Research ArticleAn Algorithm for Evaluation of Lightning ElectromagneticFields at Different Distances with respect to Lightning Channel

Mahdi Izadi12 Mohd Zainal Abidin Ab Kadir1 and Maryam Hajikhani3

1 Centre for Electromagnetic and Lightning Protection Research (CELP) Faculty of Engineering Universiti Putra Malaysia43400 Serdang Selangor Malaysia

2 Islamic Azad University Firoozkooh Branch Iran3 Aryaphase Company Tehran Iran

Correspondence should be addressed to Mahdi Izadi aryaphaseyahoocom

Received 9 September 2014 Accepted 23 October 2014 Published 12 November 2014

Academic Editor Chien-Yu Lu

Copyright copy 2014 Mahdi Izadi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The analytical field expressions are proposed to estimate the electromagnetic fields associated with vertical lightning channel at anobservation point directly in the time domain using current and geometrical parametersThe proposed expressions embrace a largenumber of channel base current functions and widely used engineering current models Moreover they can be joined with differentcoupling models for evaluation of lightning induced voltage when they provide different required field components directly in thetime domain The data from measured fields and simulated fields from previous works were employed to validate the proposedfield expressions The simulated results were compatible with both computed field values from previous studies and the measuredfields

1 Introduction

Lightning is a natural phenomenon that can give effect onthe power lines in two ways that is direct and indirect Inthe direct effects lightning strikes to power lines or towersor substations directly On the other hand the indirect effectconsiders electromagnetic fields associated with lightningchannel and induced voltage due to coupling between gener-ated electromagnetic fields and power system when lightningstrikes to the ground or any objective near to power linesor substations [1 2] This study considers the evaluationof electromagnetic fields due to lightning channel Severalmethods consider the calculation of electromagnetic fieldsdue to lightning channel while they are validated usingmeasured fields at one or a number of observation points[1] while the most common calculation methods can beconcluded by

(i) the MONOPOLE method [3 4]

(ii) the DIPOLE method [1 3]

(iii) the FDTD method [5](iv) the FDTD-HYBRID methods [6 7]

It should be mentioned that the electromagnetic fieldsassociated with the lightning channel rely basically on somegeometrical channel and ground parameters The basicinput parameters in almost all calculation methods in perfectground conductivity case are

(i) the channel base current that is usually prepared withdirect measurement [2] from the triggered lightningtechnique or obtained from inverse procedure algo-rithms from measured electromagnetic fields [8 9]

(ii) the current model for prediction of return strokecurrent wave shape along lightning channel

(iii) the radial distance from lightning channel(iv) the observation point height with respect to ground

surface(v) the return stroke current velocity along lightning

channel

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 925463 10 pageshttpdxdoiorg1011552014925463

2 Mathematical Problems in Engineering

Table 1 The widely used channel base current functions

Current model Channel base current function

Bruce and Golde [29] 119868119901[exp (minus119860119905) minus exp (minus119861119905)]

Improvement of Uman and McLain on Bruce andGolde function [30]

119868119901

120578[exp (minus119860119905) minus exp (minus119861119905)]

Improvement of Jones on Bruce and Golde function [31]

119868119901

120578[exp(minus119905

lowast

Γ1

) minus exp(minus119905lowast

Γ2

)]

119905lowast=Γ2

2

Γ1

+ 119905

Pierce and Ciones [32] 1198681199011 [exp (minus1198601119905) minus exp (minus1198611119905)] + 1198681199012 [exp (minus1198602119905) minus exp (minus1198612119905)]

Heidler function [33]119868119901

1205781

(119905Γ1)1198991

1 + (119905Γ1)1198991

exp(minus119905Γ2

)

Sum of two Heidler functions [11 34]11989401

1205781

(119905Γ11)1198991

1 + (119905Γ11)1198991

exp( minus119905

Γ12

) +11989402

1205782

(119905Γ21)1198992

1 + (119905Γ21)1198992

exp( minus119905

Γ22

)

Improvement of Nucci on Heidler function [35]11989401

1205781

(119905Γ11)1198991

1 + (119905Γ11)1198991

exp( minus119905

Γ12

) + 11989402[exp (minusΓ21119905) minus exp (minusΓ22119905)]

Two first calculation methods are expressed in the cylin-drical [1] and spherical [3 4] domains using current andcharge density [4] parameters along the lightning channelwhile they have complexity in the calculation of inductionpartial when some realistic channel base current functionsare used (as Heidler function) and they do not have ananalytical solution of integral to time [7 10] On the otherhand the FDTDmethod just is validated for closed distancesfrom lightning channel [5] Moreover the FDTD HYBRIDmethods need to estimatemagnetic fields at six points aroundobservation point for calculation of electric fields at anobservation point [6 7 11] In order to provide the gen-eral analytical electromagnetic field expressions in the timedomain the widely used channel base current functions areintroduced and categorizedTherefore after generalization ofcurrent functions based on proposed expressions the generalelectromagnetic field expressions due to vertical lightningchannel on the prefect ground are proposedThese equationscan be applied to a large number of leading current functionsand their combinations and can support different generalizedengineering current models directly in the time domain Thebasic assumptions of this study are the following

(i) The ground conductivity is infinite

(ii) The ground surface is flat

(iii) The lightning channel is perpendicular to the groundand has no branches

(iv) The corona effect is negligible

In the next section the widely used current functionsand current models are reviewed Therefore the proposedelectromagnetic field expressions are presented Finally ajustification of the proposed method and its validation bymeasuring fields and simulated fields from previous studiesis given

2 Channel Base Current Functions

Previous studies presented several channel base current func-tions to simulate return stroke current wave shape at channelbase on the ground surface where the fixed coefficients ofthe current functions were determined by the data frommeasured current or measured fields using inverse procedurealgorithms The significant channel base current functionsare tabulated in Table 1 where 119860 1198601 1198602 119861 1198611 1198612 Γ1Γ2 Γ11 Γ12 Γ21 Γ22 120578 1198991 1198992 are constant coefficients and119868119901 1198681199011 1198681199012 11986801 11986802 are current peaks Note that similarsymbols in different current functions do not have a similarconcept

Different channel base current functions in Table 1 canbe determined from (1) and (2) by changing the constantcoefficients or the combination of (1) and (2) Hence thisstudy adapts these two basic equations and considers thewidely used current functions using a combination of them

119868 (0 119905) =1198680

1205780

(1198881119890minus120572119905

minus 1198882119890minus120574119905

) (1)

119868 (0 119905) =11986801

12057801

(1199051199051)119899

1 + (1199051199051)119899 exp(

minus119905

1199052

) (2)

where

12057801 = exp[minus(11990511199052

)(1198991199052

1199051

)

1119899

] (3)

3 Return Stroke Current Models

One of the effective factors on electromagnetic field calcu-lations is the behavior of return stroke currents at differentheights along the lightning channel that is modeled via

Mathematical Problems in Engineering 3

Table 2 The internal parameters of (4) for different engineeringmodels [12]

Model 119875(1199111015840) V

Bruce and Golde (BG) [29] 1 infin

Traveling Current Source (TCS)[36]

1 minus119888

Transmission Line (TL) [1] 1 V119891

Modified Transmission Line withLinear Decay (MTLL) [1]

(1 minus1199111015840

119867119888

) V119891

Modified Transmission Line withExponential Decay (MTLE) [1]

exp(minus1199111015840

120582) V119891

current models Generally the return stroke current modelscan be classified into four groups as follows [12 13]

(i) the gas-dynamic or physical models

(ii) the current distributed models

(iii) the electromagnetic models

(iv) the engineering models

To develop general expressions for electromagnetic fieldsdue to lightning channel the engineering models are appliedin this study In the engineering models the current atdifferent heights along the lightning channel can be expressedby a special function based on the channel base currentand an attenuation height dependent factor Most of theengineering models can be presented by a general equationsuch as (4) [12 13]The internal parameters of (4) for differentengineering current models are listed in Table 2 [1 12] where119888 is considered as the speed of light in free space 120582 as a decayconstant (1sim2 km) and119867119888 as the cloud height with respect tothe ground surface [1]

119868 (1199111015840 119905) = 119868(0 119905 minus

1199111015840

V) times 119875 (119911

1015840) times 119906(119905 minus

1199111015840

V119891) (4)

where 1199111015840 is the temporary charge height along lightning

channel 119868(1199111015840 119905) is current distribution along the lightningchannel at any height 1199111015840 and any time 119905 119868(0 119905) is channel basecurrent 119875(1199111015840) is the attenuation height dependent factor Vis the current-wave propagation velocity V119891 is the upwardpropagating front velocity and 119906 is the Heaviside functiondefended as

119906(119905 minus1199111015840

V119891) =

1 for 119905 ge 1199111015840

V119891

0 for 119905 lt 1199111015840

V119891

(5)

Based on (4) return stroke current at the height of 1199111015840along the lightning channel can be estimated by using channelbase current and the height dependent attenuation factor

X

X

Y

Y

Z

Z

Z

Z

P

P

rarrr

120601

Real channel

H(t)rarrR

H998400(t)

Image channel

The perfect ground I(0 t)

rarrr2

rarrr1

rarr

R998400

1205872 minus 120601

Figure 1 Geometry of the problem

Note that in recent years several models based on (4) andusingmeasured electromagnetic fields and inverse procedurealgorithms were presented [14ndash16] Similarly different cur-rent models can be examined by the comparison betweensimulated electromagnetic fields using current model andmeasured fields at different observation points [1] In thismodel the assumed return stroke velocity is set at a constantvalue that is typically between 1198882 and 21198883 [17]

4 Electromagnetic Fields Associated withLightning Channel

In this section electromagnetic field expressions due to twoexpressed functions in (1) and (2) at an observation pointabove the ground surface are estimated whereas the currentmodel is set on (4) According to the geometry of the situation(see Figure 1) the potential vector () at the observationpoint above ground surface can be obtained from (6) whenMaxwellrsquos equation [3 18] and Lorentz gauge [19] are applied

nabla2 minus 12058301205760

1205972

1205971199052= minus1205830

119869 (6)

where 119869 is the current density 1205830 = 4120587 times 10minus7 Vsdots(Asdotm) and

1205760 = 885 times 10minus12 Fm

Moreover Figure 1 demonstrates that the solution to (6)can be obtained from (7) whereas the infinitesimal currentsource is located along 119911-axis at space in position 997888rarr1199032 from theorigin Also the observation point is located at space and 997888rarr1199031

4 Mathematical Problems in Engineering

from the origin point therefore can be estimated by =

997888rarr1199031 minus

997888rarr1199032

119889 =

1205830 [119868] 119889119871

1015840

412058710038161003816100381610038161003816

997888rarr1199031 minus

997888rarr1199032

10038161003816100381610038161003816

(7)

where 119868 is line current density and 1198891198711015840 is unit vector along

lightning channelLikewise by replacing return stroke current and 1198891199111015840 with

119868 and unit vector respectively in (7) can convert it to (8)

997888997888rarr119889119860119911 =

1205830119894 (1199111015840 119905 minus (119877119888)) 119889119911

1015840

4120587119877 (8)

where

119903 = radic1199092 + 1199102

119877 = radic1199092 + 1199102 + (119911 minus 1199111015840)2

(9)

Further the magnetic flux density and electric fields canbe obtained from potential vector as expressed in (10) and(11) respectively [3 18]

= nabla times (10)

nabla times = minus120597

120597119905= minus

120597 (nabla times )

120597119905 (11)

where is the magnetic flux density and is the electric fieldvector

Therefore based on the general current model in (4)the field components for first current function as presentedin (1) can be obtained from (12) when (8) (10) (11) thetrapezoid and FDTD methods are applied [20ndash22] Notethat the internal parameters are collected from Table 3 Also119911 RS1 (119909 119910 119911 119905119899) = 0

119909 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198651 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS1 (119909 119910 119911 119905119899)

= 119909 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198652 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

= 119910 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS1 (119909 119910 119911 119905119899)

= 119911 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198653 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198653 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(12)

where 119909 RS1 is the magnetic flux density at 119909-axis due toreturn stroke current function from (1) 119910 RS1 is themagneticflux density at 119910-axis due to return stroke current functionfrom (1) 119911 RS1 is the magnetic flux density at 119911-axis due toreturn stroke current function from (1) 119909 RS1 is the electricfield at 119909-axis due to return stroke current function from (1)119910 RS1 is the electric field at119910-axis due to return stroke currentfunction from (1) and 119911 RS1 is the electric field at 119911-axis dueto return stroke current function from (1) Δ119905 is the time step119899 is the number of time steps

119905119899 =

radic1199032 + 1199112

119888+ (119899 minus 1) Δ119905 119899 = 1 2 119899max

119886119898 =

Δℎ119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ119894

119896for others

1198861015840

119898=

Δℎ1015840

119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ1015840

119894

119896for others

(13)

119896 is division factor (ge 2)

Mathematical Problems in Engineering 5

Δℎ119894 =

1205731205942(119888119905119894 minus 119888119905119894minus1) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

+ radic(120573119888119905119894minus1 minus 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 minus 119888119905119894) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

for 119894 = 1

Δℎ1015840

119894=

1205731205942(119888119905119894minus1 minus 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

minus radic(120573119888119905119894minus1 + 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 + 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

for 119894 = 1

ℎ119898119894 =

(119898 minus 1) times Δℎ119894

119896+ ℎ119898=119896+1119894minus1

(119898 minus 1) times Δℎ119894

119896for 119894 = 1

ℎ1015840

119898119894=

(119898 minus 1) times Δℎ1015840

119894

119896+ ℎ

1015840

119898=119896+1119894minus1

(119898 minus 1) times Δℎ1015840

119894

119896for 119894 = 1

(14)

In addition by using the samemethodology that was used forthe first current function the components of electromagneticfield for the second current function as expressed in (2) areproposed by (15) Note that 119911 RS2 (119909 119910 119911 119905119899) = 0

119909 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198654 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS2 (119909 119910 119911 119905119899)

= 119909 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198655 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

= 119910 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS2 (119909 119910 119911 119905119899)

= 119911 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198656 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198656 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(15)

where 119909 RS2 is the magnetic flux density at 119909-axis due toreturn stroke current function from (2) 119910 RS2 is themagneticflux density at 119910-axis due to return stroke current functionfrom (2) 119911 RS2 is the magnetic flux density at 119911-axis due toreturn stroke current function from (2) 119909 RS2 is the electricfield at 119909-axis due to return stroke current function from (2)119910 RS2 is the electric field at119910-axis due to return stroke currentfunction from (2) and 119911 RS2 is the electric field at 119911-axis dueto return stroke current function from (2)

Consequently due to the whole return stroke currentfunctions from Table 1 the electromagnetic fields can beeasily estimated via the proposed field expressions whereasthe total fields due to two part functions are equal to the sumof the electromagnetic fields due to each part independentlyOn top of that the first part current functions in Table 3should be converted to either (1) or (2) format before gettinginto proposed expressions In other words the proposedalgorithm can support the whole current models related to(4) In the same way the components of the electromagneticfields in the Cartesian coordinates can be converted to

6 Mathematical Problems in Engineering

Table3Th

einternalp

aram

eterso

fpropo

sedele

ctromagnetic

fieldse

xpressions

Parameter

Expressio

n

1198651

10minus7119868 0119910

119875(1199111015840)

1205780

[119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1)

119888119877

2]minus[119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1)

1198773

]

1198652

119868 0119875(1199111015840)

412058712057601205780

(119911minus1199111015840)119909minus3(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

119888 11205722exp(minus120572119860

1)minus119888 21205742exp(minus120574119860

1)

1198882119877

3+

3(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

1198653

119868 0119875(1199111015840)

412058712057601205780

3(1199092+1199102)(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

2(minus119888 1120572exp(minus120572119860

1)+119888 2120574exp(minus120574119860

1))

119888119877

2+

(1199092+1199102)(minus119888 11205722exp(minus120572119860

1)+119888 21205742exp(minus120574119860

1))

1198882119877

3

minus3(1199092+1199102)(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

+2(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198773

1198654

10minus7

119868 01119875(119911

1015840)119910

1198602

12057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899

119888119905 2119877

2minus

(119860

1119905 1)119899

1198773

minus119899(119860

1119905 1)119899minus1

119888119905 1119877

2+

119899(119860

1119905 1)2119899minus1

119888119905 1119877

2[(119860

1119905 1)119899+1]

1198655

119868 01119875(119911

1015840)119909

(119911minus1199111015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899(311988821199052 2

minus3119888119905 2119877

+119877

2)

11988821199052 2119877

5+

21198992(119860

1119905 1)3119899minus2+119899(119899

minus1)(119860

1119905 1)119899minus2[(119860

1119905 1)119899+1]2+(minus

31198992+119899)(119860

1119905 1)2119899minus2[(119860

1119905 1)119899+1]

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

+3119899([(119860

1119905 1)119899+1](119860

1119905 1)119899minus1minus(119860

1119905 1)2119899minus1)

119888119905 1119877

4[(119860

1119905 1)119899+1]

+2119899((119860

1119905 1)2119899minus1minus[(119860

1119905 1)119899+1](119860

1119905 1)119899minus1)

1198882119905 1119905 2119877

3[(119860

1119905 1)119899+1]

1198656

119868 01119875(119911

1015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(1199092+1199102)[(119860

1119905 1)119899(minus31198882119905 11199052 2

+3119888119905 1119905 2119877

minus119905 1119877

2)+119899(119860

1119905 1)119899minus1(minus31198881199052 2119877

+2119905 2119877

2)]

1198882119905 11199052 2119877

5+

2(119860

1119905 1)119899(119888119905 1119905 2

minus119905 1119877)+2119899(119860

1119905 1)119899minus1119905 2119877

119888119905 1119905 2119877

3minus

119899(119899

minus1)(1199092+1199102)(119860

1119905 1)119899minus2

11988821199052 1119877

3

+

(1199092+1199102)[21198992119905 2119877(119860

1119905 1)2119899minus2+3119899119888119905 1119905 2

(119860

1119905 1)2119899minus1+119899(119899

minus1)119905 2119877(119860

1119905 1)2119899minus2minus2119899119905 1119877(119860

1119905 1)2119899minus1]minus2119899119888119905 1119905 2119877

2(119860

1119905 1)2119899minus1

11988821199052 1119905 2119877

4[(119860

1119905 1)119899+1]

minus21198992(1199092+1199102)(119860

1119905 1)3119899minus2

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

119877=

radic1199092+1199102+(119911

minus1199111015840)2

1198601=

119905minus

119877 119888minus

1003816 1003816 1003816 1003816 100381611991110158401003816 1003816 1003816 1003816 1003816

119907

1198602=exp(

(119877119888)minus119905+(1003816 1003816 1003816 1003816 1003816119911

10158401003816 1003816 1003816 1003816 1003816119907)

119905 2)

Mathematical Problems in Engineering 7

cylindrical domain using (16) and (17) for the magnetic fluxdensity and the electric field components respectively

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(16)

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(17)

where 120601 = arc cos (119910radic1199092 + 1199102) 119903 (119903 119911 120601 119905) 119903 (119903 119911 120601 119905)

is the magnetic flux densityelectric field at 119903-direction(horizontal) 120593 (119903 119911 120601 119905) 120593 (119903 119911 120601 119905) is the magnetic fluxdensityelectric field at 120601-direction and 119911 (119903 119911 120601 119905)

119911 (119903 119911 120601 119905) is the magnetic flux densityelectric field at119911-direction (vertical)

5 Result and Discussion

This section deals with the validation of the electromagneticfields due to lightning channel using some channel basecurrent functions According to the proposed method thewidely used current functions can be converted into sum ofindividual parts using (1) and (2) Therefore the electromag-netic fields due to each part can be calculated by (12) and(15) Then the values of total electromagnetic field are equalto the sum of individual components of electromagneticfield When the MTLE model is applied as current modelthe simulated magnetic flux density and the vertical electricfield caused by a sum of two Heidler function as a channelbase current function at an observation point with 46 kmdistance from lightning channel are depicted in Figures 2and 3 respectively The measured magnetic flux density andvertical electric field with similar primary conditions as theones in Figures 2 and 3 are demonstrated in Figures 4 and 5respectively Therefore comparisons between the measuredand simulated fields for magnetic flux density and verticaleclectic field are tabulated in Tables 4 and 5 respectively Itshould be noted that the current parameters are 11989401 = 17 kA11989402 = 8 kA Γ11 = 04 times 10

minus6 Γ12 = 4 times 10minus6 Γ21 = 4 times 10

minus6Γ22 = 50 times 10

minus6 1198991 = 2 1198992 = 2 V = 1 times 108 and 120582 = 1500

[23]

10 15 20 25 30 35 40 45 50 55 60 650

05

1

15

2

25

3times10minus7

Time (120583s)

Mag

netic

flux

den

sity

(Wb

m2)

Figure 2 The simulated magnetic flux density using proposedmethod (119911 = 10m 119903 = 46 km)

10 15 20 25 30 35 40 45 50 55 60 650

102030405060708090

Vert

ical

elec

tric

fiel

d (V

m)

Time (120583s)

Figure 3 The simulated vertical electric field using proposedmethod (119911 = 10m 119903 = 46 km)

0 10 20 30 40 50 60 700

05

1

15

2

25times10minus7

Time (120583s)

B120601

(Wb

m2)

Figure 4 The measured magnetic flux density for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

Additionally the sum of two Heidler functions as a chan-nel base current function can be expressed by the sum of twogeneral current functions based on (2) Therefore the totalelectromagnetic fields are estimated by using the proposedmethod and considering each current part separately Thecomparison between simulated and measured fields showsthe overall good agreement with experimental measurementfor proposed field expressions Similarly the simulated fields

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Table 1 The widely used channel base current functions

Current model Channel base current function

Bruce and Golde [29] 119868119901[exp (minus119860119905) minus exp (minus119861119905)]

Improvement of Uman and McLain on Bruce andGolde function [30]

119868119901

120578[exp (minus119860119905) minus exp (minus119861119905)]

Improvement of Jones on Bruce and Golde function [31]

119868119901

120578[exp(minus119905

lowast

Γ1

) minus exp(minus119905lowast

Γ2

)]

119905lowast=Γ2

2

Γ1

+ 119905

Pierce and Ciones [32] 1198681199011 [exp (minus1198601119905) minus exp (minus1198611119905)] + 1198681199012 [exp (minus1198602119905) minus exp (minus1198612119905)]

Heidler function [33]119868119901

1205781

(119905Γ1)1198991

1 + (119905Γ1)1198991

exp(minus119905Γ2

)

Sum of two Heidler functions [11 34]11989401

1205781

(119905Γ11)1198991

1 + (119905Γ11)1198991

exp( minus119905

Γ12

) +11989402

1205782

(119905Γ21)1198992

1 + (119905Γ21)1198992

exp( minus119905

Γ22

)

Improvement of Nucci on Heidler function [35]11989401

1205781

(119905Γ11)1198991

1 + (119905Γ11)1198991

exp( minus119905

Γ12

) + 11989402[exp (minusΓ21119905) minus exp (minusΓ22119905)]

Two first calculation methods are expressed in the cylin-drical [1] and spherical [3 4] domains using current andcharge density [4] parameters along the lightning channelwhile they have complexity in the calculation of inductionpartial when some realistic channel base current functionsare used (as Heidler function) and they do not have ananalytical solution of integral to time [7 10] On the otherhand the FDTDmethod just is validated for closed distancesfrom lightning channel [5] Moreover the FDTD HYBRIDmethods need to estimatemagnetic fields at six points aroundobservation point for calculation of electric fields at anobservation point [6 7 11] In order to provide the gen-eral analytical electromagnetic field expressions in the timedomain the widely used channel base current functions areintroduced and categorizedTherefore after generalization ofcurrent functions based on proposed expressions the generalelectromagnetic field expressions due to vertical lightningchannel on the prefect ground are proposedThese equationscan be applied to a large number of leading current functionsand their combinations and can support different generalizedengineering current models directly in the time domain Thebasic assumptions of this study are the following

(i) The ground conductivity is infinite

(ii) The ground surface is flat

(iii) The lightning channel is perpendicular to the groundand has no branches

(iv) The corona effect is negligible

In the next section the widely used current functionsand current models are reviewed Therefore the proposedelectromagnetic field expressions are presented Finally ajustification of the proposed method and its validation bymeasuring fields and simulated fields from previous studiesis given

2 Channel Base Current Functions

Previous studies presented several channel base current func-tions to simulate return stroke current wave shape at channelbase on the ground surface where the fixed coefficients ofthe current functions were determined by the data frommeasured current or measured fields using inverse procedurealgorithms The significant channel base current functionsare tabulated in Table 1 where 119860 1198601 1198602 119861 1198611 1198612 Γ1Γ2 Γ11 Γ12 Γ21 Γ22 120578 1198991 1198992 are constant coefficients and119868119901 1198681199011 1198681199012 11986801 11986802 are current peaks Note that similarsymbols in different current functions do not have a similarconcept

Different channel base current functions in Table 1 canbe determined from (1) and (2) by changing the constantcoefficients or the combination of (1) and (2) Hence thisstudy adapts these two basic equations and considers thewidely used current functions using a combination of them

119868 (0 119905) =1198680

1205780

(1198881119890minus120572119905

minus 1198882119890minus120574119905

) (1)

119868 (0 119905) =11986801

12057801

(1199051199051)119899

1 + (1199051199051)119899 exp(

minus119905

1199052

) (2)

where

12057801 = exp[minus(11990511199052

)(1198991199052

1199051

)

1119899

] (3)

3 Return Stroke Current Models

One of the effective factors on electromagnetic field calcu-lations is the behavior of return stroke currents at differentheights along the lightning channel that is modeled via

Mathematical Problems in Engineering 3

Table 2 The internal parameters of (4) for different engineeringmodels [12]

Model 119875(1199111015840) V

Bruce and Golde (BG) [29] 1 infin

Traveling Current Source (TCS)[36]

1 minus119888

Transmission Line (TL) [1] 1 V119891

Modified Transmission Line withLinear Decay (MTLL) [1]

(1 minus1199111015840

119867119888

) V119891

Modified Transmission Line withExponential Decay (MTLE) [1]

exp(minus1199111015840

120582) V119891

current models Generally the return stroke current modelscan be classified into four groups as follows [12 13]

(i) the gas-dynamic or physical models

(ii) the current distributed models

(iii) the electromagnetic models

(iv) the engineering models

To develop general expressions for electromagnetic fieldsdue to lightning channel the engineering models are appliedin this study In the engineering models the current atdifferent heights along the lightning channel can be expressedby a special function based on the channel base currentand an attenuation height dependent factor Most of theengineering models can be presented by a general equationsuch as (4) [12 13]The internal parameters of (4) for differentengineering current models are listed in Table 2 [1 12] where119888 is considered as the speed of light in free space 120582 as a decayconstant (1sim2 km) and119867119888 as the cloud height with respect tothe ground surface [1]

119868 (1199111015840 119905) = 119868(0 119905 minus

1199111015840

V) times 119875 (119911

1015840) times 119906(119905 minus

1199111015840

V119891) (4)

where 1199111015840 is the temporary charge height along lightning

channel 119868(1199111015840 119905) is current distribution along the lightningchannel at any height 1199111015840 and any time 119905 119868(0 119905) is channel basecurrent 119875(1199111015840) is the attenuation height dependent factor Vis the current-wave propagation velocity V119891 is the upwardpropagating front velocity and 119906 is the Heaviside functiondefended as

119906(119905 minus1199111015840

V119891) =

1 for 119905 ge 1199111015840

V119891

0 for 119905 lt 1199111015840

V119891

(5)

Based on (4) return stroke current at the height of 1199111015840along the lightning channel can be estimated by using channelbase current and the height dependent attenuation factor

X

X

Y

Y

Z

Z

Z

Z

P

P

rarrr

120601

Real channel

H(t)rarrR

H998400(t)

Image channel

The perfect ground I(0 t)

rarrr2

rarrr1

rarr

R998400

1205872 minus 120601

Figure 1 Geometry of the problem

Note that in recent years several models based on (4) andusingmeasured electromagnetic fields and inverse procedurealgorithms were presented [14ndash16] Similarly different cur-rent models can be examined by the comparison betweensimulated electromagnetic fields using current model andmeasured fields at different observation points [1] In thismodel the assumed return stroke velocity is set at a constantvalue that is typically between 1198882 and 21198883 [17]

4 Electromagnetic Fields Associated withLightning Channel

In this section electromagnetic field expressions due to twoexpressed functions in (1) and (2) at an observation pointabove the ground surface are estimated whereas the currentmodel is set on (4) According to the geometry of the situation(see Figure 1) the potential vector () at the observationpoint above ground surface can be obtained from (6) whenMaxwellrsquos equation [3 18] and Lorentz gauge [19] are applied

nabla2 minus 12058301205760

1205972

1205971199052= minus1205830

119869 (6)

where 119869 is the current density 1205830 = 4120587 times 10minus7 Vsdots(Asdotm) and

1205760 = 885 times 10minus12 Fm

Moreover Figure 1 demonstrates that the solution to (6)can be obtained from (7) whereas the infinitesimal currentsource is located along 119911-axis at space in position 997888rarr1199032 from theorigin Also the observation point is located at space and 997888rarr1199031

4 Mathematical Problems in Engineering

from the origin point therefore can be estimated by =

997888rarr1199031 minus

997888rarr1199032

119889 =

1205830 [119868] 119889119871

1015840

412058710038161003816100381610038161003816

997888rarr1199031 minus

997888rarr1199032

10038161003816100381610038161003816

(7)

where 119868 is line current density and 1198891198711015840 is unit vector along

lightning channelLikewise by replacing return stroke current and 1198891199111015840 with

119868 and unit vector respectively in (7) can convert it to (8)

997888997888rarr119889119860119911 =

1205830119894 (1199111015840 119905 minus (119877119888)) 119889119911

1015840

4120587119877 (8)

where

119903 = radic1199092 + 1199102

119877 = radic1199092 + 1199102 + (119911 minus 1199111015840)2

(9)

Further the magnetic flux density and electric fields canbe obtained from potential vector as expressed in (10) and(11) respectively [3 18]

= nabla times (10)

nabla times = minus120597

120597119905= minus

120597 (nabla times )

120597119905 (11)

where is the magnetic flux density and is the electric fieldvector

Therefore based on the general current model in (4)the field components for first current function as presentedin (1) can be obtained from (12) when (8) (10) (11) thetrapezoid and FDTD methods are applied [20ndash22] Notethat the internal parameters are collected from Table 3 Also119911 RS1 (119909 119910 119911 119905119899) = 0

119909 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198651 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS1 (119909 119910 119911 119905119899)

= 119909 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198652 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

= 119910 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS1 (119909 119910 119911 119905119899)

= 119911 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198653 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198653 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(12)

where 119909 RS1 is the magnetic flux density at 119909-axis due toreturn stroke current function from (1) 119910 RS1 is themagneticflux density at 119910-axis due to return stroke current functionfrom (1) 119911 RS1 is the magnetic flux density at 119911-axis due toreturn stroke current function from (1) 119909 RS1 is the electricfield at 119909-axis due to return stroke current function from (1)119910 RS1 is the electric field at119910-axis due to return stroke currentfunction from (1) and 119911 RS1 is the electric field at 119911-axis dueto return stroke current function from (1) Δ119905 is the time step119899 is the number of time steps

119905119899 =

radic1199032 + 1199112

119888+ (119899 minus 1) Δ119905 119899 = 1 2 119899max

119886119898 =

Δℎ119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ119894

119896for others

1198861015840

119898=

Δℎ1015840

119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ1015840

119894

119896for others

(13)

119896 is division factor (ge 2)

Mathematical Problems in Engineering 5

Δℎ119894 =

1205731205942(119888119905119894 minus 119888119905119894minus1) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

+ radic(120573119888119905119894minus1 minus 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 minus 119888119905119894) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

for 119894 = 1

Δℎ1015840

119894=

1205731205942(119888119905119894minus1 minus 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

minus radic(120573119888119905119894minus1 + 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 + 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

for 119894 = 1

ℎ119898119894 =

(119898 minus 1) times Δℎ119894

119896+ ℎ119898=119896+1119894minus1

(119898 minus 1) times Δℎ119894

119896for 119894 = 1

ℎ1015840

119898119894=

(119898 minus 1) times Δℎ1015840

119894

119896+ ℎ

1015840

119898=119896+1119894minus1

(119898 minus 1) times Δℎ1015840

119894

119896for 119894 = 1

(14)

In addition by using the samemethodology that was used forthe first current function the components of electromagneticfield for the second current function as expressed in (2) areproposed by (15) Note that 119911 RS2 (119909 119910 119911 119905119899) = 0

119909 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198654 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS2 (119909 119910 119911 119905119899)

= 119909 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198655 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

= 119910 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS2 (119909 119910 119911 119905119899)

= 119911 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198656 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198656 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(15)

where 119909 RS2 is the magnetic flux density at 119909-axis due toreturn stroke current function from (2) 119910 RS2 is themagneticflux density at 119910-axis due to return stroke current functionfrom (2) 119911 RS2 is the magnetic flux density at 119911-axis due toreturn stroke current function from (2) 119909 RS2 is the electricfield at 119909-axis due to return stroke current function from (2)119910 RS2 is the electric field at119910-axis due to return stroke currentfunction from (2) and 119911 RS2 is the electric field at 119911-axis dueto return stroke current function from (2)

Consequently due to the whole return stroke currentfunctions from Table 1 the electromagnetic fields can beeasily estimated via the proposed field expressions whereasthe total fields due to two part functions are equal to the sumof the electromagnetic fields due to each part independentlyOn top of that the first part current functions in Table 3should be converted to either (1) or (2) format before gettinginto proposed expressions In other words the proposedalgorithm can support the whole current models related to(4) In the same way the components of the electromagneticfields in the Cartesian coordinates can be converted to

6 Mathematical Problems in Engineering

Table3Th

einternalp

aram

eterso

fpropo

sedele

ctromagnetic

fieldse

xpressions

Parameter

Expressio

n

1198651

10minus7119868 0119910

119875(1199111015840)

1205780

[119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1)

119888119877

2]minus[119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1)

1198773

]

1198652

119868 0119875(1199111015840)

412058712057601205780

(119911minus1199111015840)119909minus3(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

119888 11205722exp(minus120572119860

1)minus119888 21205742exp(minus120574119860

1)

1198882119877

3+

3(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

1198653

119868 0119875(1199111015840)

412058712057601205780

3(1199092+1199102)(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

2(minus119888 1120572exp(minus120572119860

1)+119888 2120574exp(minus120574119860

1))

119888119877

2+

(1199092+1199102)(minus119888 11205722exp(minus120572119860

1)+119888 21205742exp(minus120574119860

1))

1198882119877

3

minus3(1199092+1199102)(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

+2(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198773

1198654

10minus7

119868 01119875(119911

1015840)119910

1198602

12057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899

119888119905 2119877

2minus

(119860

1119905 1)119899

1198773

minus119899(119860

1119905 1)119899minus1

119888119905 1119877

2+

119899(119860

1119905 1)2119899minus1

119888119905 1119877

2[(119860

1119905 1)119899+1]

1198655

119868 01119875(119911

1015840)119909

(119911minus1199111015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899(311988821199052 2

minus3119888119905 2119877

+119877

2)

11988821199052 2119877

5+

21198992(119860

1119905 1)3119899minus2+119899(119899

minus1)(119860

1119905 1)119899minus2[(119860

1119905 1)119899+1]2+(minus

31198992+119899)(119860

1119905 1)2119899minus2[(119860

1119905 1)119899+1]

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

+3119899([(119860

1119905 1)119899+1](119860

1119905 1)119899minus1minus(119860

1119905 1)2119899minus1)

119888119905 1119877

4[(119860

1119905 1)119899+1]

+2119899((119860

1119905 1)2119899minus1minus[(119860

1119905 1)119899+1](119860

1119905 1)119899minus1)

1198882119905 1119905 2119877

3[(119860

1119905 1)119899+1]

1198656

119868 01119875(119911

1015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(1199092+1199102)[(119860

1119905 1)119899(minus31198882119905 11199052 2

+3119888119905 1119905 2119877

minus119905 1119877

2)+119899(119860

1119905 1)119899minus1(minus31198881199052 2119877

+2119905 2119877

2)]

1198882119905 11199052 2119877

5+

2(119860

1119905 1)119899(119888119905 1119905 2

minus119905 1119877)+2119899(119860

1119905 1)119899minus1119905 2119877

119888119905 1119905 2119877

3minus

119899(119899

minus1)(1199092+1199102)(119860

1119905 1)119899minus2

11988821199052 1119877

3

+

(1199092+1199102)[21198992119905 2119877(119860

1119905 1)2119899minus2+3119899119888119905 1119905 2

(119860

1119905 1)2119899minus1+119899(119899

minus1)119905 2119877(119860

1119905 1)2119899minus2minus2119899119905 1119877(119860

1119905 1)2119899minus1]minus2119899119888119905 1119905 2119877

2(119860

1119905 1)2119899minus1

11988821199052 1119905 2119877

4[(119860

1119905 1)119899+1]

minus21198992(1199092+1199102)(119860

1119905 1)3119899minus2

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

119877=

radic1199092+1199102+(119911

minus1199111015840)2

1198601=

119905minus

119877 119888minus

1003816 1003816 1003816 1003816 100381611991110158401003816 1003816 1003816 1003816 1003816

119907

1198602=exp(

(119877119888)minus119905+(1003816 1003816 1003816 1003816 1003816119911

10158401003816 1003816 1003816 1003816 1003816119907)

119905 2)

Mathematical Problems in Engineering 7

cylindrical domain using (16) and (17) for the magnetic fluxdensity and the electric field components respectively

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(16)

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(17)

where 120601 = arc cos (119910radic1199092 + 1199102) 119903 (119903 119911 120601 119905) 119903 (119903 119911 120601 119905)

is the magnetic flux densityelectric field at 119903-direction(horizontal) 120593 (119903 119911 120601 119905) 120593 (119903 119911 120601 119905) is the magnetic fluxdensityelectric field at 120601-direction and 119911 (119903 119911 120601 119905)

119911 (119903 119911 120601 119905) is the magnetic flux densityelectric field at119911-direction (vertical)

5 Result and Discussion

This section deals with the validation of the electromagneticfields due to lightning channel using some channel basecurrent functions According to the proposed method thewidely used current functions can be converted into sum ofindividual parts using (1) and (2) Therefore the electromag-netic fields due to each part can be calculated by (12) and(15) Then the values of total electromagnetic field are equalto the sum of individual components of electromagneticfield When the MTLE model is applied as current modelthe simulated magnetic flux density and the vertical electricfield caused by a sum of two Heidler function as a channelbase current function at an observation point with 46 kmdistance from lightning channel are depicted in Figures 2and 3 respectively The measured magnetic flux density andvertical electric field with similar primary conditions as theones in Figures 2 and 3 are demonstrated in Figures 4 and 5respectively Therefore comparisons between the measuredand simulated fields for magnetic flux density and verticaleclectic field are tabulated in Tables 4 and 5 respectively Itshould be noted that the current parameters are 11989401 = 17 kA11989402 = 8 kA Γ11 = 04 times 10

minus6 Γ12 = 4 times 10minus6 Γ21 = 4 times 10

minus6Γ22 = 50 times 10

minus6 1198991 = 2 1198992 = 2 V = 1 times 108 and 120582 = 1500

[23]

10 15 20 25 30 35 40 45 50 55 60 650

05

1

15

2

25

3times10minus7

Time (120583s)

Mag

netic

flux

den

sity

(Wb

m2)

Figure 2 The simulated magnetic flux density using proposedmethod (119911 = 10m 119903 = 46 km)

10 15 20 25 30 35 40 45 50 55 60 650

102030405060708090

Vert

ical

elec

tric

fiel

d (V

m)

Time (120583s)

Figure 3 The simulated vertical electric field using proposedmethod (119911 = 10m 119903 = 46 km)

0 10 20 30 40 50 60 700

05

1

15

2

25times10minus7

Time (120583s)

B120601

(Wb

m2)

Figure 4 The measured magnetic flux density for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

Additionally the sum of two Heidler functions as a chan-nel base current function can be expressed by the sum of twogeneral current functions based on (2) Therefore the totalelectromagnetic fields are estimated by using the proposedmethod and considering each current part separately Thecomparison between simulated and measured fields showsthe overall good agreement with experimental measurementfor proposed field expressions Similarly the simulated fields

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Table 2 The internal parameters of (4) for different engineeringmodels [12]

Model 119875(1199111015840) V

Bruce and Golde (BG) [29] 1 infin

Traveling Current Source (TCS)[36]

1 minus119888

Transmission Line (TL) [1] 1 V119891

Modified Transmission Line withLinear Decay (MTLL) [1]

(1 minus1199111015840

119867119888

) V119891

Modified Transmission Line withExponential Decay (MTLE) [1]

exp(minus1199111015840

120582) V119891

current models Generally the return stroke current modelscan be classified into four groups as follows [12 13]

(i) the gas-dynamic or physical models

(ii) the current distributed models

(iii) the electromagnetic models

(iv) the engineering models

To develop general expressions for electromagnetic fieldsdue to lightning channel the engineering models are appliedin this study In the engineering models the current atdifferent heights along the lightning channel can be expressedby a special function based on the channel base currentand an attenuation height dependent factor Most of theengineering models can be presented by a general equationsuch as (4) [12 13]The internal parameters of (4) for differentengineering current models are listed in Table 2 [1 12] where119888 is considered as the speed of light in free space 120582 as a decayconstant (1sim2 km) and119867119888 as the cloud height with respect tothe ground surface [1]

119868 (1199111015840 119905) = 119868(0 119905 minus

1199111015840

V) times 119875 (119911

1015840) times 119906(119905 minus

1199111015840

V119891) (4)

where 1199111015840 is the temporary charge height along lightning

channel 119868(1199111015840 119905) is current distribution along the lightningchannel at any height 1199111015840 and any time 119905 119868(0 119905) is channel basecurrent 119875(1199111015840) is the attenuation height dependent factor Vis the current-wave propagation velocity V119891 is the upwardpropagating front velocity and 119906 is the Heaviside functiondefended as

119906(119905 minus1199111015840

V119891) =

1 for 119905 ge 1199111015840

V119891

0 for 119905 lt 1199111015840

V119891

(5)

Based on (4) return stroke current at the height of 1199111015840along the lightning channel can be estimated by using channelbase current and the height dependent attenuation factor

X

X

Y

Y

Z

Z

Z

Z

P

P

rarrr

120601

Real channel

H(t)rarrR

H998400(t)

Image channel

The perfect ground I(0 t)

rarrr2

rarrr1

rarr

R998400

1205872 minus 120601

Figure 1 Geometry of the problem

Note that in recent years several models based on (4) andusingmeasured electromagnetic fields and inverse procedurealgorithms were presented [14ndash16] Similarly different cur-rent models can be examined by the comparison betweensimulated electromagnetic fields using current model andmeasured fields at different observation points [1] In thismodel the assumed return stroke velocity is set at a constantvalue that is typically between 1198882 and 21198883 [17]

4 Electromagnetic Fields Associated withLightning Channel

In this section electromagnetic field expressions due to twoexpressed functions in (1) and (2) at an observation pointabove the ground surface are estimated whereas the currentmodel is set on (4) According to the geometry of the situation(see Figure 1) the potential vector () at the observationpoint above ground surface can be obtained from (6) whenMaxwellrsquos equation [3 18] and Lorentz gauge [19] are applied

nabla2 minus 12058301205760

1205972

1205971199052= minus1205830

119869 (6)

where 119869 is the current density 1205830 = 4120587 times 10minus7 Vsdots(Asdotm) and

1205760 = 885 times 10minus12 Fm

Moreover Figure 1 demonstrates that the solution to (6)can be obtained from (7) whereas the infinitesimal currentsource is located along 119911-axis at space in position 997888rarr1199032 from theorigin Also the observation point is located at space and 997888rarr1199031

4 Mathematical Problems in Engineering

from the origin point therefore can be estimated by =

997888rarr1199031 minus

997888rarr1199032

119889 =

1205830 [119868] 119889119871

1015840

412058710038161003816100381610038161003816

997888rarr1199031 minus

997888rarr1199032

10038161003816100381610038161003816

(7)

where 119868 is line current density and 1198891198711015840 is unit vector along

lightning channelLikewise by replacing return stroke current and 1198891199111015840 with

119868 and unit vector respectively in (7) can convert it to (8)

997888997888rarr119889119860119911 =

1205830119894 (1199111015840 119905 minus (119877119888)) 119889119911

1015840

4120587119877 (8)

where

119903 = radic1199092 + 1199102

119877 = radic1199092 + 1199102 + (119911 minus 1199111015840)2

(9)

Further the magnetic flux density and electric fields canbe obtained from potential vector as expressed in (10) and(11) respectively [3 18]

= nabla times (10)

nabla times = minus120597

120597119905= minus

120597 (nabla times )

120597119905 (11)

where is the magnetic flux density and is the electric fieldvector

Therefore based on the general current model in (4)the field components for first current function as presentedin (1) can be obtained from (12) when (8) (10) (11) thetrapezoid and FDTD methods are applied [20ndash22] Notethat the internal parameters are collected from Table 3 Also119911 RS1 (119909 119910 119911 119905119899) = 0

119909 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198651 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS1 (119909 119910 119911 119905119899)

= 119909 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198652 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

= 119910 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS1 (119909 119910 119911 119905119899)

= 119911 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198653 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198653 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(12)

where 119909 RS1 is the magnetic flux density at 119909-axis due toreturn stroke current function from (1) 119910 RS1 is themagneticflux density at 119910-axis due to return stroke current functionfrom (1) 119911 RS1 is the magnetic flux density at 119911-axis due toreturn stroke current function from (1) 119909 RS1 is the electricfield at 119909-axis due to return stroke current function from (1)119910 RS1 is the electric field at119910-axis due to return stroke currentfunction from (1) and 119911 RS1 is the electric field at 119911-axis dueto return stroke current function from (1) Δ119905 is the time step119899 is the number of time steps

119905119899 =

radic1199032 + 1199112

119888+ (119899 minus 1) Δ119905 119899 = 1 2 119899max

119886119898 =

Δℎ119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ119894

119896for others

1198861015840

119898=

Δℎ1015840

119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ1015840

119894

119896for others

(13)

119896 is division factor (ge 2)

Mathematical Problems in Engineering 5

Δℎ119894 =

1205731205942(119888119905119894 minus 119888119905119894minus1) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

+ radic(120573119888119905119894minus1 minus 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 minus 119888119905119894) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

for 119894 = 1

Δℎ1015840

119894=

1205731205942(119888119905119894minus1 minus 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

minus radic(120573119888119905119894minus1 + 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 + 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

for 119894 = 1

ℎ119898119894 =

(119898 minus 1) times Δℎ119894

119896+ ℎ119898=119896+1119894minus1

(119898 minus 1) times Δℎ119894

119896for 119894 = 1

ℎ1015840

119898119894=

(119898 minus 1) times Δℎ1015840

119894

119896+ ℎ

1015840

119898=119896+1119894minus1

(119898 minus 1) times Δℎ1015840

119894

119896for 119894 = 1

(14)

In addition by using the samemethodology that was used forthe first current function the components of electromagneticfield for the second current function as expressed in (2) areproposed by (15) Note that 119911 RS2 (119909 119910 119911 119905119899) = 0

119909 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198654 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS2 (119909 119910 119911 119905119899)

= 119909 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198655 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

= 119910 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS2 (119909 119910 119911 119905119899)

= 119911 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198656 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198656 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(15)

where 119909 RS2 is the magnetic flux density at 119909-axis due toreturn stroke current function from (2) 119910 RS2 is themagneticflux density at 119910-axis due to return stroke current functionfrom (2) 119911 RS2 is the magnetic flux density at 119911-axis due toreturn stroke current function from (2) 119909 RS2 is the electricfield at 119909-axis due to return stroke current function from (2)119910 RS2 is the electric field at119910-axis due to return stroke currentfunction from (2) and 119911 RS2 is the electric field at 119911-axis dueto return stroke current function from (2)

Consequently due to the whole return stroke currentfunctions from Table 1 the electromagnetic fields can beeasily estimated via the proposed field expressions whereasthe total fields due to two part functions are equal to the sumof the electromagnetic fields due to each part independentlyOn top of that the first part current functions in Table 3should be converted to either (1) or (2) format before gettinginto proposed expressions In other words the proposedalgorithm can support the whole current models related to(4) In the same way the components of the electromagneticfields in the Cartesian coordinates can be converted to

6 Mathematical Problems in Engineering

Table3Th

einternalp

aram

eterso

fpropo

sedele

ctromagnetic

fieldse

xpressions

Parameter

Expressio

n

1198651

10minus7119868 0119910

119875(1199111015840)

1205780

[119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1)

119888119877

2]minus[119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1)

1198773

]

1198652

119868 0119875(1199111015840)

412058712057601205780

(119911minus1199111015840)119909minus3(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

119888 11205722exp(minus120572119860

1)minus119888 21205742exp(minus120574119860

1)

1198882119877

3+

3(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

1198653

119868 0119875(1199111015840)

412058712057601205780

3(1199092+1199102)(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

2(minus119888 1120572exp(minus120572119860

1)+119888 2120574exp(minus120574119860

1))

119888119877

2+

(1199092+1199102)(minus119888 11205722exp(minus120572119860

1)+119888 21205742exp(minus120574119860

1))

1198882119877

3

minus3(1199092+1199102)(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

+2(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198773

1198654

10minus7

119868 01119875(119911

1015840)119910

1198602

12057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899

119888119905 2119877

2minus

(119860

1119905 1)119899

1198773

minus119899(119860

1119905 1)119899minus1

119888119905 1119877

2+

119899(119860

1119905 1)2119899minus1

119888119905 1119877

2[(119860

1119905 1)119899+1]

1198655

119868 01119875(119911

1015840)119909

(119911minus1199111015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899(311988821199052 2

minus3119888119905 2119877

+119877

2)

11988821199052 2119877

5+

21198992(119860

1119905 1)3119899minus2+119899(119899

minus1)(119860

1119905 1)119899minus2[(119860

1119905 1)119899+1]2+(minus

31198992+119899)(119860

1119905 1)2119899minus2[(119860

1119905 1)119899+1]

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

+3119899([(119860

1119905 1)119899+1](119860

1119905 1)119899minus1minus(119860

1119905 1)2119899minus1)

119888119905 1119877

4[(119860

1119905 1)119899+1]

+2119899((119860

1119905 1)2119899minus1minus[(119860

1119905 1)119899+1](119860

1119905 1)119899minus1)

1198882119905 1119905 2119877

3[(119860

1119905 1)119899+1]

1198656

119868 01119875(119911

1015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(1199092+1199102)[(119860

1119905 1)119899(minus31198882119905 11199052 2

+3119888119905 1119905 2119877

minus119905 1119877

2)+119899(119860

1119905 1)119899minus1(minus31198881199052 2119877

+2119905 2119877

2)]

1198882119905 11199052 2119877

5+

2(119860

1119905 1)119899(119888119905 1119905 2

minus119905 1119877)+2119899(119860

1119905 1)119899minus1119905 2119877

119888119905 1119905 2119877

3minus

119899(119899

minus1)(1199092+1199102)(119860

1119905 1)119899minus2

11988821199052 1119877

3

+

(1199092+1199102)[21198992119905 2119877(119860

1119905 1)2119899minus2+3119899119888119905 1119905 2

(119860

1119905 1)2119899minus1+119899(119899

minus1)119905 2119877(119860

1119905 1)2119899minus2minus2119899119905 1119877(119860

1119905 1)2119899minus1]minus2119899119888119905 1119905 2119877

2(119860

1119905 1)2119899minus1

11988821199052 1119905 2119877

4[(119860

1119905 1)119899+1]

minus21198992(1199092+1199102)(119860

1119905 1)3119899minus2

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

119877=

radic1199092+1199102+(119911

minus1199111015840)2

1198601=

119905minus

119877 119888minus

1003816 1003816 1003816 1003816 100381611991110158401003816 1003816 1003816 1003816 1003816

119907

1198602=exp(

(119877119888)minus119905+(1003816 1003816 1003816 1003816 1003816119911

10158401003816 1003816 1003816 1003816 1003816119907)

119905 2)

Mathematical Problems in Engineering 7

cylindrical domain using (16) and (17) for the magnetic fluxdensity and the electric field components respectively

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(16)

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(17)

where 120601 = arc cos (119910radic1199092 + 1199102) 119903 (119903 119911 120601 119905) 119903 (119903 119911 120601 119905)

is the magnetic flux densityelectric field at 119903-direction(horizontal) 120593 (119903 119911 120601 119905) 120593 (119903 119911 120601 119905) is the magnetic fluxdensityelectric field at 120601-direction and 119911 (119903 119911 120601 119905)

119911 (119903 119911 120601 119905) is the magnetic flux densityelectric field at119911-direction (vertical)

5 Result and Discussion

This section deals with the validation of the electromagneticfields due to lightning channel using some channel basecurrent functions According to the proposed method thewidely used current functions can be converted into sum ofindividual parts using (1) and (2) Therefore the electromag-netic fields due to each part can be calculated by (12) and(15) Then the values of total electromagnetic field are equalto the sum of individual components of electromagneticfield When the MTLE model is applied as current modelthe simulated magnetic flux density and the vertical electricfield caused by a sum of two Heidler function as a channelbase current function at an observation point with 46 kmdistance from lightning channel are depicted in Figures 2and 3 respectively The measured magnetic flux density andvertical electric field with similar primary conditions as theones in Figures 2 and 3 are demonstrated in Figures 4 and 5respectively Therefore comparisons between the measuredand simulated fields for magnetic flux density and verticaleclectic field are tabulated in Tables 4 and 5 respectively Itshould be noted that the current parameters are 11989401 = 17 kA11989402 = 8 kA Γ11 = 04 times 10

minus6 Γ12 = 4 times 10minus6 Γ21 = 4 times 10

minus6Γ22 = 50 times 10

minus6 1198991 = 2 1198992 = 2 V = 1 times 108 and 120582 = 1500

[23]

10 15 20 25 30 35 40 45 50 55 60 650

05

1

15

2

25

3times10minus7

Time (120583s)

Mag

netic

flux

den

sity

(Wb

m2)

Figure 2 The simulated magnetic flux density using proposedmethod (119911 = 10m 119903 = 46 km)

10 15 20 25 30 35 40 45 50 55 60 650

102030405060708090

Vert

ical

elec

tric

fiel

d (V

m)

Time (120583s)

Figure 3 The simulated vertical electric field using proposedmethod (119911 = 10m 119903 = 46 km)

0 10 20 30 40 50 60 700

05

1

15

2

25times10minus7

Time (120583s)

B120601

(Wb

m2)

Figure 4 The measured magnetic flux density for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

Additionally the sum of two Heidler functions as a chan-nel base current function can be expressed by the sum of twogeneral current functions based on (2) Therefore the totalelectromagnetic fields are estimated by using the proposedmethod and considering each current part separately Thecomparison between simulated and measured fields showsthe overall good agreement with experimental measurementfor proposed field expressions Similarly the simulated fields

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

from the origin point therefore can be estimated by =

997888rarr1199031 minus

997888rarr1199032

119889 =

1205830 [119868] 119889119871

1015840

412058710038161003816100381610038161003816

997888rarr1199031 minus

997888rarr1199032

10038161003816100381610038161003816

(7)

where 119868 is line current density and 1198891198711015840 is unit vector along

lightning channelLikewise by replacing return stroke current and 1198891199111015840 with

119868 and unit vector respectively in (7) can convert it to (8)

997888997888rarr119889119860119911 =

1205830119894 (1199111015840 119905 minus (119877119888)) 119889119911

1015840

4120587119877 (8)

where

119903 = radic1199092 + 1199102

119877 = radic1199092 + 1199102 + (119911 minus 1199111015840)2

(9)

Further the magnetic flux density and electric fields canbe obtained from potential vector as expressed in (10) and(11) respectively [3 18]

= nabla times (10)

nabla times = minus120597

120597119905= minus

120597 (nabla times )

120597119905 (11)

where is the magnetic flux density and is the electric fieldvector

Therefore based on the general current model in (4)the field components for first current function as presentedin (1) can be obtained from (12) when (8) (10) (11) thetrapezoid and FDTD methods are applied [20ndash22] Notethat the internal parameters are collected from Table 3 Also119911 RS1 (119909 119910 119911 119905119899) = 0

119909 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198651 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198651 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS1 (119909 119910 119911 119905119899)

= 119909 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198652 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS1 (119909 119910 119911 119905119899)

= 119910 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198652 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS1 (119909 119910 119911 119905119899)

= 119911 RS1 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198653 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198653 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(12)

where 119909 RS1 is the magnetic flux density at 119909-axis due toreturn stroke current function from (1) 119910 RS1 is themagneticflux density at 119910-axis due to return stroke current functionfrom (1) 119911 RS1 is the magnetic flux density at 119911-axis due toreturn stroke current function from (1) 119909 RS1 is the electricfield at 119909-axis due to return stroke current function from (1)119910 RS1 is the electric field at119910-axis due to return stroke currentfunction from (1) and 119911 RS1 is the electric field at 119911-axis dueto return stroke current function from (1) Δ119905 is the time step119899 is the number of time steps

119905119899 =

radic1199032 + 1199112

119888+ (119899 minus 1) Δ119905 119899 = 1 2 119899max

119886119898 =

Δℎ119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ119894

119896for others

1198861015840

119898=

Δℎ1015840

119894

2 times 119896for 119898 = 1 119898 = 119896 + 1

Δℎ1015840

119894

119896for others

(13)

119896 is division factor (ge 2)

Mathematical Problems in Engineering 5

Δℎ119894 =

1205731205942(119888119905119894 minus 119888119905119894minus1) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

+ radic(120573119888119905119894minus1 minus 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 minus 119888119905119894) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

for 119894 = 1

Δℎ1015840

119894=

1205731205942(119888119905119894minus1 minus 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

minus radic(120573119888119905119894minus1 + 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 + 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

for 119894 = 1

ℎ119898119894 =

(119898 minus 1) times Δℎ119894

119896+ ℎ119898=119896+1119894minus1

(119898 minus 1) times Δℎ119894

119896for 119894 = 1

ℎ1015840

119898119894=

(119898 minus 1) times Δℎ1015840

119894

119896+ ℎ

1015840

119898=119896+1119894minus1

(119898 minus 1) times Δℎ1015840

119894

119896for 119894 = 1

(14)

In addition by using the samemethodology that was used forthe first current function the components of electromagneticfield for the second current function as expressed in (2) areproposed by (15) Note that 119911 RS2 (119909 119910 119911 119905119899) = 0

119909 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198654 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS2 (119909 119910 119911 119905119899)

= 119909 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198655 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

= 119910 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS2 (119909 119910 119911 119905119899)

= 119911 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198656 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198656 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(15)

where 119909 RS2 is the magnetic flux density at 119909-axis due toreturn stroke current function from (2) 119910 RS2 is themagneticflux density at 119910-axis due to return stroke current functionfrom (2) 119911 RS2 is the magnetic flux density at 119911-axis due toreturn stroke current function from (2) 119909 RS2 is the electricfield at 119909-axis due to return stroke current function from (2)119910 RS2 is the electric field at119910-axis due to return stroke currentfunction from (2) and 119911 RS2 is the electric field at 119911-axis dueto return stroke current function from (2)

Consequently due to the whole return stroke currentfunctions from Table 1 the electromagnetic fields can beeasily estimated via the proposed field expressions whereasthe total fields due to two part functions are equal to the sumof the electromagnetic fields due to each part independentlyOn top of that the first part current functions in Table 3should be converted to either (1) or (2) format before gettinginto proposed expressions In other words the proposedalgorithm can support the whole current models related to(4) In the same way the components of the electromagneticfields in the Cartesian coordinates can be converted to

6 Mathematical Problems in Engineering

Table3Th

einternalp

aram

eterso

fpropo

sedele

ctromagnetic

fieldse

xpressions

Parameter

Expressio

n

1198651

10minus7119868 0119910

119875(1199111015840)

1205780

[119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1)

119888119877

2]minus[119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1)

1198773

]

1198652

119868 0119875(1199111015840)

412058712057601205780

(119911minus1199111015840)119909minus3(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

119888 11205722exp(minus120572119860

1)minus119888 21205742exp(minus120574119860

1)

1198882119877

3+

3(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

1198653

119868 0119875(1199111015840)

412058712057601205780

3(1199092+1199102)(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

2(minus119888 1120572exp(minus120572119860

1)+119888 2120574exp(minus120574119860

1))

119888119877

2+

(1199092+1199102)(minus119888 11205722exp(minus120572119860

1)+119888 21205742exp(minus120574119860

1))

1198882119877

3

minus3(1199092+1199102)(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

+2(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198773

1198654

10minus7

119868 01119875(119911

1015840)119910

1198602

12057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899

119888119905 2119877

2minus

(119860

1119905 1)119899

1198773

minus119899(119860

1119905 1)119899minus1

119888119905 1119877

2+

119899(119860

1119905 1)2119899minus1

119888119905 1119877

2[(119860

1119905 1)119899+1]

1198655

119868 01119875(119911

1015840)119909

(119911minus1199111015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899(311988821199052 2

minus3119888119905 2119877

+119877

2)

11988821199052 2119877

5+

21198992(119860

1119905 1)3119899minus2+119899(119899

minus1)(119860

1119905 1)119899minus2[(119860

1119905 1)119899+1]2+(minus

31198992+119899)(119860

1119905 1)2119899minus2[(119860

1119905 1)119899+1]

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

+3119899([(119860

1119905 1)119899+1](119860

1119905 1)119899minus1minus(119860

1119905 1)2119899minus1)

119888119905 1119877

4[(119860

1119905 1)119899+1]

+2119899((119860

1119905 1)2119899minus1minus[(119860

1119905 1)119899+1](119860

1119905 1)119899minus1)

1198882119905 1119905 2119877

3[(119860

1119905 1)119899+1]

1198656

119868 01119875(119911

1015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(1199092+1199102)[(119860

1119905 1)119899(minus31198882119905 11199052 2

+3119888119905 1119905 2119877

minus119905 1119877

2)+119899(119860

1119905 1)119899minus1(minus31198881199052 2119877

+2119905 2119877

2)]

1198882119905 11199052 2119877

5+

2(119860

1119905 1)119899(119888119905 1119905 2

minus119905 1119877)+2119899(119860

1119905 1)119899minus1119905 2119877

119888119905 1119905 2119877

3minus

119899(119899

minus1)(1199092+1199102)(119860

1119905 1)119899minus2

11988821199052 1119877

3

+

(1199092+1199102)[21198992119905 2119877(119860

1119905 1)2119899minus2+3119899119888119905 1119905 2

(119860

1119905 1)2119899minus1+119899(119899

minus1)119905 2119877(119860

1119905 1)2119899minus2minus2119899119905 1119877(119860

1119905 1)2119899minus1]minus2119899119888119905 1119905 2119877

2(119860

1119905 1)2119899minus1

11988821199052 1119905 2119877

4[(119860

1119905 1)119899+1]

minus21198992(1199092+1199102)(119860

1119905 1)3119899minus2

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

119877=

radic1199092+1199102+(119911

minus1199111015840)2

1198601=

119905minus

119877 119888minus

1003816 1003816 1003816 1003816 100381611991110158401003816 1003816 1003816 1003816 1003816

119907

1198602=exp(

(119877119888)minus119905+(1003816 1003816 1003816 1003816 1003816119911

10158401003816 1003816 1003816 1003816 1003816119907)

119905 2)

Mathematical Problems in Engineering 7

cylindrical domain using (16) and (17) for the magnetic fluxdensity and the electric field components respectively

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(16)

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(17)

where 120601 = arc cos (119910radic1199092 + 1199102) 119903 (119903 119911 120601 119905) 119903 (119903 119911 120601 119905)

is the magnetic flux densityelectric field at 119903-direction(horizontal) 120593 (119903 119911 120601 119905) 120593 (119903 119911 120601 119905) is the magnetic fluxdensityelectric field at 120601-direction and 119911 (119903 119911 120601 119905)

119911 (119903 119911 120601 119905) is the magnetic flux densityelectric field at119911-direction (vertical)

5 Result and Discussion

This section deals with the validation of the electromagneticfields due to lightning channel using some channel basecurrent functions According to the proposed method thewidely used current functions can be converted into sum ofindividual parts using (1) and (2) Therefore the electromag-netic fields due to each part can be calculated by (12) and(15) Then the values of total electromagnetic field are equalto the sum of individual components of electromagneticfield When the MTLE model is applied as current modelthe simulated magnetic flux density and the vertical electricfield caused by a sum of two Heidler function as a channelbase current function at an observation point with 46 kmdistance from lightning channel are depicted in Figures 2and 3 respectively The measured magnetic flux density andvertical electric field with similar primary conditions as theones in Figures 2 and 3 are demonstrated in Figures 4 and 5respectively Therefore comparisons between the measuredand simulated fields for magnetic flux density and verticaleclectic field are tabulated in Tables 4 and 5 respectively Itshould be noted that the current parameters are 11989401 = 17 kA11989402 = 8 kA Γ11 = 04 times 10

minus6 Γ12 = 4 times 10minus6 Γ21 = 4 times 10

minus6Γ22 = 50 times 10

minus6 1198991 = 2 1198992 = 2 V = 1 times 108 and 120582 = 1500

[23]

10 15 20 25 30 35 40 45 50 55 60 650

05

1

15

2

25

3times10minus7

Time (120583s)

Mag

netic

flux

den

sity

(Wb

m2)

Figure 2 The simulated magnetic flux density using proposedmethod (119911 = 10m 119903 = 46 km)

10 15 20 25 30 35 40 45 50 55 60 650

102030405060708090

Vert

ical

elec

tric

fiel

d (V

m)

Time (120583s)

Figure 3 The simulated vertical electric field using proposedmethod (119911 = 10m 119903 = 46 km)

0 10 20 30 40 50 60 700

05

1

15

2

25times10minus7

Time (120583s)

B120601

(Wb

m2)

Figure 4 The measured magnetic flux density for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

Additionally the sum of two Heidler functions as a chan-nel base current function can be expressed by the sum of twogeneral current functions based on (2) Therefore the totalelectromagnetic fields are estimated by using the proposedmethod and considering each current part separately Thecomparison between simulated and measured fields showsthe overall good agreement with experimental measurementfor proposed field expressions Similarly the simulated fields

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Δℎ119894 =

1205731205942(119888119905119894 minus 119888119905119894minus1) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

+ radic(120573119888119905119894minus1 minus 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 minus 119888119905119894) minus

radic(120573119888119905119894 minus 119911)2+ (

119903

120594)

2

for 119894 = 1

Δℎ1015840

119894=

1205731205942(119888119905119894minus1 minus 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

minus radic(120573119888119905119894minus1 + 119911)2+ (

119903

120594)

2

1205731205942minus (120573119911 + 119888119905119894) +

radic(120573119888119905119894 + 119911)2+ (

119903

120594)

2

for 119894 = 1

ℎ119898119894 =

(119898 minus 1) times Δℎ119894

119896+ ℎ119898=119896+1119894minus1

(119898 minus 1) times Δℎ119894

119896for 119894 = 1

ℎ1015840

119898119894=

(119898 minus 1) times Δℎ1015840

119894

119896+ ℎ

1015840

119898=119896+1119894minus1

(119898 minus 1) times Δℎ1015840

119894

119896for 119894 = 1

(14)

In addition by using the samemethodology that was used forthe first current function the components of electromagneticfield for the second current function as expressed in (2) areproposed by (15) Note that 119911 RS2 (119909 119910 119911 119905119899) = 0

119909 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198654 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

=

119899

sum

119894=1

119896+1

sum

119898=1

(minus119909

119910) 1198861198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198654 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119909 RS2 (119909 119910 119911 119905119899)

= 119909 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198655 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119910 RS2 (119909 119910 119911 119905119899)

= 119910 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

(119910

119909) 1198861198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ119898119894)

minus 1198861015840

1198981198655 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

119911 RS2 (119909 119910 119911 119905119899)

= 119911 RS2 (119909 119910 119911 119905119899minus1) + Δ119905

times

119899

sum

119894=1

119896+1

sum

119898=1

1198861198981198656 (119909 119910 119911 119905 = 119905119899 1199111015840= ℎ119898119894)

minus 1198861015840

1198981198656 (119909 119910 119911 119905 = 119905119899 119911

1015840= ℎ

1015840

119898119894)

(15)

where 119909 RS2 is the magnetic flux density at 119909-axis due toreturn stroke current function from (2) 119910 RS2 is themagneticflux density at 119910-axis due to return stroke current functionfrom (2) 119911 RS2 is the magnetic flux density at 119911-axis due toreturn stroke current function from (2) 119909 RS2 is the electricfield at 119909-axis due to return stroke current function from (2)119910 RS2 is the electric field at119910-axis due to return stroke currentfunction from (2) and 119911 RS2 is the electric field at 119911-axis dueto return stroke current function from (2)

Consequently due to the whole return stroke currentfunctions from Table 1 the electromagnetic fields can beeasily estimated via the proposed field expressions whereasthe total fields due to two part functions are equal to the sumof the electromagnetic fields due to each part independentlyOn top of that the first part current functions in Table 3should be converted to either (1) or (2) format before gettinginto proposed expressions In other words the proposedalgorithm can support the whole current models related to(4) In the same way the components of the electromagneticfields in the Cartesian coordinates can be converted to

6 Mathematical Problems in Engineering

Table3Th

einternalp

aram

eterso

fpropo

sedele

ctromagnetic

fieldse

xpressions

Parameter

Expressio

n

1198651

10minus7119868 0119910

119875(1199111015840)

1205780

[119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1)

119888119877

2]minus[119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1)

1198773

]

1198652

119868 0119875(1199111015840)

412058712057601205780

(119911minus1199111015840)119909minus3(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

119888 11205722exp(minus120572119860

1)minus119888 21205742exp(minus120574119860

1)

1198882119877

3+

3(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

1198653

119868 0119875(1199111015840)

412058712057601205780

3(1199092+1199102)(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

2(minus119888 1120572exp(minus120572119860

1)+119888 2120574exp(minus120574119860

1))

119888119877

2+

(1199092+1199102)(minus119888 11205722exp(minus120572119860

1)+119888 21205742exp(minus120574119860

1))

1198882119877

3

minus3(1199092+1199102)(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

+2(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198773

1198654

10minus7

119868 01119875(119911

1015840)119910

1198602

12057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899

119888119905 2119877

2minus

(119860

1119905 1)119899

1198773

minus119899(119860

1119905 1)119899minus1

119888119905 1119877

2+

119899(119860

1119905 1)2119899minus1

119888119905 1119877

2[(119860

1119905 1)119899+1]

1198655

119868 01119875(119911

1015840)119909

(119911minus1199111015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899(311988821199052 2

minus3119888119905 2119877

+119877

2)

11988821199052 2119877

5+

21198992(119860

1119905 1)3119899minus2+119899(119899

minus1)(119860

1119905 1)119899minus2[(119860

1119905 1)119899+1]2+(minus

31198992+119899)(119860

1119905 1)2119899minus2[(119860

1119905 1)119899+1]

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

+3119899([(119860

1119905 1)119899+1](119860

1119905 1)119899minus1minus(119860

1119905 1)2119899minus1)

119888119905 1119877

4[(119860

1119905 1)119899+1]

+2119899((119860

1119905 1)2119899minus1minus[(119860

1119905 1)119899+1](119860

1119905 1)119899minus1)

1198882119905 1119905 2119877

3[(119860

1119905 1)119899+1]

1198656

119868 01119875(119911

1015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(1199092+1199102)[(119860

1119905 1)119899(minus31198882119905 11199052 2

+3119888119905 1119905 2119877

minus119905 1119877

2)+119899(119860

1119905 1)119899minus1(minus31198881199052 2119877

+2119905 2119877

2)]

1198882119905 11199052 2119877

5+

2(119860

1119905 1)119899(119888119905 1119905 2

minus119905 1119877)+2119899(119860

1119905 1)119899minus1119905 2119877

119888119905 1119905 2119877

3minus

119899(119899

minus1)(1199092+1199102)(119860

1119905 1)119899minus2

11988821199052 1119877

3

+

(1199092+1199102)[21198992119905 2119877(119860

1119905 1)2119899minus2+3119899119888119905 1119905 2

(119860

1119905 1)2119899minus1+119899(119899

minus1)119905 2119877(119860

1119905 1)2119899minus2minus2119899119905 1119877(119860

1119905 1)2119899minus1]minus2119899119888119905 1119905 2119877

2(119860

1119905 1)2119899minus1

11988821199052 1119905 2119877

4[(119860

1119905 1)119899+1]

minus21198992(1199092+1199102)(119860

1119905 1)3119899minus2

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

119877=

radic1199092+1199102+(119911

minus1199111015840)2

1198601=

119905minus

119877 119888minus

1003816 1003816 1003816 1003816 100381611991110158401003816 1003816 1003816 1003816 1003816

119907

1198602=exp(

(119877119888)minus119905+(1003816 1003816 1003816 1003816 1003816119911

10158401003816 1003816 1003816 1003816 1003816119907)

119905 2)

Mathematical Problems in Engineering 7

cylindrical domain using (16) and (17) for the magnetic fluxdensity and the electric field components respectively

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(16)

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(17)

where 120601 = arc cos (119910radic1199092 + 1199102) 119903 (119903 119911 120601 119905) 119903 (119903 119911 120601 119905)

is the magnetic flux densityelectric field at 119903-direction(horizontal) 120593 (119903 119911 120601 119905) 120593 (119903 119911 120601 119905) is the magnetic fluxdensityelectric field at 120601-direction and 119911 (119903 119911 120601 119905)

119911 (119903 119911 120601 119905) is the magnetic flux densityelectric field at119911-direction (vertical)

5 Result and Discussion

This section deals with the validation of the electromagneticfields due to lightning channel using some channel basecurrent functions According to the proposed method thewidely used current functions can be converted into sum ofindividual parts using (1) and (2) Therefore the electromag-netic fields due to each part can be calculated by (12) and(15) Then the values of total electromagnetic field are equalto the sum of individual components of electromagneticfield When the MTLE model is applied as current modelthe simulated magnetic flux density and the vertical electricfield caused by a sum of two Heidler function as a channelbase current function at an observation point with 46 kmdistance from lightning channel are depicted in Figures 2and 3 respectively The measured magnetic flux density andvertical electric field with similar primary conditions as theones in Figures 2 and 3 are demonstrated in Figures 4 and 5respectively Therefore comparisons between the measuredand simulated fields for magnetic flux density and verticaleclectic field are tabulated in Tables 4 and 5 respectively Itshould be noted that the current parameters are 11989401 = 17 kA11989402 = 8 kA Γ11 = 04 times 10

minus6 Γ12 = 4 times 10minus6 Γ21 = 4 times 10

minus6Γ22 = 50 times 10

minus6 1198991 = 2 1198992 = 2 V = 1 times 108 and 120582 = 1500

[23]

10 15 20 25 30 35 40 45 50 55 60 650

05

1

15

2

25

3times10minus7

Time (120583s)

Mag

netic

flux

den

sity

(Wb

m2)

Figure 2 The simulated magnetic flux density using proposedmethod (119911 = 10m 119903 = 46 km)

10 15 20 25 30 35 40 45 50 55 60 650

102030405060708090

Vert

ical

elec

tric

fiel

d (V

m)

Time (120583s)

Figure 3 The simulated vertical electric field using proposedmethod (119911 = 10m 119903 = 46 km)

0 10 20 30 40 50 60 700

05

1

15

2

25times10minus7

Time (120583s)

B120601

(Wb

m2)

Figure 4 The measured magnetic flux density for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

Additionally the sum of two Heidler functions as a chan-nel base current function can be expressed by the sum of twogeneral current functions based on (2) Therefore the totalelectromagnetic fields are estimated by using the proposedmethod and considering each current part separately Thecomparison between simulated and measured fields showsthe overall good agreement with experimental measurementfor proposed field expressions Similarly the simulated fields

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table3Th

einternalp

aram

eterso

fpropo

sedele

ctromagnetic

fieldse

xpressions

Parameter

Expressio

n

1198651

10minus7119868 0119910

119875(1199111015840)

1205780

[119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1)

119888119877

2]minus[119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1)

1198773

]

1198652

119868 0119875(1199111015840)

412058712057601205780

(119911minus1199111015840)119909minus3(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

119888 11205722exp(minus120572119860

1)minus119888 21205742exp(minus120574119860

1)

1198882119877

3+

3(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

1198653

119868 0119875(1199111015840)

412058712057601205780

3(1199092+1199102)(119888 1120572exp(minus120572119860

1)minus119888 2120574exp(minus120574119860

1))

119888119877

4+

2(minus119888 1120572exp(minus120572119860

1)+119888 2120574exp(minus120574119860

1))

119888119877

2+

(1199092+1199102)(minus119888 11205722exp(minus120572119860

1)+119888 21205742exp(minus120574119860

1))

1198882119877

3

minus3(1199092+1199102)(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198775

+2(119888 1exp(minus120572119860

1)minus119888 2exp(minus120574119860

1))

1198773

1198654

10minus7

119868 01119875(119911

1015840)119910

1198602

12057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899

119888119905 2119877

2minus

(119860

1119905 1)119899

1198773

minus119899(119860

1119905 1)119899minus1

119888119905 1119877

2+

119899(119860

1119905 1)2119899minus1

119888119905 1119877

2[(119860

1119905 1)119899+1]

1198655

119868 01119875(119911

1015840)119909

(119911minus1199111015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(119860

1119905 1)119899(311988821199052 2

minus3119888119905 2119877

+119877

2)

11988821199052 2119877

5+

21198992(119860

1119905 1)3119899minus2+119899(119899

minus1)(119860

1119905 1)119899minus2[(119860

1119905 1)119899+1]2+(minus

31198992+119899)(119860

1119905 1)2119899minus2[(119860

1119905 1)119899+1]

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

+3119899([(119860

1119905 1)119899+1](119860

1119905 1)119899minus1minus(119860

1119905 1)2119899minus1)

119888119905 1119877

4[(119860

1119905 1)119899+1]

+2119899((119860

1119905 1)2119899minus1minus[(119860

1119905 1)119899+1](119860

1119905 1)119899minus1)

1198882119905 1119905 2119877

3[(119860

1119905 1)119899+1]

1198656

119868 01119875(119911

1015840)119860

2

4120587120576012057801[(119860

1119905 1)119899+1]

(1199092+1199102)[(119860

1119905 1)119899(minus31198882119905 11199052 2

+3119888119905 1119905 2119877

minus119905 1119877

2)+119899(119860

1119905 1)119899minus1(minus31198881199052 2119877

+2119905 2119877

2)]

1198882119905 11199052 2119877

5+

2(119860

1119905 1)119899(119888119905 1119905 2

minus119905 1119877)+2119899(119860

1119905 1)119899minus1119905 2119877

119888119905 1119905 2119877

3minus

119899(119899

minus1)(1199092+1199102)(119860

1119905 1)119899minus2

11988821199052 1119877

3

+

(1199092+1199102)[21198992119905 2119877(119860

1119905 1)2119899minus2+3119899119888119905 1119905 2

(119860

1119905 1)2119899minus1+119899(119899

minus1)119905 2119877(119860

1119905 1)2119899minus2minus2119899119905 1119877(119860

1119905 1)2119899minus1]minus2119899119888119905 1119905 2119877

2(119860

1119905 1)2119899minus1

11988821199052 1119905 2119877

4[(119860

1119905 1)119899+1]

minus21198992(1199092+1199102)(119860

1119905 1)3119899minus2

11988821199052 1119877

3[(119860

1119905 1)119899+1]2

119877=

radic1199092+1199102+(119911

minus1199111015840)2

1198601=

119905minus

119877 119888minus

1003816 1003816 1003816 1003816 100381611991110158401003816 1003816 1003816 1003816 1003816

119907

1198602=exp(

(119877119888)minus119905+(1003816 1003816 1003816 1003816 1003816119911

10158401003816 1003816 1003816 1003816 1003816119907)

119905 2)

Mathematical Problems in Engineering 7

cylindrical domain using (16) and (17) for the magnetic fluxdensity and the electric field components respectively

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(16)

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(17)

where 120601 = arc cos (119910radic1199092 + 1199102) 119903 (119903 119911 120601 119905) 119903 (119903 119911 120601 119905)

is the magnetic flux densityelectric field at 119903-direction(horizontal) 120593 (119903 119911 120601 119905) 120593 (119903 119911 120601 119905) is the magnetic fluxdensityelectric field at 120601-direction and 119911 (119903 119911 120601 119905)

119911 (119903 119911 120601 119905) is the magnetic flux densityelectric field at119911-direction (vertical)

5 Result and Discussion

This section deals with the validation of the electromagneticfields due to lightning channel using some channel basecurrent functions According to the proposed method thewidely used current functions can be converted into sum ofindividual parts using (1) and (2) Therefore the electromag-netic fields due to each part can be calculated by (12) and(15) Then the values of total electromagnetic field are equalto the sum of individual components of electromagneticfield When the MTLE model is applied as current modelthe simulated magnetic flux density and the vertical electricfield caused by a sum of two Heidler function as a channelbase current function at an observation point with 46 kmdistance from lightning channel are depicted in Figures 2and 3 respectively The measured magnetic flux density andvertical electric field with similar primary conditions as theones in Figures 2 and 3 are demonstrated in Figures 4 and 5respectively Therefore comparisons between the measuredand simulated fields for magnetic flux density and verticaleclectic field are tabulated in Tables 4 and 5 respectively Itshould be noted that the current parameters are 11989401 = 17 kA11989402 = 8 kA Γ11 = 04 times 10

minus6 Γ12 = 4 times 10minus6 Γ21 = 4 times 10

minus6Γ22 = 50 times 10

minus6 1198991 = 2 1198992 = 2 V = 1 times 108 and 120582 = 1500

[23]

10 15 20 25 30 35 40 45 50 55 60 650

05

1

15

2

25

3times10minus7

Time (120583s)

Mag

netic

flux

den

sity

(Wb

m2)

Figure 2 The simulated magnetic flux density using proposedmethod (119911 = 10m 119903 = 46 km)

10 15 20 25 30 35 40 45 50 55 60 650

102030405060708090

Vert

ical

elec

tric

fiel

d (V

m)

Time (120583s)

Figure 3 The simulated vertical electric field using proposedmethod (119911 = 10m 119903 = 46 km)

0 10 20 30 40 50 60 700

05

1

15

2

25times10minus7

Time (120583s)

B120601

(Wb

m2)

Figure 4 The measured magnetic flux density for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

Additionally the sum of two Heidler functions as a chan-nel base current function can be expressed by the sum of twogeneral current functions based on (2) Therefore the totalelectromagnetic fields are estimated by using the proposedmethod and considering each current part separately Thecomparison between simulated and measured fields showsthe overall good agreement with experimental measurementfor proposed field expressions Similarly the simulated fields

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

cylindrical domain using (16) and (17) for the magnetic fluxdensity and the electric field components respectively

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(16)

[[

[

119903 (119903 119911 120601 119905119899)

120593 (119903 119911 120601 119905119899)

119911 (119903 119911 120601 119905119899)

]]

]

=

[[[[[[

[

cos(1205872minus 120601) sin(120587

2minus 120601) 0

minus sin(1205872minus 120601) cos(120587

2minus 120601) 0

0 0 1

]]]]]]

]

times[[

[

119909 (119909 119910 119911 119905119899)

119910 (119909 119910 119911 119905119899)

119911 (119909 119910 119911 119905119899)

]]

]

(17)

where 120601 = arc cos (119910radic1199092 + 1199102) 119903 (119903 119911 120601 119905) 119903 (119903 119911 120601 119905)

is the magnetic flux densityelectric field at 119903-direction(horizontal) 120593 (119903 119911 120601 119905) 120593 (119903 119911 120601 119905) is the magnetic fluxdensityelectric field at 120601-direction and 119911 (119903 119911 120601 119905)

119911 (119903 119911 120601 119905) is the magnetic flux densityelectric field at119911-direction (vertical)

5 Result and Discussion

This section deals with the validation of the electromagneticfields due to lightning channel using some channel basecurrent functions According to the proposed method thewidely used current functions can be converted into sum ofindividual parts using (1) and (2) Therefore the electromag-netic fields due to each part can be calculated by (12) and(15) Then the values of total electromagnetic field are equalto the sum of individual components of electromagneticfield When the MTLE model is applied as current modelthe simulated magnetic flux density and the vertical electricfield caused by a sum of two Heidler function as a channelbase current function at an observation point with 46 kmdistance from lightning channel are depicted in Figures 2and 3 respectively The measured magnetic flux density andvertical electric field with similar primary conditions as theones in Figures 2 and 3 are demonstrated in Figures 4 and 5respectively Therefore comparisons between the measuredand simulated fields for magnetic flux density and verticaleclectic field are tabulated in Tables 4 and 5 respectively Itshould be noted that the current parameters are 11989401 = 17 kA11989402 = 8 kA Γ11 = 04 times 10

minus6 Γ12 = 4 times 10minus6 Γ21 = 4 times 10

minus6Γ22 = 50 times 10

minus6 1198991 = 2 1198992 = 2 V = 1 times 108 and 120582 = 1500

[23]

10 15 20 25 30 35 40 45 50 55 60 650

05

1

15

2

25

3times10minus7

Time (120583s)

Mag

netic

flux

den

sity

(Wb

m2)

Figure 2 The simulated magnetic flux density using proposedmethod (119911 = 10m 119903 = 46 km)

10 15 20 25 30 35 40 45 50 55 60 650

102030405060708090

Vert

ical

elec

tric

fiel

d (V

m)

Time (120583s)

Figure 3 The simulated vertical electric field using proposedmethod (119911 = 10m 119903 = 46 km)

0 10 20 30 40 50 60 700

05

1

15

2

25times10minus7

Time (120583s)

B120601

(Wb

m2)

Figure 4 The measured magnetic flux density for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

Additionally the sum of two Heidler functions as a chan-nel base current function can be expressed by the sum of twogeneral current functions based on (2) Therefore the totalelectromagnetic fields are estimated by using the proposedmethod and considering each current part separately Thecomparison between simulated and measured fields showsthe overall good agreement with experimental measurementfor proposed field expressions Similarly the simulated fields

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Numerical comparison between the simulated magnetic flux density and the corresponding measured fields from Figures 2 and 4after subtracting initial decay time

Time (120583s) 2 5 8 10 15119861120593 (measured) Wbm2

275 times 10minus7

2 times 10minus7

17 times 10minus7

15 times 10minus7

13 times 10minus7

119861120593 (simulated) Wbm227 times 10

minus7194 times 10

minus716 times 10

minus714 times 10

minus7125 times 10

minus7

Percentage difference 2 3 6 6 4

Table 5 Numerical comparison between the simulated verticalelectric field and the corresponding measured fields from Figures3 and 5 after subtracting initial decay time

Time (120583s) 3 10 15 20 25minus119864119911 (measured) vm 81 53 53 58 62minus119864119911 (simulated) vm 79 51 50 54 58Percentage difference 3 4 6 7 6

0 10 20 30 40 50 60 70Time (120583s)

0102030405060708090

Ez (V

m)

Figure 5 The measured vertical electric field for an observationpoint (119911 = 10m 119903 = 46 km) [23 24]

are in accord with other previous method of calculation in[23 24]

The proposed method is also validated by Nucci currentfunction in Table 1 which is defined by the combination of (1)and (2) when the MTLL model is applied as a current modelThe simulatedmagnetic flux density and vertical electric fieldare formed as in Figures 6 and 7 respectively By the sametoken Figure 8 depicts the measured vertical electric field

In addition the numerical comparison between themeasured fields and simulated vertical fields is listed inTable 6 suggesting that the simulated field is in agreementwith the corresponding measured field Moreover magneticflux densities simulated using the proposed field expressionsare compatible with the simulated field calculated using othermethods given in the reference [5] Note that the currentparameters are 11989401 = 325 kA 11989402 = 895 kA Γ11 = 0072 times

10minus6 Γ12 = 1667 times 10

minus6 Γ21 = 100 times 10minus6 Γ22 = 05 times 10

minus61198991 = 2 and V = 15 times 10

8 ms [5] The proposed expressionscan directly be applied towidely used current functions in thetime domain just by substituting the current and geometricalparameters in the proposed field expressions without needingto apply any extra algorithms and extra conversions (such asFourier transform) when the realistic current functions areapplied [1] Besides the proposed algorithm expressions can

0 5 10 15 20 25 30 35 400

051

152

253

354

455

times10minus5

Time (120583s)B

(Wb

m2)

Figure 6 The simulated magnetic flux density using Nucci currentfunction (119911 = 0m 119903 = 50m)

0123456789

10

Vert

ical

elec

tric

fiel

d (V

m)

times104

0 5 10 15 20 25 30 35 40Time (120583s)

Figure 7 The simulated vertical electric field using Nucci currentfunction (119911 = 0m 119903 = 15m)

0 5 10 15 20 25 30 35 40

0

Due to return strokeDue toleader

minus100

minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

minus10 minus5

Time (120583s)

Ez (k

Vm

)

Figure 8Themeasured vertical electric field at an observation point(119911 = 0m 119903 = 15m) [5]

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 6 Numerical comparison between the simulated magneticflux density and the corresponding measured fields from Figures 7and 8 after subtracting initial decay time

Time (120583s) 5 10 15 20 25 30minus119864119911 (measured) kvm 92 93 945 955 96 975minus119864119911 (simulated) kvm 905 91 92 935 94 955Percentage difference 16 21 26 2 2 2

support different engineering current models provided thatthey follow (4)

The proposed method can corroborate most of the cou-pling models that compute lightning induced overvoltage(LIOV) directly in the time domain because not onlyare all of the electromagnetic field components accessiblebut also the fundamental equations are in the Cartesiancoordinates This adds a new dimension to the developmentof LIOV coupling models which are restricted in the presentcontext to apply a fast and direct computational block inthe time domain for estimation of electromagnetic fieldsgenerated by lightning channel without needing to use extracomputational stage as Fourier transform for a large numberof channel base current functions and engineering currentmodels Likewise the proposed field expressions can be usedin the inverse procedure algorithms to evaluate lightningreturn stroke current using measured electromagnetic fieldsdirectly in the time domain [8 9] Moreover the proposedfield expressions can be developed for the case of nonperfectground conductivity using Cooray-Rubinstein equation [25]

Consequently the evaluation of electromagnetic fields atdifferent points along the power line can easily be computedwith different values of 119909 or 119910 [25ndash28] The accuracy of theresults can be increased by changing the values of 119896 andΔ119905 However the processing time will increase with suchincreasing of 119896 Further the proposed field expressions canbe used for close and medium distances from the lightningchannel however some of the calculation methods in thetime domain are merely confined to close distances from thelightning channel

6 Conclusion

In this study the general electromagnetic field expressionsdue to vertical lightning channel at an observation point areproposed directly in the time domain just by substitutingof current and geometrical parameters in the proposedfield expressions without needing to apply any algorithmsand extra conversions The proposed field expressions cansupport the widely used current functions and currentmodels Besides the proposed method was validated withtwo return stroke current samples The results showed thatthere is a good agreement between the electromagnetic fieldscomputed by proposed method and both the measured fieldsand those calculated by previous simulation methods Inaddition the proposed method is applicable for both closeand intermediate distances from the lightning channel andit is completely in accordance with many coupling modelsin evaluating the induced lightning voltage and also inverse

procedure algorithms in the time domain This would bevery useful especially for the utility in assessing the lightningperformance of an overhead line particularly the distributionline

Conflict of Interests

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

Acknowledgments

The authors would like to thank Professor V Cooray ofUppsala University Sweden who read the paper and pro-vided valuable comments and suggestions Universiti PutraMalaysia is also acknowledged for providing the financialsupport in this work

References

[1] 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

[2] VA RakovMAUman andK J Rambo ldquoA reviewof ten yearsof triggered-lightning experiments at Camp Blanding FloridardquoAtmospheric Research vol 76 pp 503ndash517 2005

[3] 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

[4] R Thottappillil and V A Rakov ldquoDistribution of charge alongthe lightning channel relation to remote electric and magneticfields and to return-stroke modelsrdquo Journal of GeophysicalResearch D Atmospheres vol 102 no 6 Article ID 96JD03344pp 6987ndash7006 1997

[5] C Yang and B Zhou ldquoCalculation methods of electromagneticfields very close to lightningrdquo IEEE Transactions on Electromag-netic Compatibility vol 46 no 1 pp 133ndash141 2004

[6] C Sartori and J R Cardoso ldquoAnalytical-FDTDmethod for nearLEMP calculationrdquo IEEE Transactions onMagnetics vol 36 no4 pp 1631ndash1634 2000

[7] 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

[8] A Andreotti D Assante S Falco and L Verolino ldquoAnimproved procedure for the return stroke current identifica-tionrdquo IEEE Transactions on Magnetics vol 41 no 5 pp 1872ndash1875 2005

[9] A Ceclan D D Micu and L Czumbil ldquoOn a return strokelightning identification procedure by inverse formulation andregularizationrdquo in Proceedings of the 14th Biennial IEEE Con-ference on Electromagnetic Field Computation (CEFC rsquo10) IEEEChicago Ill USA May 2010

[10] Z Feizhou and L Shanghe ldquoA new function to representthe lightning return-stroke currentsrdquo IEEE Transactions onElectromagnetic Compatibility vol 44 no 4 pp 595ndash597 2002

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[11] M Izadi and M Z A A Kadir ldquoNew algorithm for evaluationof electric fields due to indirect lightning strikerdquo ComputerModeling in Engineering and Sciences vol 67 no 1 pp 1ndash122010

[12] V A Rakov and M A Uman ldquoReview and evaluation oflightning return stroke models including some aspects of theirapplicationrdquo IEEE Transactions on Electromagnetic Compatibil-ity vol 40 no 4 pp 403ndash426 1998

[13] F Delfino R Procopio A Andreotti and L Verolino ldquoLight-ning return stroke current identification via field measure-mentsrdquo Electrical Engineering vol 84 no 1 pp 41ndash50 2002

[14] A Andreotti U De Martinis and L Verolino ldquoAn inverseprocedure for the return stroke current identificationrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp155ndash160 2001

[15] A Andreotti F Delfino P Girdinio and L Verolino ldquoA field-based inverse algorithm for the identification of different heightlightning return strokesrdquo COMPEL The International Journalfor Computation and Mathematics in Electrical and ElectronicEngineering vol 20 no 3 pp 724ndash731 2001

[16] A Andreotti F Delfino P Girdinio and L Verolino ldquoAnidentification procedure for lightning return strokesrdquo Journal ofElectrostatics vol 51 pp 326ndash332 2001

[17] V Rakov ldquoLightning return stroke speedrdquo Journal of LightningResearch vol 1 2007

[18] X L Zhou ldquoOn independence completeness of Maxwellrsquosequations and uniqueness theorems in electromagneticsrdquoProgress in Electromagnetics Research vol 64 pp 117ndash134 2006

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

[20] E Kreyszig Advanced Engineering Mathematics John Wiley ampSons New Delhi India 2007

[21] M N O Sadiku ldquoFinite difference methodsrdquo in NumericalTechniques in Electromagnetics chapter 3 CRC Press 2ndedition 2001

[22] F Rachidi Effects electromagnetiques de la foudre sur les lignesde transmission aerienne modelisation et simulation [PhDthesis] Ecole Polytechnique Federale de Lausanne LausanneSwitzerland 1991

[23] N Mrsquoziou L Mokhnache A Boubakeur and R Kattan ldquoVal-idation of the Simpson-finite-difference time domain methodfor evaluating the electromagnetic field in the vicinity of thelightning channel initiated at ground levelrdquo IET GenerationTransmission and Distribution vol 3 no 3 pp 279ndash285 2009

[24] M Paolone C Nucci and F Rachidi ldquoA new finite differencetime domain scheme for the evaluation of lightning inducedovervoltage on multiconductor overhead linesrdquo in Proceedingsof the International Conference on Power System Transients(IPST rsquo01) pp 596ndash602 2001

[25] C Caligaris F Delfino and R Procopio ldquoCooray-Rubinsteinformula for the evaluation of lightning radial electric fieldsderivation and implementation in the time domainrdquo IEEETransactions on Electromagnetic Compatibility vol 50 no 1 pp194ndash197 2008

[26] 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

[27] 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

[28] C A Nucci ldquoLightning-induced voltages on overhead powerlines Part II coupling models for the evaluation of the inducedvoltagesrdquo Electra vol 162 pp 121ndash145 1995

[29] C E R Bruce and R H Golde ldquoThe lightning dischargerdquo TheJournal of the Institution of Electrical Engineers vol 2 p 88 1941

[30] M Uman and D K McLain ldquoMagnetic field of lightning returnstrokerdquo Journal of Geophysical Research vol 74 no 28 pp6899ndash6910 1969

[31] R D Jones ldquoOn the use of tailored return-stroke currentrepresentations to simplify the analysis of lightning effects onsystemsrdquo IEEE Transactions on Electromagnetic Compatibilityvol 19 no 2 pp 95ndash96 1977

[32] E T Pierce ldquoTriggered lightning and some unsuspectedlightning hazards (Lightning triggered by man and lightninghazards)rdquo ONR Naval Research Reviews vol 25 1972

[33] F Heidler ldquoAnalytische blitzstromfunktion zur LEMP-bere-chnungrdquo in Proceedings of the 18th International Conference onLightning Protection (ICLP rsquo85) Munich Germany 1985

[34] G Diendorfer and M A Uman ldquoAn improved return strokemodel with specified channel-base currentrdquo Journal of Geophys-ical Research vol 95 no 9 pp 13ndash644 1990

[35] C A Nucci G Diendorfer M Uman F Rachidi and CMazzetti ldquoLightning return-stroke models with channel-basespecified current a review and comparisonrdquo Journal of Geophys-ical Research vol 95 no 12 pp 20395ndash20408 1990

[36] F Heidler ldquoTravelling current source model for LEMP calcula-tionrdquo in Proceedings of the 6th SymposiumAnd Technical Exhibi-tion On Electromagnetic Compability Zurich Switzerland 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of