)

=

V1+ (

V2

2

)2 minus exp(minus (1199111015840minus 1)

1205821

) minus exp(minus (1199111015840minus 1)

1205822

)

1 le 1199111015840le 50

V3exp(minus119911

1015840

1205823

) minus V4exp(minus119911

1015840

1205824

) 1199111015840ge 50

(5)

Furthermore the velocity profiles along a lightning channelfor a number of current peaks are illustrated in Figure 1 wherethe initial parameters are obtained from Table 2

Figure 2 illustrates the average and maximum valuesof the velocities of different lightning channels based onthe velocity profiles from Figure 1 Figure 2 shows that byincreasing the current peak the maximum value of thevelocity is increased while the average velocity along thechannel has an increasing trend against an increasing currentpeak up to 21 kA This increasing trend can be seen againin the current peak ranges from 24 kA to 30 kA It shouldbe mentioned that the average values were calculated infirst 500m of channel as an effective part for the peak ofelectromagnetic field components



119868119901(kA) V

1(times108) V

2(times108) V

3(times108) V

4(times108) 120582

11205822

1205823

1205824

3 072 118 019 171 18 62 400 9004 078 123 0201 181 16 66 120 12006 086 129 086 129 16 68 370 22009 095 134 0687 160 14 74 320 200012 102 135 0711 166 14 74 400 210015 107 137 0488 195 12 80 330 200018 112 136 0496 198 12 78 400 210021 116 136 0504 202 12 80 400 220024 12 134 0254 129 12 76 340 210027 123 134 0257 131 12 78 400 210030 126 133 0259 133 12 78 400 2200

0 50 100 150 200 250 300 350 400 450 5000608

112141618

2222426

Channel height (m)

Retu

rn st

roke

velo

city

(ms

)

times108







nabla times = minus

120597

120597119905

(6)


120597

120597119905

(7)


nabla sdot = 0 (9)

5 10 15 20 25 3012

14

16

18

2

22

24

26

Retu

rn st

roke

velo

city

(ms

)

times108

Ip (kA)

avemax




= nabla times (10)


= 120583 (11)

= 120576 (12)


nabla times = 120583 119869 + 120583120576

120597

120597119905

(13)

nabla times = minus

120597

120597119905

= minus


120597119905

(14)



nabla times ( +

120597

120597119905

) = 0

+

120597

120597119905

= minusnablaV119890

(15)




120597

120597119905

(16)


120597 (nablaV119890)

120597119905

minus 120583120576

1205972

1205971199052 (17)



1205972


120597V119890

120597119905

) (18)



120597V119890

120597119905

(19)


0 120576 = 120576


following equation

nabla2 minus 120583

01205760

1205972

1205971199052= minus1205830119869 (20)


position997888rarr




997888rarr

1199031015840





where 997888rarr



119889 =

1205830[

997888rarr

119869119894] 119889V1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[] 119889119878

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[ 119868] 119889119871

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

(21)



997888997888rarr

119889119860 =

1205830119894 (1199111015840 119905119899minus 119877 (119911

1015840) 119888) 119889119911

1015840

4120587119877 (1199111015840)

(22)


119905119899=

119899

sum

119894=1

Δℎ

V119894

+

radic(119899Δℎ)2+ 1199032

119888

119877 (1199111015840= 119899Δℎ) = radic(119899Δℎ)

2+ 1199032

119894 (1199111015840 119905119899minus

119877 (1199111015840)

119888

)

= 119875 (1199111015840= 119899Δℎ) times 119894(0 119905

119899minus

119899

sum

119894=1

Δℎ

V119894

minus

radic(119899Δℎ)2+ 1199032

119888

)

(23)



119861119899

120593=


119899

sum

119895=1

1198651(119895) 119899 gt 0

0 119899 = 0

(24)


where

1198651(119895)

=

119896 times [

119875 (Δℎ)


times

119889 119894 (0 119905119899minus ΔℎV


119889119903


times 119894(0 119905119899minus

Δℎ

V1

minus


119888

)

times [Δℎ2+ 1199032]

minus32

]

+[

1

119903

times

119889 119894 (0 119905119899minus 119903119888)

119889119903

minus

119894 (0 119905119899minus 119903119888)

1199032

]

if 119895 = 1

119896 times[

[

[

119875 (119895Δℎ)

radic(119895Δℎ)2

+ 1199032

times119889

119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times (119889119903)minus1

minus119903 times 119875 (119895Δℎ)

times119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times [(119895Δℎ)2

+ 1199032]

minus32]

]

]

if 119895 gt 1

119896 =

1 119895 = 119899

2 119895 = 119899

(25)


119889119864119899

119911

119889119905

=

minusΔℎ

41205871199031205760

119899

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(26)

119864119899

119911=

minusΔℎ

41205871199031205760

119899

sum

119904=1

1198961015840

119904

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(27)

y

z

x

rarr

R998400

P

drarrA

rarrr1

rarrr2


rP

Real channel

Image channel

The perfect ground

Δh

Δh

Δh

Δh

R1

R2

1

1

2

2


where

1198961015840=

1

2

[

Δℎ

V1

+


119888

] 119904 = 1

1

2

[

[

[

(

119904

sum

119894=1

Δℎ

V119894

minus

119904minus1

sum

119894=1

Δℎ

V119894

)

+


2+ 1199032

119888

]

]

]

119904 = 1

(28)




11989401

(kA)11989402

(kA)12059111

(120583s)12059112

(120583s)12059121

(120583s)12059122

(120583s) 1198991

1198992

17793 10753 0434 1775 2611 5764 2 2

0 2 4 6 8 10 12 14 16 18 200

02040608

112141618

2

Time (120583s)

Curr

ent (

A)


times104




0 05 1 15 2 25 3 35 4 45 50

1

2

Time (120583s)

times10minus4

B120601

(Wb

m2)

Measured field



0 05 1 15 2 25 3 35 4 45 50

5

10

15times104

Time (120583s)

Ez

(Vm

)

Measured field








0 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

minus050 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

Time (120583s)

minus05

dEzdt

(Vm

s)



119903 = 15m

0 05 1 15 2 25 3 35 4 45 50123456789

10times104

Time (120583s)


Er

(Vm

)




5 Conclusion


1 2 3 4 5 6 7 8 9 10250725082509

25125112512251325142515


times10minus4

B120601

(Wb

m2)


1 2 3 4 5 6 7 8 9 10100

105

110

115

120

125

130


Ez

(kV

m)


1 2 3 4 5 6 7 8 9 10102030405060708090

100


Er

(kV

m)







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868119901(kA) V

1(times108) V

2(times108) V

3(times108) V

4(times108) 120582

11205822

1205823

1205824

3 072 118 019 171 18 62 400 9004 078 123 0201 181 16 66 120 12006 086 129 086 129 16 68 370 22009 095 134 0687 160 14 74 320 200012 102 135 0711 166 14 74 400 210015 107 137 0488 195 12 80 330 200018 112 136 0496 198 12 78 400 210021 116 136 0504 202 12 80 400 220024 12 134 0254 129 12 76 340 210027 123 134 0257 131 12 78 400 210030 126 133 0259 133 12 78 400 2200

0 50 100 150 200 250 300 350 400 450 5000608

112141618

2222426

Channel height (m)

Retu

rn st

roke

velo

city

(ms

)

times108







nabla times = minus

120597

120597119905

(6)


120597

120597119905

(7)


nabla sdot = 0 (9)

5 10 15 20 25 3012

14

16

18

2

22

24

26

Retu

rn st

roke

velo

city

(ms

)

times108

Ip (kA)

avemax




= nabla times (10)


= 120583 (11)

= 120576 (12)


nabla times = 120583 119869 + 120583120576

120597

120597119905

(13)

nabla times = minus

120597

120597119905

= minus


120597119905

(14)



nabla times ( +

120597

120597119905

) = 0

+

120597

120597119905

= minusnablaV119890

(15)




120597

120597119905

(16)


120597 (nablaV119890)

120597119905

minus 120583120576

1205972

1205971199052 (17)



1205972


120597V119890

120597119905

) (18)



120597V119890

120597119905

(19)


0 120576 = 120576


following equation

nabla2 minus 120583

01205760

1205972

1205971199052= minus1205830119869 (20)


position997888rarr




997888rarr

1199031015840





where 997888rarr



119889 =

1205830[

997888rarr

119869119894] 119889V1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[] 119889119878

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[ 119868] 119889119871

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

(21)



997888997888rarr

119889119860 =

1205830119894 (1199111015840 119905119899minus 119877 (119911

1015840) 119888) 119889119911

1015840

4120587119877 (1199111015840)

(22)


119905119899=

119899

sum

119894=1

Δℎ

V119894

+

radic(119899Δℎ)2+ 1199032

119888

119877 (1199111015840= 119899Δℎ) = radic(119899Δℎ)

2+ 1199032

119894 (1199111015840 119905119899minus

119877 (1199111015840)

119888

)

= 119875 (1199111015840= 119899Δℎ) times 119894(0 119905

119899minus

119899

sum

119894=1

Δℎ

V119894

minus

radic(119899Δℎ)2+ 1199032

119888

)

(23)



119861119899

120593=


119899

sum

119895=1

1198651(119895) 119899 gt 0

0 119899 = 0

(24)


where

1198651(119895)

=

119896 times [

119875 (Δℎ)


times

119889 119894 (0 119905119899minus ΔℎV


119889119903


times 119894(0 119905119899minus

Δℎ

V1

minus


119888

)

times [Δℎ2+ 1199032]

minus32

]

+[

1

119903

times

119889 119894 (0 119905119899minus 119903119888)

119889119903

minus

119894 (0 119905119899minus 119903119888)

1199032

]

if 119895 = 1

119896 times[

[

[

119875 (119895Δℎ)

radic(119895Δℎ)2

+ 1199032

times119889

119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times (119889119903)minus1

minus119903 times 119875 (119895Δℎ)

times119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times [(119895Δℎ)2

+ 1199032]

minus32]

]

]

if 119895 gt 1

119896 =

1 119895 = 119899

2 119895 = 119899

(25)


119889119864119899

119911

119889119905

=

minusΔℎ

41205871199031205760

119899

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(26)

119864119899

119911=

minusΔℎ

41205871199031205760

119899

sum

119904=1

1198961015840

119904

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(27)

y

z

x

rarr

R998400

P

drarrA

rarrr1

rarrr2


rP

Real channel

Image channel

The perfect ground

Δh

Δh

Δh

Δh

R1

R2

1

1

2

2


where

1198961015840=

1

2

[

Δℎ

V1

+


119888

] 119904 = 1

1

2

[

[

[

(

119904

sum

119894=1

Δℎ

V119894

minus

119904minus1

sum

119894=1

Δℎ

V119894

)

+


2+ 1199032

119888

]

]

]

119904 = 1

(28)




11989401

(kA)11989402

(kA)12059111

(120583s)12059112

(120583s)12059121

(120583s)12059122

(120583s) 1198991

1198992

17793 10753 0434 1775 2611 5764 2 2

0 2 4 6 8 10 12 14 16 18 200

02040608

112141618

2

Time (120583s)

Curr

ent (

A)


times104




0 05 1 15 2 25 3 35 4 45 50

1

2

Time (120583s)

times10minus4

B120601

(Wb

m2)

Measured field



0 05 1 15 2 25 3 35 4 45 50

5

10

15times104

Time (120583s)

Ez

(Vm

)

Measured field








0 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

minus050 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

Time (120583s)

minus05

dEzdt

(Vm

s)



119903 = 15m

0 05 1 15 2 25 3 35 4 45 50123456789

10times104

Time (120583s)


Er

(Vm

)




5 Conclusion


1 2 3 4 5 6 7 8 9 10250725082509

25125112512251325142515


times10minus4

B120601

(Wb

m2)


1 2 3 4 5 6 7 8 9 10100

105

110

115

120

125

130


Ez

(kV

m)


1 2 3 4 5 6 7 8 9 10102030405060708090

100


Er

(kV

m)







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











nabla times ( +

120597

120597119905

) = 0

+

120597

120597119905

= minusnablaV119890

(15)




120597

120597119905

(16)


120597 (nablaV119890)

120597119905

minus 120583120576

1205972

1205971199052 (17)



1205972


120597V119890

120597119905

) (18)



120597V119890

120597119905

(19)


0 120576 = 120576


following equation

nabla2 minus 120583

01205760

1205972

1205971199052= minus1205830119869 (20)


position997888rarr




997888rarr

1199031015840





where 997888rarr



119889 =

1205830[

997888rarr

119869119894] 119889V1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[] 119889119878

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

119889 =

1205830[ 119868] 119889119871

1015840

4120587

10038161003816100381610038161003816100381610038161003816

119903 minus

997888rarr

1199031015840

10038161003816100381610038161003816100381610038161003816

(21)



997888997888rarr

119889119860 =

1205830119894 (1199111015840 119905119899minus 119877 (119911

1015840) 119888) 119889119911

1015840

4120587119877 (1199111015840)

(22)


119905119899=

119899

sum

119894=1

Δℎ

V119894

+

radic(119899Δℎ)2+ 1199032

119888

119877 (1199111015840= 119899Δℎ) = radic(119899Δℎ)

2+ 1199032

119894 (1199111015840 119905119899minus

119877 (1199111015840)

119888

)

= 119875 (1199111015840= 119899Δℎ) times 119894(0 119905

119899minus

119899

sum

119894=1

Δℎ

V119894

minus

radic(119899Δℎ)2+ 1199032

119888

)

(23)



119861119899

120593=


119899

sum

119895=1

1198651(119895) 119899 gt 0

0 119899 = 0

(24)


where

1198651(119895)

=

119896 times [

119875 (Δℎ)


times

119889 119894 (0 119905119899minus ΔℎV


119889119903


times 119894(0 119905119899minus

Δℎ

V1

minus


119888

)

times [Δℎ2+ 1199032]

minus32

]

+[

1

119903

times

119889 119894 (0 119905119899minus 119903119888)

119889119903

minus

119894 (0 119905119899minus 119903119888)

1199032

]

if 119895 = 1

119896 times[

[

[

119875 (119895Δℎ)

radic(119895Δℎ)2

+ 1199032

times119889

119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times (119889119903)minus1

minus119903 times 119875 (119895Δℎ)

times119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times [(119895Δℎ)2

+ 1199032]

minus32]

]

]

if 119895 gt 1

119896 =

1 119895 = 119899

2 119895 = 119899

(25)


119889119864119899

119911

119889119905

=

minusΔℎ

41205871199031205760

119899

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(26)

119864119899

119911=

minusΔℎ

41205871199031205760

119899

sum

119904=1

1198961015840

119904

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(27)

y

z

x

rarr

R998400

P

drarrA

rarrr1

rarrr2


rP

Real channel

Image channel

The perfect ground

Δh

Δh

Δh

Δh

R1

R2

1

1

2

2


where

1198961015840=

1

2

[

Δℎ

V1

+


119888

] 119904 = 1

1

2

[

[

[

(

119904

sum

119894=1

Δℎ

V119894

minus

119904minus1

sum

119894=1

Δℎ

V119894

)

+


2+ 1199032

119888

]

]

]

119904 = 1

(28)




11989401

(kA)11989402

(kA)12059111

(120583s)12059112

(120583s)12059121

(120583s)12059122

(120583s) 1198991

1198992

17793 10753 0434 1775 2611 5764 2 2

0 2 4 6 8 10 12 14 16 18 200

02040608

112141618

2

Time (120583s)

Curr

ent (

A)


times104




0 05 1 15 2 25 3 35 4 45 50

1

2

Time (120583s)

times10minus4

B120601

(Wb

m2)

Measured field



0 05 1 15 2 25 3 35 4 45 50

5

10

15times104

Time (120583s)

Ez

(Vm

)

Measured field








0 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

minus050 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

Time (120583s)

minus05

dEzdt

(Vm

s)



119903 = 15m

0 05 1 15 2 25 3 35 4 45 50123456789

10times104

Time (120583s)


Er

(Vm

)




5 Conclusion


1 2 3 4 5 6 7 8 9 10250725082509

25125112512251325142515


times10minus4

B120601

(Wb

m2)


1 2 3 4 5 6 7 8 9 10100

105

110

115

120

125

130


Ez

(kV

m)


1 2 3 4 5 6 7 8 9 10102030405060708090

100


Er

(kV

m)







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

1198651(119895)

=

119896 times [

119875 (Δℎ)


times

119889 119894 (0 119905119899minus ΔℎV


119889119903


times 119894(0 119905119899minus

Δℎ

V1

minus


119888

)

times [Δℎ2+ 1199032]

minus32

]

+[

1

119903

times

119889 119894 (0 119905119899minus 119903119888)

119889119903

minus

119894 (0 119905119899minus 119903119888)

1199032

]

if 119895 = 1

119896 times[

[

[

119875 (119895Δℎ)

radic(119895Δℎ)2

+ 1199032

times119889

119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times (119889119903)minus1

minus119903 times 119875 (119895Δℎ)

times119894(0 119905119899minus

119895

sum

119894=1

Δℎ

V119894

minus

radic(119895Δℎ)2

+ 1199032

119888

)

times [(119895Δℎ)2

+ 1199032]

minus32]

]

]

if 119895 gt 1

119896 =

1 119895 = 119899

2 119895 = 119899

(25)


119889119864119899

119911

119889119905

=

minusΔℎ

41205871199031205760

119899

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(26)

119864119899

119911=

minusΔℎ

41205871199031205760

119899

sum

119904=1

1198961015840

119904

sum

119895=1

119889 [1199031198651(119895)]

119889119903

119899 gt 0

0 119899 = 0

(27)

y

z

x

rarr

R998400

P

drarrA

rarrr1

rarrr2


rP

Real channel

Image channel

The perfect ground

Δh

Δh

Δh

Δh

R1

R2

1

1

2

2


where

1198961015840=

1

2

[

Δℎ

V1

+


119888

] 119904 = 1

1

2

[

[

[

(

119904

sum

119894=1

Δℎ

V119894

minus

119904minus1

sum

119894=1

Δℎ

V119894

)

+


2+ 1199032

119888

]

]

]

119904 = 1

(28)




11989401

(kA)11989402

(kA)12059111

(120583s)12059112

(120583s)12059121

(120583s)12059122

(120583s) 1198991

1198992

17793 10753 0434 1775 2611 5764 2 2

0 2 4 6 8 10 12 14 16 18 200

02040608

112141618

2

Time (120583s)

Curr

ent (

A)


times104




0 05 1 15 2 25 3 35 4 45 50

1

2

Time (120583s)

times10minus4

B120601

(Wb

m2)

Measured field



0 05 1 15 2 25 3 35 4 45 50

5

10

15times104

Time (120583s)

Ez

(Vm

)

Measured field








0 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

minus050 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

Time (120583s)

minus05

dEzdt

(Vm

s)



119903 = 15m

0 05 1 15 2 25 3 35 4 45 50123456789

10times104

Time (120583s)


Er

(Vm

)




5 Conclusion


1 2 3 4 5 6 7 8 9 10250725082509

25125112512251325142515


times10minus4

B120601

(Wb

m2)


1 2 3 4 5 6 7 8 9 10100

105

110

115

120

125

130


Ez

(kV

m)


1 2 3 4 5 6 7 8 9 10102030405060708090

100


Er

(kV

m)







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11989401

(kA)11989402

(kA)12059111

(120583s)12059112

(120583s)12059121

(120583s)12059122

(120583s) 1198991

1198992

17793 10753 0434 1775 2611 5764 2 2

0 2 4 6 8 10 12 14 16 18 200

02040608

112141618

2

Time (120583s)

Curr

ent (

A)


times104




0 05 1 15 2 25 3 35 4 45 50

1

2

Time (120583s)

times10minus4

B120601

(Wb

m2)

Measured field



0 05 1 15 2 25 3 35 4 45 50

5

10

15times104

Time (120583s)

Ez

(Vm

)

Measured field








0 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

minus050 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

Time (120583s)

minus05

dEzdt

(Vm

s)



119903 = 15m

0 05 1 15 2 25 3 35 4 45 50123456789

10times104

Time (120583s)


Er

(Vm

)




5 Conclusion


1 2 3 4 5 6 7 8 9 10250725082509

25125112512251325142515


times10minus4

B120601

(Wb

m2)


1 2 3 4 5 6 7 8 9 10100

105

110

115

120

125

130


Ez

(kV

m)


1 2 3 4 5 6 7 8 9 10102030405060708090

100


Er

(kV

m)







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

minus050 05 1 15 2 25 3 35 4 45 5

0

05

1

15

2

25times1011

Time (120583s)

minus05

dEzdt

(Vm

s)



119903 = 15m

0 05 1 15 2 25 3 35 4 45 50123456789

10times104

Time (120583s)


Er

(Vm

)




5 Conclusion


1 2 3 4 5 6 7 8 9 10250725082509

25125112512251325142515


times10minus4

B120601

(Wb

m2)


1 2 3 4 5 6 7 8 9 10100

105

110

115

120

125

130


Ez

(kV

m)


1 2 3 4 5 6 7 8 9 10102030405060708090

100


Er

(kV

m)







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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