Lightning-Induced Over Voltages in Power Lines Validity of Various Approximations

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 4, NOVEMBER 1998 355

Lightning-Induced Overvoltages in PowerLines: Validity of Various Approximations

Made in Overvoltage CalculationsVernon Cooray and Viktor Scuka

Abstract—In this paper, the validity of different approxima-tions used in the calculation of induced overvoltages in powerlines are investigated. These approximations are as follows: 1) ne-glect the distortions introduced by the finitely conducting groundon the electromagnetic (EM) fields; 2) the horizontal electric fieldat ground level is calculated by using the wavetilt approximation,which is valid for radiation fields and for grazing incidence;3) the horizontal field at line height is obtained by adding thehorizontal field calculated at ground level to the horizontal fieldat line height calculated over perfectly conducting ground; 4)the transmission line equations derived by assuming that theground is perfectly conducting are used with the horizontal fieldpresent over finitely conducting ground as a source term incalculating the induced overvoltages; and 5) the propagationeffects on the transients as they propagate along the line areeither neglected or modeled by replacing the line impedance dueto ground by a constant resistance. The results presented in thispaper show that in the calculation of induced overvoltages theapproximation 3) is justified and approximation 2) is justifiedif the interest is to estimate the peak value of the inducedovervoltage. Approximation 4) is probably justified for shortlines and/or for highly conducting grounds. But it can introducesignificant errors if the line is long and ground conductivity islow. Approximations 1) and 5) may lead to significant errors inthe peak value, risetime, and derivative of the lightning inducedovervoltages.

Index Terms—Coupling models, lightning-induced voltage,lossy ground.

I. INTRODUCTION

DUE to technical difficulties associated with measuringthe characteristics of lightning induced overvoltages in

power lines, many researchers have resorted to numericalmodels to derive the features of overvoltages produced bylightning. The main steps in these numerical models areas follows [1]–[3]: 1) find a reliable return stroke modelcapable of generating fields similar to those generated bylightning return strokes; 2) estimate the distortions introducedinto these fields as they propagate over finitely conductingground and then obtain the vertical and horizontal electricfields at ground level and at line height; 3) use a reliablecoupling model to simulate the interaction of electromagnetic(EM) fields with power lines; and 4) include the effects of

Manuscript received May 16, 1995; revised May 22, 1998.The authors are with the Institute of High Voltage Research, University of

Uppsala, S-752 28 Sweden.Publisher Item Identifier S 0018-9375(98)06340-6.

propagation on the transients as they propagate along thepower line. In executing these steps in the numerical modelmany researchers have used different approximations. Someof the main approximations are as follows: 1) neglect thedistortions introduced by the finitely conducting ground onthe EM fields; 2) the horizontal electric field at ground level iscalculated by using the wavetilt approximation, which is validfor radiation fields and for grazing incidence; 3) the horizontalfield at line height is obtained by adding the horizontal fieldcalculated at ground level to the horizontal field at line heightcalculated over perfectly conducting ground—in the literature,this is known as Cooray–Rubinstein approximation [4]; 4)the transmission line equations derived by assuming that theground is perfectly conducting are used with the horizontalfield present over finitely conducting ground as a source termin calculating the induced overvoltages; and 5) the propagationeffects on the transients as they propagate along the line areeither neglected or modeled by replacing the line impedancedue to ground by a constant resistance. In this paper, theeffect of these different assumptions on the calculated inducedovervoltages are investigated. The results show that some ofthese approximations are justified, whereas others can lead tosignificant errors.

II. THE ANALYSIS

A. Electric Fields Generated by Lightning ReturnStrokes over Finitely Conducting Ground

The electric fields generated by lightning return strokescan be obtained if the temporal and spatial variation of thereturn stroke current and return stroke velocity along the returnstroke channel are known. In the literature there are severalreturn stroke models that specify these parameters. In thispaper, a model introduced by Cooray [5] was used to generatethe lightning return stroke fields. The model is capable ofgenerating the return stroke current and return stroke velocityas a function of height. These model predictions are inreasonable agreement with the experimental observations (seealso Nucci [6] for a description and evaluation of channelbase current models, which are also suitable for EM fieldcalculations). In this paper, the calculations are presented fora peak return stroke current of 13 kA. Once the temporal andspatial variation of the return stroke current is known, the

0018–9375/98$10.00 1998 IEEE

356 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 4, NOVEMBER 1998

electric and magnetic fields over finitely conducting groundcan be calculated by using the field expressions for electricdipoles over finitely conducting ground. The electric field dueto a dipole over finitely conducting ground was first evaluatedby Sommerfeld [7], and the results were given in terms ofintegrals. Many researchers have considered different approx-imations for these integrals. Recently, Zeddam and Degauque[8] evaluated these Sommerfeld integrals and compared theresults with these well known approximations. Their resultsshow that the fields over finitely conducting ground can be wellrepresented by the results of Norton [9] among others. In thispaper, expressions for the dipole fields over finitely conductingground (as published by Norton [9]) were used in calculatingthe electric and magnetic fields. Cooray and Lundquist [10]showed that the calculation of lightning generated EM fieldsover finitely conducting ground can be simplified to a largeextent by replacing the attenuation function in the equationsby the attenuation function corresponding to a dipole atground level. The results presented by Cooray [11], Mingand Cooray [12], and Cooray and Ming [13] show that thisapproximation is valid for distances as close as 1 km fromthe lightning channel. Calculations done recently by Coorayshow that this approximation is valid even for distances asclose as 200 m. For example, the vertical electric field at 200m calculated over finitely conducting ground of conductivity0.002 S/m and relative dielectric constant five by replacing theattenuation function in equations by the attenuation functioncorresponding to a dipole at ground level is shown by a dashedline in Fig. 1(a). The vertical electric field at 200 m calculatedwithout making this approximation is shown by a solid line inFig. 1(a). Note that the maximum error is about 7%.

B. Horizontal Field at Ground Level

The horizontal field at ground level can be calculatedby making use of the surface impedance relationship [14].According to this relationship, the horizontal electric field atground level in frequency domain is given by

(1)

where is the horizontal magnetic field in frequencydomain at ground level, the relative dielectric constantof the ground, the conductivity of the ground, theangular frequency, and the permittivity of free-space. Forfrequencies and ground conductivities of interest in power linecalculations the second term in the right-hand side of theequation can be neglected (see the discussion by Wait [15]on this topic) Then the horizontal electric field in timedomain at ground level can be written as

(2)

with

(3)

(a)

(b)

Fig. 1. (a) The vertical electric field at 200 m calculated over finitelyconducting ground of conductivity 0.002 s/m and relative dielectric constantfive by replacing the attenuation function in equations by the attenuationfunction corresponding to a dipole at ground level is shown by a dashedline. The vertical electric field at 200 m calculated without making thisapproximation is shown by a solid line. (b) Variation of the ratio of thehorizontal field at line height calculated by using the approximation III to theexact horizontal field at line height:D = 200 m, ! = 10

6 rad/s,� = 0:01

S/m," = 5; D = 200 m, ! = 106 rad/s,� = 0:001 S/m," = 5; D = 200

m, ! = 107 rad/s,� = 0:01 S/m, " = 5; D = 200 m, ! = 10

7 rad/s,� = 0:001 S/m, " = 5. D is the horizontal distance between the dipole andpoint of observation.

where denotes the inverse Fourier transformation, isthe horizontal magnetic field in time domain, and ,

, and are modified Bessel functions of zeroand first order. Cooray [14] showed that this relationshipis capable of generating horizontal fields more accuratelythan the Wavetilt approximation can. He compared the fieldscalculated by using both surface impedance and wavetiltexpression with the horizontal field obtained directly fromNorton’s [9] equations and showed that the surface impedanceexpression is capable of accurately predicting the horizontalfield for distances down to about 200 m from the return strokechannel. Now let us consider the horizontal electric field thatexists at line height.

COORAY AND SCUKA: LIGHTNING-INDUCED OVERVOLTAGES IN POWER LINES 357

C. Horizontal Electric Field at Line Height

As mentioned in the introduction to this paper, severalresearchers have calculated the horizontal field at line heightby adding the horizontal field calculated at line height overperfectly conducting ground to the horizontal field calculatedat ground level over finitely conducting ground. In the lit-erature this is known as Cooray–Rubinstein approximation[4], [14], ,[16], [17]. Let us investigate the validity of thisapproximation. Norton [8] published the expressions for theelectric dipole fields in frequency domain for dipoles atdifferent heights over finitely conducting ground and fordifferent heights of observation points. To test the validityof the above approximation, the horizontal electric field 10 mabove ground was calculated by using the expressions givenby Norton [9] for different frequencies and for differentdipole heights. These calculations were performed for angularfrequencies (i.e., ) in the range of 10–10 rad/s and fordipole heights in the range of 0–2000 m. The distance to thepoint of observation was changed from 200 to 5000 m, andthe conductivity was changed over the range from 0.01 to0.001 S/m. The results are compared with the electric fieldat line height calculated by using the above approximation.Several examples of the calculations are given in Fig. 1(b). Inthis figure “ratio” is the ratio of the horizontal electric field atline height to the horizontal field calculated using the aboveapproximation. The results show that the maximum error onecan expect due to the above approximation is not more than20% for the range of parameters considered. Of course, thesecalculations are made in frequency domain and therefore itis difficult to pinpoint exactly the magnitude of error onemay encounter in time-domain calculations. But, it would notbe much larger than the above estimate. The return strokeis an extended source that can be divided into elementarydipoles located at different heights. Since the length of thereturn stroke channel of interest when calculating inducedovervoltages in power lines is less than 2000 m, the resultsshow that this approximation is reasonable in calculatinghorizontal fields at line height for times longer than about0.1 s.

D. The Coupling Model

Several coupling models available in the literature describethe interaction of EM waves with overhead power lines. Thestudies of Nucciet al. [18], Cooray [19], and Nucciet al.[20] show that many of the models available in the literatureare special cases of a model proposed by Agrawalet al. [21].The latter authors derived transmission line equations for amulticonductor system including a reference conductor. Toderive the transmission line equations over finitely conductingground we will use an approach similar to that of Agrawaletal. [21] but modified to take into account the finite conductivityof the ground. Let us consider a single conductor line overfinitely conducting ground. The relevant geometry is shownin Fig. 2(a). Consider the Maxwell’s equation related to theFaraday’s law. This can be written as

(4)

(a)

(b)

(c)

Fig. 2. Geometry relevant to the calculations presented in this paper.

Writing the above equation in integral form and performing theintegration over the closed path shown in Fig. 2(a) we obtain

(5)

where is a depth in the ground at which the amplitude ofthe electric and magnetic fields can be assumed to be zero.In general, is on the order of a skin-depth corresponding tothe frequency under consideration. For most of the frequenciesof interest, the lower limit of the integral in the case of thevertical field can be replaced by zero. The reasons for thisare the following: first, the vertical electric field inside theground is smaller by a factor of compared with the electricfield in air; second, the relaxation time of the ground forconductivity 0.001 S/m and for relative dielectric constant of 5is about 45 ns and the relaxation time decreases with increasingconductivity. Therefore, for variations of the electric fieldswhich take place over times longer than about 50 ns thevertical electric field in the ground can be neglected. For mostof the lightning generated EM fields that have propagated oversea water, the risetimes are longer than 50 ns and the risetime


of the EM fields increases with increasing propagation distanceover finitely conducting ground. Therefore, for frequencies thatwill be encountered in lightning EM fields and for typicalground conductivities it is reasonable to neglect the verticalelectric field in the ground. Now, dividing each term in (5) by

and taking the limit as

(6)

Dividing each field value into incident and scattered compo-nents

(7)

(8)

(9)

and substituting (7) through (9) in (6), we obtain

(10)

Assuming the scattered fields are transverse magnetic, thescattered line voltage can be defined as

(11)

and the integral of the scattered magnetic field can be relatedto the inductance per unit length of the line and the currentin the conductor

(12)

Substituting (11) and (12) in (10), we obtain

(13)

The right-hand side of this equation can be expressed as thecomponent of the incident horizontal field in the direction ofthe conductor at the line height, i.e.,

(14)

Substituting (14) into (13), we obtain

(15)

However

(16)

where is the series resistance per unit length of line. Finally,substituting (16) into (15) yields the first transmission lineequation

(17)

In most practical situations, the constant resistance is negligi-ble compared with and this transmission line equationcan be written as

(18)

Consider the Maxwell’s equation related to the Ampere’s law.This can be written in frequency domain as

(19)

However, ; therefore

(20)

Let us apply this equation to the closed volume around theconductor [refer to Fig. 2(b)]. The current flows parallel tothe axis of the conductor, therefore

(21)

(22)

Substitution of (21) and (22) into (20) gives

(23)

Dividing (23) by and taking the limit as we obtain

(24)

The charge per unit length can be related to the capacitanceper unit length of the line and the scattered voltage through

(25)

Substituting (25) into (24) yields the second transmission lineequation

(26)

The total voltage on the line is the sum of the voltages due tothe incident and scattered waves; that is

(27)

In deriving the transmission line equations we neglected theskin-effect resistance of the conductor. This is a reasonableapproximation for frequencies encountered in lightning EMfields.

An expression for the inductance of the line whenthe ground is finitely conducting can be obtained by using theconcept of images. Complex image theory represents a finitelyconductive ground as a perfectly conductive plane located at


a complex depth below the surface [22]. Using this theoryBannister [23] derived an expression for the inductance of theline. This expression is given by

(28)

where is the radius of the conductor. Hermossilo [24]showed that this expression is identical to an expression givenby Sunde [25] for the inductance of the line. Furthermore,Chang and Damrau [26], by comparing the above equationwith a more exact formulation, showed that this expressionaccurately describe the inductance of the line in the presenceof a finitely conducting ground plane.

E. Solution of the Transmission Line Equations

The propagation constant for a single conductor line over afinitely conductive ground is given by [24]

(29)

(30)

where . It is of interest to note that the expressionderived by Sunde [26] for the propagation constant of a lineover finitely conducting ground reduces to the above equationunder conditions in which the skin-effect resistance of the linecan be neglected.

The transmission line equations (17) and (26) predict thatthe incident EM field excites freely propagating waves at eachconductor segment. Consider a small elementlocated atpoint on the line. The voltage induced at this element dueto the incident EM field is given by

(31)

where is the component of the horizontal field inthe direction of the overhead conductor. This voltage gives riseto two traveling waves propagating in opposite directions onthe line. Therefore, the contribution from this small elementto the total scattered voltage at pointon the line is

(32)

where and is the speed of light in free-space.The scattered voltage in time domain at the point of intereston the line due to the element is given by the followingconvolution integral:

(33)

In this equation, is the impulse response of the line,which is given by

(34)

and is given by

(35)

The total scattered voltage at the point of interest canbe obtained by summing the contribution from all the elementson the line. The total induced voltage at the point of interest

can be obtained by adding the integral of the incidentvertical electric field at the point of interest over the line heightto the total scattered voltage, i.e.,

(36)

F. Transmission Line Equation in Time Domainover Finitely Conducting Ground

The two transmission line equations (17) and (26) can bedirectly transformed into time domain, the result being

(37)

(38)

(39)

where is the inverse Fourier transformation of .From these equations one can see that if it is possible to neglect

compared with then the above transmissionline equations reduce to

(40)

(41)

These equations are identical to the transmission line equationscorresponding to a lossless line over perfectly conductingground except that the horizontal electric field driving the lineis the one present over finitely conducting ground. This maybe a good approximation for short lines in which the effects ofthe ground on the transients propagating along the line can beneglected. Under these conditions the approximation 4) madein calculating induced overvoltages is justified. On the otherhand, if can be replaced by a constant resistance,

, then the transmission line equations reduce to

(42)

(43)

These equations are identical to the transmission line equationscorresponding to a lossy line over perfectly conducting groundexcept that the horizontal electric field that drives the line isthe horizontal electric field present over finitely conductingground. Under these conditions as well the approximation 4)


(a)

(b)

Fig. 3. (a) Induced voltage at termination B when the strike point isat P1 [Fig. 2(c)]. Curve 1: voltage calculated over perfectly conductingground. Curve 2: voltage calculated over finitely conducting ground. Curve3: voltage calculated over finitely conducting ground by using the wavetiltapproximation. (b) Induced voltage at termination B when the strike pointis at P2 [Fig. 2(c)]. Curve 1: voltage calculated over perfectly conductingground. Curve 2: voltage calculated over finitely conducting ground. Curve3: voltage calculated over finitely conducting ground by using the wavetiltapproximation.

made in calculating, induced overvoltages is justified. The nextsection includes a discussion of the errors and problems thatcan be expected in the induced overvoltage calculations whenone attempts to replace the line impedance associated withfinitely conducting ground by a constant resistance.

III. RESULTS AND DISCUSSION

To illustrate the effects of different assumptions on thecalculated induced overvoltages, consider a 5-km-long lineat a height of 10 m and terminated with it’s characteristicimpedance. The ground is assumed to be finitely conductingwith conductivity of 0.002 S/m and the relative dielectricconstant of the ground is assumed to be five. The geometryrelevant to this situation is shown in Fig. 2(c). The inducedvoltage and its derivative at the termination B, calculated byusing different approximations when the lightning strikes atP1, are depicted in Figs. 3(a), 4(a), and 5(a). The correspond-

(a)

(b)

Fig. 4. (a) Induced voltage at termination B when the strike point is atP1 [Fig. 2(c)]. Curve 1: voltage calculated over finitely conducting ground.Curve 2: voltage calculated over finitely conducting ground by neglecting thepropagation effects on the transients traveling along the line. Curve 3: voltagecalculated over finitely conducting ground by neglecting the propagationeffects on both the transients and fields. (b) Induced voltage at terminationB when the strike point is at P2 [Fig. 2(c)]. Curve 1: voltage calculatedover finitely conducting ground. Curve 2: voltage calculated over finitelyconducting ground by neglecting the propagation effects on the transientstraveling along the line. Curve 3: voltage calculated over finitely conductingground by neglecting the propagation effects on both the transients and fields.

ing waveforms when lightning strikes point P2 are depicted inFigs. 3(b), 4(b), and 5(b). Consider Fig. 3(a) and (b). In thisfigure, curve 1 is the voltage calculated when the ground isperfectly conducting; curve 2 is the voltage calculated overfinitely conducting ground, and curve 3 is the voltage calcu-lated by using wavetilt approximation. First, note the influenceof the ground conductivity on the calculated induced voltages.The finite conductivity of the ground changes the polarity,the shape, and the amplitude of the induced overvoltages.Note also that as far as the peak of the induced voltageis concerned, the wavetilt approximation does not introducesignificant errors. However, the wavetilt approximation maylead to significant errors in the tail part of the inducedovervoltages. This point is important in cases where it isnecessary to quantify the amount of energy dissipation inprotective devices.


(a)

(b)

Fig. 5. (a) The derivative of the induced voltage at termination B whenthe strike point is at P1 [Fig. 2(c)]. Curve 1: voltage calculated over finitelyconducting ground. Curve 2: voltage calculated over finitely conductingground by neglecting the propagation effects on the transients traveling alongthe line. Curve 3: voltage calculated over finitely conducting ground byneglecting the propagation effects on both the transients and fields. (b) Thederivative of the induced voltage at termination B when the strike point is atP2 [Fig. 2(c)]. Curve 1: voltage calculated over finitely conducting ground.Curve 2: voltage calculated over finitely conducting ground by neglecting thepropagation effects on the transients traveling along the line. Curve 3: voltagecalculated over finitely conducting ground by neglecting the propagationeffects on both the transients and fields.

Fig. 4(a) and (b) depicts the effects of approximations 1)and 5). In these figures, curve 1 is the induced voltagecalculated without any approximations, curve 2 is calculatedby neglecting the propagation effects as they propagate alongthe line, but taking into account the propagation effects onthe EM fields, and curve 3 is calculated by neglecting thepropagation effects on both the transients and fields. Thederivatives of these voltages are shown in Fig. 5(a) and (b),respectively. First, let us consider the effects of approximation1): neglecting the propagation effects on the EM fields asthey propagate along the line can lead to an overestimateof the peak of the induced overvoltage. The reason for thisis that for a given conductivity the peak of the horizontalfield at ground level increases as the risetime of the magneticfield decreases. Neglecting the propagation effects on the

EM fields leads to an underestimation of the risetime of themagnetic field, which, in turn, leads to an increase in thecalculated horizontal electric field. The peak value of thederivative is more sensitive to this approximation than is thepeak of the induced overvoltage. Furthermore, note that theeffect of this approximation influences the results in differentways depending on the location of the lightning strike. Ofcourse, it is important to note that in the example consideredthe maximum distance of propagation encountered by theEM fields is 5 km. For lightning flashes located at longerdistances from the line and for ground conductivities less than0.002 S/m the effects of this approximation would be largerthan those shown in the Figs. 4(a)–5(b). On the other hand, forshort lines and for high ground conductivities this may be avalid approximation [27]. Second, consider the approximation5): The results in Figs. 4(a)–5(b) show very clearly that thepropagation effects on the transients as they propagate alongthe line make a significant difference in the peak of the inducedvoltage. Note that the peak voltage calculated with propagationeffects on the transients is about half the voltage calculatedwithout propagation effects along the line. The change in thederivative of the voltage is even more significant. Note thatthe derivative of the voltage is several times smaller than thevoltage calculated without taking into account the propagationeffects along the ground. The results also show that theinfluence of the propagation effects depends on the locationof the strike point with respect to the line. For example, theinfluence of the propagation effects along the line on thevoltage measured at B is larger for strike point P1 than forstrike point P2. The reason is that in the case of strike point P2the transients contributing to the peak voltage travel a longerdistance along the line compared with those corresponding tostrike point P2. Furthermore, the importance of these effectsincrease with decreasing conductivity and increasing linelength. However, for shorter lines (500 m) over sufficientlyconducting ground it may be a reasonable approximation toneglect the propagation effects on the transients on the line.The results show that for lines of several kilometers lengthover moderately conducting ground the effects of propagationon the transients as they propagate along the line may notbe neglected. In these cases the approximation 5) is onlyvalid if the impedance due to ground can be replaced by aconstant resistance. To test this possibility the quantityin the equations was replaced by a constant resistance. Thevalue of this resistance was changed until the peak value ofthe calculated induced overvoltage agreed with the peak ofthe induced overvoltage in curve 2 in Fig. 3(a) and (b). Theresults for strike point P2 are shown in Fig. 6. The effectiveresistance required to obtain this agreement was 0.36/m.Note that even though the peak values are equal there is asignificant difference between the risetime and the derivativeof the induced overvoltage. The results for strike point P1are shown in Fig. 7. The effective resistance required to bringthe peak values of the two waveforms into agreement is 0.24

/m, which differs from the value we had to use in the casewhere the strike point was P2. For comparative purposes, thewaveform obtained when the resistance is 0.36/m (i.e., thevalue corresponding to strike point P2 is also shown in the


Fig. 6. Voltage induced at termination B when the strike point is at P2.Curve 1: voltage calculated over finitely conducting ground. Curve 2: voltagecalculated over finitely conducting ground by replacingj!�L by a constantresistance. The value of the constant resistance is 0.36/m.

Fig. 7. Voltage induced at termination B when the strike point is at P1.Curve 1: voltage calculated over finitely conducting ground. Curve 2: voltagecalculated over finitely conducting ground replacingj!�L by a constantresistance. The value of the constant resistance is 0.24/m. Curve 3: thevoltage obtained when the constant resistance was assumed to be equal to0.36 /m.

figure. Note again that the risetimes and the half widths of theinduced voltages calculated by simulating ground effects bya constant resistance differ from those calculated without thisapproximation. The effect of this assumption on the derivativesof the induced voltages is even more significant. For example,the derivatives of the waveforms in Fig. 7 are shown in Fig. 8.Note that the constant resistance required to obtain agreementbetween the peaks of the induced voltages may over-estimatethe derivative by about a factor two.

The most important points we have gathered from thisexercise are as follows. First, simulation of the ground effectsby a constant resistance leads to an overestimation of deriva-tives and an underestimation of the risetimes of the inducedvoltages. Second, the magnitude of the resistance that gives thecorrect results for the peak of the induced voltage depends onthe point of strike and probably on the length of the line. Thereason for this is that the frequency content of the inducedvoltages changes in different ways depending on the strike

Fig. 8. Derivative of the voltage induced at termination B when the strikepoint is at P1. Curve 1: voltage calculated over finitely conducting ground.Curve 2: voltage calculated over finitely conducting ground by replacingj!�L by a constant resistance. The value of the constant resistance is 0.24/m. Curve 3: the voltage obtained when the constant resistance was assumedto be equal to 0.36/m.

point and probably on the length of the line. This makes itdifficult to simulate ground effects with a single resistancewhen calculating induced voltages in power lines.

IV. CONCLUSIONS

In this paper, the validity of different approximations usedin the calculation of induced overvoltages in power lines hasbeen investigated. These approximations are the following: 1)neglect the distortions introduced by the finitely conductingground on the EM fields; 2) the horizontal electric field atground level is calculated by using the wavetilt approximation,which is valid for radiation fields and for grazing incidence;3) the horizontal field at line height is obtained by addingthe horizontal field calculated at ground level to the horizontalfield at line height calculated over perfectly conducting groundi.e., Cooray–Rubinstein approximation; 4) the transmissionline equations derived by assuming that the ground is perfectlyconducting are used with the horizontal field present overfinitely conducting ground as a source term in calculatingthe induced overvoltages; and 5) the propagation effects onthe transients as they propagate along the line are eitherneglected or modeled by replacing the line impedance dueto ground by a constant resistance. The results presented inthis paper show that in the calculation of lightning inducedovervoltages approximation 3) is justified and approximation2) is justified if the aim is to estimate the peak value of theinduced overvoltage. Approximation 4) is probably justifiedfor short lines and/or for highly conducting grounds, but itcan introduce significant errors if the line is long and groundconductivity is low. Approximations 1) and 5) may lead tosignificant errors in the peak value, risetime, and derivative ofthe lightning induced overvoltages.

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[21] A. K. Agrawal, H. J. Price, and S. J. Gurbaxani, “Transient response ofmulticonductor transmission lines excited by a nonuniform electromag-netic field,” IEEE Trans. Electromagn. Compat., vol. 22, pp. 119–129,May 1980.

[22] P. R. Bannister, “Applications of complex image theory,”Radio Sci.,vol. 21, no. 4, pp. 605–616, 1986.

[23] , “On the impedance of a finite-length horizontal wire located nearthe Earth’s surface,”IEEE Trans. Antennas Propagat., pp. 244–245,Mar. 1976.

[24] V. F. Hermosillo, “Attenuation and distortion of transient surges prop-agating on a single horizontal overhead line over a finitely conduc-tive plane,” Internal Rep. UURIE: 223-89, Univ. Uppsala, Sweden,UURIE:223-89, 1989.

[25] E. D. Sunde,Earth Conduction Effects in Transmission Systems.NewYork: Dover, 1948, p. 274.

[26] K. C. Chen and K. M. Damrau, “Accuracy of approximate transmissionline formulas for overhead wires,”IEEE Trans. Electromagn. Compat.,vol. 31, pp. 396–397, Nov. 1989.

[27] C. A. Nucci, F. Rachidi, M. Ianoz, and C. Mazzetti, “Lightning-inducedover-voltages on overhead lines,”IEEE Trans. Electromagn. Compat.,vol. 35, pp. 75–86, Feb. 1993.

Vernon Cooray received the B.Sc. degree in physics (first class honors) fromthe University of Colombo, Sri Lanka, in 1976, and the Ph.D. degree inelectricity science from the University of Uppala, Sweden, in 1982.

He is currently an Associate Professor at the Institute of High-VoltageResearch, Uppsala University. He has conducted experimental and theoreticalresearch work in electromagnetic compatibility, electromagnetic wave prop-agation, lightning physics, and discharge physics in relation to high-voltageengineering. He has authored and coauthored more than 100 scientific papers.

Dr. Cooray is the Swedish member of Task Force 1 (lightning location),Task Force 2 (interaction of electromagnetic fields with power lines), andTask Force 3 (lightning interception) of CIGRE-WG 33. He holds the cochairof the working group onTerrestrial and Planetary Noise of Man-Made andNatural Origin of the URSI Commission E.

Viktor Scuka received the B.Sc. degree from the University of Ljubljana,Slovenia, in 1958, and the Philosophy Licentiate and Ph.D. degrees inelectricity science from Sweden, 1969 and 1975, respectively.

In 1989, he became a Professor at Uppsala University and, since 1990,has been the Department Head. From 1975 to 1983 he was employed asa Scientist in electrophysics at Ericsson Cables, Sundyber, Sweden, doingresearch in the field of high-voltage power cables and optical fibers fortelecommunications. He is the author of more than 200 scientific publicationsdealing with atmospheric electricity, lightning physics, lightning and transientprotection, high-voltage techniques, discharge physics, and electromagneticcompatibility.

Dr. Scuka was Vice Chairman from 1990 to 1993 and chairman from 1993to 1996 of URSI Commission E—electromagnetic noise and interference andis a member of several international scientific organizations.

Documents

Lightning-Induced Over Voltages in Power Lines Validity of Various Approximations