Nucci Et Al (1993).pdf

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL 35. NO. I , FEBRUARY 1993

75

Lightning-Induced Voltages on Overhead Lines Carlo Albert0 Nucci, Member, IEEE, Farhad Rachidi, Michel V. Ianoz, Senior Member, IEEE, and Carlo Mazzetti, Member, IEEE

Abstract- The paper discusses a modeling procedure that permits calculation of lightning-induced voltages on overhead lines starting from the channel-base current. The procedure makes use of 1) a lightning return-stroke model proposed by the authors for the calculation of the lightning electromagnetic field; and 2) a coupling model already presented in the literature based on the transmission line theory for field-to-overhead line coupling calculations. Both models are discussed and tested with experimental results. The hypothesis of perfect conducting ground, generally adopted in studies on the subject, is discussed in order to better assess its validity limits. The procedure is applied for the analysis of the voltages induced on an overhead line by a nearby lightning return stroke with a striking point equidistant from the line terminations. The analysis shows that the vertical and horizontal components of the electric field are both to be taken into account in the coupling mechanism. The peak value and the maximum time derivative of the channel-base current are shown to affect both the peak value and the maximum front steepness of the induced voltages while, for the examined case, the returnstroke velocity affects practically only the front steepness of the induced voltages. A comparison with other models proposed for the same purpose is presented. Peak value and maximum front steepness of the induced voltages calculated using other lightning return-stroke models differ; these differences are of the same order of magnitude as those that would result from different sets of characteristic parameters of the lightning discharge. It is also shown that a different coupling model used in the power-lightning literature by several other authors may result in a less accurate estimation of the induced voltages.

have been reported concerning the intensity and the waveshape of the induced voltage. This may essentially be due to: different approaches in modeling the distribution of the current along the channel during the return stroke phase when calculating the electromagnetic field; and different approaches in modeling the coupling between the electromagnetic field and the conductors. In addition to the theoretical studies, several measurement campaigns on lightning electromagnetic fields [9], [IO] and many tests with voltages induced on experimental lines [2], [ 1 I]-[ 141 have been performed; from these, data have been collected that result in a better comprehension of the phenomena involved. The aim of this paper is to 1 ) discuss the adequateness of a modeling procedure that permits calculation of the lightning induced voltages starting from the channel-base current; 2 ) apply it for the analysis of the voltages induced on a overhead line by a typical subsequent return stroke; and 3) point out the differences with other methods proposed for the same purpose. The induced voltages are calculated in the following way: Starting from the lightning current at the channel base, the electromagnetic field at different distances is calculated making use of a lightning return-stroke model proposed by the authors, which specifies the spatial-temporal distribution of the current along the channel. Then use is made of the electromagnetic field in order I. INTRODUCTION to calculate the induced voltages using a coupling model OLTAGES caused by lightning on overhead power lines already presented in the literature based on the transmiscan cause damages to either the power system (in parsion line theory for field-to-transmission line coupling ticular to distribution transformers), the electronic control and calculations. management system, or both [1]-[3]. Eriksson et al. [2] have In the first part of the paper (Sections I1 and III), the adopted evidenced that indirect lightning return strokes, hitting the procedure is examined. The hypothesis of perfect conductground in the vicinity of overhead lines, constitute a more ing ground, generally assumed in studies on the subject, is dangerous cause of damage than direct strikes, because of their discussed in order to better assess its validity limits. Some more frequent occurrence. comparisons between calculations and available experimental Several theoretical studies have been performed and differ- results are presented, even though a complete and adequate ent models have been proposed in order to estimate the severity comparison is limited by an insufficient set of simultaneof voltages induced by indirect lightning return strokes [4]- ously recorded parameters (namely, channel-base current wave [8]. In spite of some general agreement, quite different results shapes, velocity of the return stroke, vertical and horizontal electric field at different distances from the striking point, and Manuscript received February 18, 1992; revised July IO, 1992. This work line-induced voltages). was sponsored by the Italian Ministry of University and Scientific and In the second part of the paper (Section IV), the analysis Technological Research, Electrical Utilities of Switzerland, and the Swiss National Science Foundation. of the voltages is performed. Part of the analysis is aimed at C. A. Nucci is with the 1st. di Elettrotecnica Induwiale, Universiti di quantifying the contribution of both the horizontal and vertical Bologna, viale Risorgimento 2, 40136 Bologna. Italy. F. Rachidi and M. Ianoz are with the Lah. de Reseaux d‘Energie Electrique, electric field components to the induced voltages. In addition, Ecole Polytechnique FCdCrale de Lausanne, CH- 1015 Lausanne, Switzerland. we discuss the influence of the main parameters pertaining to C. Mazzetti is with the Dip. lngegneria Elettrica. Universita di Roma “La the lightning discharge (maximum front steepness and peak Sapienm,” via Eudoasiana 18, 0184 Roma, Italy. value of the channel-base current, return-stroke velocity) on IEEE Log Number 9204745.

V

00 IX-9375/93$03.00 0 I093 IEEE

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. 35, NO. I . FEBRUARY 1993

t

TABLE I PARAMETERS OF TWO FUNCTIONS (2) TH.4T REPRODUCE THE CHANNEL-BASE OF A TYPICAL SUBSEQUENT RETURN STROKE CURRENT WAVESHAPE I01\0

rI

(W

(11s)

((1s)

10.7

0.25

2.5

721

1

it1

102

2

(kA) 6.5

r.22

(11s)

(11s)

2.1

230

122

2

as expressed by i ( d .t ) = exp ( - z ’ / X ) i ( O . t - Z ’ / Y )

Fig. I .

Geometry used for the calculation of voltages induced by a lightning return stroke on a nearby overhead line.

the induced voltages. The analysis is completed by a comparison of the voltages calculated adopting different lightning return-stroke models, in order to assess how much the differences in these models may affect the calculated voltages. The comparison between the adopted field-to-overhead line coupling model and one that is widely used in the literature (e.g., 151, [6], [13], 1151) is also given. In this paper we shall concentrate on voltages induced during the lightning return-stroke phase; it is generally accepted that voltages induced during the preceding leader phase are consistently lower 141, 171. The calculation results presented in this paper were obtained by means of computer programs developed by the authors.

(1)

where X decay constant introduced in order to take into account the effect of the vertical distribution of charge stored in the corona sheath of the leader and subsequently discharged during the return stroke phase; it has already been determined to be in the range of 1-2 km [21]; and v velocity of the return stroke. This engineering model represents a modification of the transmission line (TL) model introduced by Uman and McLain [ 171, and, as will be shown, results in a better agreement with experimental results. For the current at the channel base i ( O , t ) , we adopt an analytical expression described by a sum of two functions of the following type [ 191 i(0.t) = -

(2)

where 71

= rxp [ - ( ‘ 1 / ~ 2 ) ( 1 2 r 2 / r 1 ) ( l ’ ” ) ]

(3)

and

10 amplitude of the channel-base current; front time constant; decay time constant; amplitude correction factor; and exponent (2 . . . 10). 11 Function (2) has been preferred to the commonly used double-exponential function since it exhibits a time-derivative equal to zero at t = 0, consistent with measured return-stroke current waveshapes. Further, it allows for the adjustment of the current amplitude, maximum current derivative, and charge transferred nearly independently by varying 10,r1, and r2, respectively. A sum of two functions has been chosen in order to better reproduce the overall waveshape of the current as observed in typical experimental results 1221. We have chosen the current parameters reported in Table I in order to reproduce the typical features of the lightning current at the channel base-namely the peak value, maximum time derivative, and decay time-of a typical subsequent return stroke, in accordance with Berger et al. observations [22] (see Fig. 2). TI

11. CALCULATION OF LIGHTNING RETURN STROKEELECTROMAGNETIC FIELDS A. Spatial-Temporal Distribution of the Lightning Return-Stroke Current

Fig. 1 presents the geometry of the problem: the lightning channel is assumed to be a vertical unidimensional antenna above a ground plane; no channel branches are considered, which means that the results, as computed, are mainly applicable to subsequent return strokes [7]. For the calculation of the lightning return-stroke electromagnetic field, a spatial-temporal distribution of the current along the channel i ( z ’ . t ) must be assumed. To this purpose several models have been proposed in the past years [ 16]-[20]; it is to be observed that only models in which the return-stroke current i(z’.t) can be simply related to a specified channelbase (ground-level) current i ( 0 , t ) are suitable, since only the channel-base current can be measured directly and only for it experimental results are available. The model we adopt for the present study is the modified transmission line (MTL) model, proposed and tested by the authors in 1201 and 1211; in it the lightning current is allowed to decrease with height while propagating the channel upward,

r2 rj

B. Lightning Return-Stroke Electromagnetic Field The coupling model adopted in this paper for the calculation of lightning-induced voltages requires the determination of the horizontal and vertical components of the electric field.

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NUCCI et al.: LIGHTNING-INDUCED VOLTAGES

14

77

I

+ -r (c’z -Rs

2’)

exp (-z’/A)

ai(& t - z ‘ / v - R / c ) at

where t o vacuum permittivity; c speed of light in vacuum; and

R distance from the single dipole to the observation point, as defined in Fig. 1. In (4) and ( 5 ) , the first term is called the electrostatic field, the second the electric induction or intermediate field, and 0 10 20 30 40 the third the electric radiation field. The total vertical and f [PI 50 horizontal electric fields are obtained by integrating (4) and ( 5 ) along the channel and its image. We now discuss the assumption regarding the perfect conductivity of the ground for the calculation of the horizontal and vertical electric field components. The incident electric field Ei,which excites the line, is due to the contribution of field originated by the channel dipoles and the response of the ground. This problem has been treated by Sommerfeld [24] in the general case of a dipole above an imperfectly conducting ground. The expressions derived by him for the electromagnetic field are in terms of slowly converging integrals. Simplified approaches valid for different distance ranges from the lightning channel have been proposed, some of which 0 0.4 0.8 1.2 1.6 2.0 are discussed in [25]. f [PI For the vertical component of the electric field, at distances from the lightning channel that do not exceed a few kilometers, Fig. 2. Channel-base (ground level) current for a typical negative subsequent return stroke: (a) solid line--experimental data (from [22]); dashed its intensity can be calculated with reasonable approximation line--analytical approximation using a sum of two functions; (b) derivative assuming the ground as a perfect conductor, as discussed by of the analytical approximation for the first 2 11s. several authors (e.g., [25]). At greater distances, attenuations and distortions of the vertical field, when propagating along an By assuming the ground as a perfect conductor (a hypothesis imperfectly conducting ground, are no longer negligible [26]. discussed later in this section), Master and Uman [23] have Regarding the horizontal component of the electric field, derived the equations for the vertical electric field dE,(r. z , t ) its intensity is more affected by the finite conductivity of and the horizontal electric field dE,(r, z . t ) at altitude z and the ground than the vertical one [25].However, for distance distance r originated by a vertical dipole of infinitesimal length ranges not exceeding a few hundred meters, the assumption dz‘ at height z’ along the channel by solving Maxwell’s of perfect ground conductivity can still be considered as equations in terms of retarded scalar and vector potentials. reasonable. This is shown by Fig. 3(a), where results published Their equations, adapted to the MTL lightning return stroke in [25], which have been obtained using accurate numerical current model expressed by (l), become solutions of the Sommerfeld integrals assuming a Bround conductivity of 1OW2 R-l/m and a relative ground permittivity of 10, are compared with calculations performed by the 2 ( z - 2’)’ - r 2 dE,(r. z . t ) = authors assuming an infinite ground conductivity. Note that, as 47rto R5 discussed in [8], the horizontal component of the field above a perfect conducting ground is due to the different distances and different retardation times of the fields produced by the 2(2 - z ’ ) ~- r 2 current in each channel segment and its image. exp (-z’/A)I(O. t - z’/u - R / r ) For observation points far from the channel, these difcR4 ferences tend to equalize and the horizontal field above a perfect conducting ground becomes negligible; the assumption ,of perfect conducting ground is no more reasonable for real soils as shown in Fig. 3(b) and (c). In order to calculate the horizontal component of the electric field above an imperfectly conducting ground at observation points far from the lightning exp (-.’/A) I(0, ‘T - z ’ / u - R / c )d7 channel, use is made by some authors of the wave-tilt function 37.(z - z ’ ) [7], [27]. This function relates the Fourier transform of the cxp ( - z ’ / A ) i ( O . t - z’/v - R / c ) horizontal electric field E , ( j w ) to that of the vertical electric cR4

[

+

i’

+

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 35, NO. 1, FEBRUARY 1993

...-......-. e-------. -4

r=12km z=6m

- - - - .. from 1251

-

.

Wavetilt

I~"'1""

l""I""I""1''"TT"

and e,- = 10, same as in 1251, have been adopted.

field E Z ( j w )

where erg and og are the relative permittivity and the conductivity of the ground, respectively. Equation (6) applies rigorously to the case of plane waves with grazing angle of incidence with respect to the ground plane. For the case of lightning electromagnetic fields, its application has been shown to be reasonable provided that the observation point is a few kilometer distant from the lightning channel [7], 1271.

far distance range from the channel ( T > 1 km) 191. The second have been obtained through firing small rockets trailing a grounded wire upward a few hundred meters during thunderstorms. With this technique, simultaneous measurements of channel-base current and vertical electric field at close distance (e.g., T = 50 m) have been obtained [lo]. No measurements of the horizontal electric field above ground at close distances have been published. In Fig. 4 the typical vertical electric field at 2 km from the lightning channel for a natural subsequent return stroke,

already been presented in 1201 and [21]. Fig. 5 compares the triggered lightning electric field measured at 50 m from the channel reported in [lo] with calculations using the MTL model. The channel-base current reported in [lo] has been digitized and used as input for the calculation. The presence of the rocket-launching metallic structure has been modeled by considering two current pulses traveling in opposite directions. as suggested in [lo], starting from a junction point at 15 m above ground; the value of the return stroke velocity near ground level has been fixed at 2.46 . lo8 m/s as derived in [IO], and reflections of the downward-traveling current pulse at the base of the launching structure (ground level) have not been taken into account. As far as the above assumption can be considered reasonable, the comparison shows a satisfactory agreement.

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NUCCI et al.: LIGHTNING-INDUCED VOLTAGES

19

The equations that describe the coupling between the electromagnetic field produced by an indirect lightning stroke and a multiconductor overhead line above ground expressed in the time domain are

d

-[[U;(z,t)] + dz

'

3 . at

+

[ i ; ( z : t ) ] [L:j]-[[zi(z> t)] =

d

+

+ [ C : ~dt] , [ U ~ ( Xt,) ]

- [ Z i ( ~ , t ) ] [Gij]. [ u ; ( z , t ) ] a:l:

[E:(z. hi,t ) ](7)

d

=o 0

10

20

30

40

r IN

50

Fig. 4. Typical vertical electric field at 2 km from the lightning channel for a natural subsequent return stroke. Solid line--experimental data (from [9]). Dotted line--calculation adopting the MTL model (A = 1.7 km).

where

[ E k ( z h,, , t ) ] vector of the horizontal component of the incident electric field along the .r axis at the conductor's height hi, where the subindex i denotes the particular wire of the multi-wire line; [ R l j ] [ L l j ] ,[ G ! , j ] ,and [ C i j ] resistance, inductance, conductance, and capacitance matrices respectively, per unit length of the line, where the subindex zj denotes the mutual resistance, inductance, conductance, and capacitance between the Sith and the jth wires, respectively; [ i i ( x ,t ) ] line current vector; [uf(z. t ) ] scattered voltage vector, related to [u,(z,t ) ] , total line voltage vector, by the following expression. [ ' U L ( Z ,t ) ] =

0

0.2

0.4

0.6

(8)

+ [,u:(z.t ) ]

[7g(z.t ) ]

1.0

0.8

lwl Fig. 5. Comparison between calculated and measured vertical electric field at 50 m from a triggered lightning return stroke channel. Solid line--experimental data (from [lo]). Dotted line--calculation adopting the MTL model ( A = 1.7 km).

where: E l ( z .z , t ) incident (or inducing) vertical electric field that can be considered as unvarying in the height range h 0 < z < hi; and [ui(x, t)]= E:(z, z , t)dz] % -[hi . E i ( z ,0 . t ) ] incident (or inducing) voltage vector. The boundary conditions for the scattered voltage vector [ u B ( z . t ) ]which , take into account the coupling between the incident field and the vertical terminations-the "risers"-are

-[So

The above justifies the adoption of the MTL model for the calculation of the electromagnetic field radiated by lightning return strokes. 111. CALCULATION OF INDUCED VOLTAGES

A. Coupling between Electromagnetic Field and Overhead Line

In this paper, use is made of the model in the time domain proposed by Agrawal et al. [29], based on the transmission line theory for field-to-transmission line coupling calculations. According to this model, the forcing functions that excite the line are the horizontal and vertical components of the incident electric field (E: and E:), that is, as already defined in Section 11-B, the sum of the field radiated by the lightning return stroke and the ground-reflected field in absence of the line wires. Note that the total field E is given by the sum of the incident field El and the scattered field E S . A time-domain approach permits to deal with the problem in a more straightforward way and to take more easily into account nonlinearities. Its main disadvantage lies in the more complex treatment of frequency-dependant parameters, such as the ground resistance [3O]. The convolution theorem in the time domain, however, can be used for this purpose (e.g., [3I]).

+

[u;'(z0.t ) ] = -[zA4] . ["(zo. t ) ] [hi . El(zo.0. tu (IO) [u;(3!o 1. t ) ]= [Z,] . ["(zo 1. t ) ] [hi . Ei(ao 1 ; 0: t ) ]

+

+

+

+

(1 1 ) where [Z,] and [Z,] are the matrices of the termination impedances. Using the above mathematical formalism, the equivalent circuit for the case of a lossless single wire above a perfect conducting ground excited by an incident electromagnetic field is shown in Fig. 6.

B. Comparison with Experimental Results Equations (7)-( 11) have been solved by means of a computer code using the point-centered finite-difference technique. Calculated values have been compared with experimental results recorded in Florida during the measurement campaigns carried out in 1985. The experimental data consist of simultaneous measurements of vertical electric field and voltage at

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL 35, NO. I , FEBRUARY 1993

'9 i

- h El(x.0)

-

0

-

20

40

60

Measured E, field

Fig. 6 . Equivalent circuit of a lossless single-wire overhead line excited by lightning return-stroke electromagnetic field.

100

80 1

r)

[MI

0

20

60

40

80 I

100

[PI

E, field

A

.EO

\I

-0.061

0

20 40 60 80 Measured voltage (*)

'

100

Calculated voltage

IP'

-0.08 0

20

40

60

Measured E, field

80

r)

I

100

0

40

20

60

80

/Wl

100 /PI

1

E, field

Fig. 8. Example of comparison between measurements and calculations of voltages induced by lightning return stroke. o = 6'; line open circuited at both ends. *Experimental data taken from 1141.

of error: nonuniformity of ground conductivity, modeling of the terminations as pure resistances, and the fact that the ground impedance has been calculated neglecting its frequency dependence. 0

20

40

60

80 1

Measured voltage

r)

100

/PI

0

20

40

60

80 1

100

/PI

Calculated voltage

Fig. 7. Example of comparison between measurements and calculations of voltages induced by lightning retum stroke. 0 = 4>.4', line open circuited at both ends. *Experimental data provided by the Lightning Research Group of the University of Florida.

one end of the top conductor of a 450-m-long three-phase unenergized overhead distribution line. The azimuth angle of the incident field was also measured (see [I41 for further detail about the test facility). For all the examined cases (about 20), we have obtained satisfactory agreement for the waveshapes, but less good agreement for the amplitudes. Two examples of comparison are shown in Figs. 7 and 8. We summarize briefly hereunder the procedure used for the comparison. The low value of the measured vertical field E,, as can be seen in Figs. 7(a) and 8(a), indicates that the lightning struck the ground at a distance far from the line and hence the field intensity can be considered as constant along the line; for the same reason the horizontal field can be calculated with reasonable approximation from the digitized values of the vertical one using (6) [see Figs. 7(b) and 8(b)]. In (6), the ground conductivity cghas been fixed at 1 . G . 1 V 2 R-l/m, as resulted from measurements, and the relative ground permittivity c,. has been assumed equal to 10. As suggested in [14], a value of 0.05 R/m has been adopted for the resistance per unit length of the line (conductor plus ground). It can be seen that the calculated voltages at the line termination [Figs. 7(d) and 8(d)] are in reasonable agreement with the measured ones [Figs. 7(c) and 8(c), especially if considering the various sources

IV. Analysis of the Induced Voltages

We consider a 1-km-long 10-m-high single wire overhead line, matched at both ends. Matching the line avoids the effect of reflections on the induced voltages at the line ends that would make the discussion of the results less straightforward; for the same reason, the line has been considered as single phase. The relative position of the line with respect to the striking point is defined by 20 and y o (see Fig. 1). The striking point is considered equidistant from the line terminations and at a distance of 50 m from the line center (3.0 = -500 m and yo = 50 m). Lower values of yo are assumed to result in a direct strike to the line. Since the electromagnetic field that excites the line is at a distance range lower than about 500 m, for the following analysis the hypothesis of an infinite conducting ground has been assumed as reasonable. The line resistance as well as the line conductance are neglected; further, no corona effect is taken into account, which means that the induced voltages are presumably overestimated. A base case to which the analysis is referred has been specified, as described next.

A. Buse Cuse For the base case, the lightning channel-base current of Fig. 2 has been adopted. This current has a peak value of 12 kA and a maximum time-derivative of 50 kA/ps, corresponding, as already mentioned, to the average experimental data for a subsequent retum stroke reported in [22]. For the MTL model, a retum-stroke velocity of 1.3 . lo8 m/s and a decay constant X of 1.7 km (211 have been assumed. The channel

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NUCCI et al.: LIGHTNING-INDUCED VOLTAGES

height has been fixed for all the calculations equal to 7.5 km 1181; however, this parameter has been: shown not to affect 1 ? significantly the radiated electromagnetic fields [32], [33] and, hence, the induced voltages. The voltage induced at three points of the line (zol = -500 m; 2 0 2 = -250 m; 2 0 3 = 0 m) is presented in Fig. 9. The contribution to the total voltage of 1) the incident voltage; 2) the voltage due to the horizontal field; and 3) the voltage due to the coupling of the vertical field to the vertical line terminations (risers contribution), is represented in the same figure. The latter two components of the voltage are the two terms that form the scattered voltage. Note that the voltage due to the vertical field is the sum of the first and the third terms (incident voltage plus risers contribution). From Fig. 9(a), (b), and (c), it can be seen that according to the employed coupling model, both the vertical and horizontal components of the electric field are of fundamental importance in the coupling mechanism. However, it is worth noting that the contribution to the total voltage of the vertical field (incident voltage plus risers) decreases along the line moving away from the point closest to the striking point, in agreement with what is observed by Diendorfer [SI. For the examined case, the total voltage shows a positive and unipolar waveshape for each of the considered observation points. It is interesting to observe that while the contribution of the vertical field to the total voltage at each point is of positive polarity, the horizontal field coupling may result in a contribution of positive [Fig. 9(a)], bipolar [Fig. 9(b)], or negative [Fig. 9(c)] polarity. Different locations of the striking point, as well as different line terminations [14], however, could result in negative or bipolar induced voltages. In what follows, the influence of some characteristic parameters related to the lightning discharge on the amplitude and waveshape of the induced voltage is examined.

B. Peak Value and Muximum Time Derivative of the Lightning Current at the Channel Base In this study, use is made of the values reported by Berger et al. [22], referring to peak value and maximum time derivative for subsequent negative return strokes (see Table 11). From (4), ( S ) , and ( 2 ) , it results that both the horizontal and the vertical field intensities are proportional to the current intensity independently from the return stroke current model used in (4) and (5); as (7)-(9) are linear equations, it follows that an increase of a factor n of the channel-base current intensity will result in the same increase for both the peak value and the maximum time derivative of the induced voltage, as confirmed by calculations. However, in order to assess separately the influence of the peak value and of the maximum time derivative, we have varied singularly each one independently from the other. In each case the suitable coefficients for function (2) have been determined in order to reproduce the values reported in Table 11. The variation of the peak value keeping constant the maximum time derivative of the current at 40 kA/ps is shown in Fig. 10 (see also Table I11 for numerical values).

81

6

.. .. -.x.-. 0.. .x. .-.Incident -o.Ex conrribulion Voltage

\$

40

20

4

----n----A--Riwrrconlribution

I

I * .....

*

*----20

2 0

1

2

3

4

5

6

7 f

-40

8

I!&

b)

>

* - .-.

.Q

~. ....e._..

-0---X---

X

- - Incident Voliagc

- - - E, convihiion

100

-1oA

’$\\

\0030

1

2

3

4

5

6

7 I

8

lPl

Fig. 9. Voltages induced at different positions along a I-kmmatched overhead line by a typical subsequent retum stroke: (a) line terminations; (b) 250 m from line terminations; (c) middle point of the line. Striking point equidistant from the line ends (xu = -500 m, yu = 50 m). Channel-base current of Fig. 2. The contributions to the total voltage of the incident voltage, of the voltage due to the horizontal electric field, and of the voltage due to the vertical electric field coupling to the risers are also shown. The first and third terms represent the contribution of the vertical field to the total voltage. The time origin is shifted by yo/c.

The influence of the maximum time derivative of the channel-base current, keeping constant the peak value at 12 kA, is shown in Fig. 11. For an increase of a factor 10 in the examined range, the induced voltage maximum front steepness (time derivative) increases by a factor of 4, whereas the peak voltage increases only by a factor of about 1.4 (see Table 111).

C. Return-Stroke Velocity In this paper we vary the return-stroke velocity, keeping constant the intensity of the current, even though correla-

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TABLE I1 TIMEDERIVATIVE OF THE TYPICAL PEAKVALUEA N D MAXIMUM CHANNEL-BASE CURRENT r ( 0. t j FOR A SUBSEQUENT RETURNSTROKE[22] Parameter

Percent of Cases Exceeding Tabulated Value 95 9%

50%

5%

Peak value (kA)

4.6

12

30

Maximum time derivative (kA/ps)

12

40

I20 0

1

2

3

4

,

.

.

5

6

7

r 140

Fig. 1 1 .

? 120i

:

8

[PI

Influence of the maximum time derivative of the channel-base current on the induced voltage.

100

i7

80 60

40 20

0 0

1

2

3

4

5

6

7

r

Fig. IO.

5

IN

Influence of the peak value of the channel-babe current on the induced voltage. f

TABLE I11 INFLUENCE OF LIGHTNING DISCHARGE PARAMETERS ON THE PEAK TIMEDERIVATIVE ( d l - / d t ),,,,,, RISE VALUEl , , MAXIMUM TIMEf:3(,p<jocA,TIME-TO-HALF VALUE OF THE INDUCED VOLTAGE.CALCULATIONS PERFORMED USINGTHE MTL MODEL Case Base 66

190

0.5

I .4

I , = 1.6 kA

28

I 05

0.3

1.1

I , = 30 kA

133

225

0.7

2. I

50

65

0.8

2.2

72

290

0.3

1.1

69

120

0.7

2.2

61

220

0.3

0.8

I'

= 2 ' lo* m / s

tions between velocity and return stroke currents have been considered by other authors (e.g. [5], 1341). Three different values for a constant return-stroke velocity along the channel, namely: 0.6. lo8, 1 . 3 . lo8,and 2.10' m/s, have been chosen,

IWI

Fig. 12. Influence of the return stroke velocity on the induced voltage.

and calculations are presented in Fig. 12. It can be seen that, for the given configuration, an increase in the return-stroke velocity results in a marked increase of the induced voltage front steepness but does not affect very much the peak value (see also numerical values in Table 111). This result can be explained, considering the contribution of the different field terms to the total electric field and hence on the induced voltage. At close distance from the lightning channel, the radiation term is responsible only for the initial fast-rising part of the field. During this early period, this term increases nearly proportionally to the return-stroke velocity; soon after the electrostatic term, the amplitude of which decreases slightly with the velocity [32],becomes predominant. Since experimental data on lightning seem to show a decrease of velocity with height for a kilometer scale [28], a velocity decreasing along the channel has been considered. However, no significant variation in voltages have been found using an exponentially decreasing velocity 'ti = '00 exp( -z'/y), with y varying from 0.5 km to infinity. D. Lightning Return-Stroke Models

Several return-stroke current models with specified channelbase current have been proposed in the past years; some of these, namely the Bruce and Golde (BG) [ 161; the transmission line (TL) [17]; the Master, Uman, Lin and Standler (MULS) [ 181; the traveling current source (TCS) [ 191, and the modified transmission line (MTL) [20,2 11 have been recently reviewed and compared in [33], assuming a common current waveshape at the channel base.

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NUCCI et al.: LIGHTNING-INDUCED VOLTAGES

0

1

2

3

4

83

5

6

7 f

lightning return-stroke models differ, and that these differences are of the same order of magnitude as those that would result from different sets of characteristic parameters of the lightning discharge (compare Tables 111 and IV). It is worth mentioning that a new model, physically oriented and consistent with experimental results, has been recently proposed by Diendorfer and Uman [35]. Due to the fact that the channel-base current according to this model is the sum of two components and their choice may not always be straightforward, this model has not been used for the comparison. Further work, however, is needed in this respect.

8

I N

Fig. 13. Induced voltages calculated adopting two different lightning return stroke models (MTL and TCS) starting from the same channel-base current of Fig. 2. TABLE IV

INDUCEDVOLTAGEPARAMETERS CALCULATED ADWTING Two LIGHTN'NG RETURN-STRoKE (MTL A N D TCS) C U R R E N T OF FIG 2 STARTING FROM T H E S A M E CHANNEL-BASE

Model

MTL TCS

(kv)

t:iii-wu

c 7 ~ l

( ~ [ - / ~ / ~ ) I ~ L z L Y

f

(Its)

66

(kV//ts) 190

~

0.5

I .4

74

260

0.3

I .2

i

(/IS) ~ ( ~

It has been shown that all of them, apart from the TL model, produce overall field shapes that are reasonable approximations to measured fields from natural lightning. Further, it has been shown that they can be classified in two groups depending on: 1) the treatment of the return-stroke wavefront either as a discontinuity (BG, TCS) or as a fast-rising current waveshape with finite rise time equal the rise time assumed for the specified current at the channel base (TL, MULS, MTL); and 2) the spatial and temporal distribution of charge removed from the leader channel. In order to show how the above-mentioned differences can have an influence on the calculated voltages, one representative model from each group, namely the TCS and the MTL models, have been selected for comparison. The spatial-temporal distribution of the lightning current for the TCS model is reported in Appendix A. The induced voltages calculated starting from the same channel-base current represented in Fig. 2 and assuming the same line configuration and same impact point defined in Section IV-A, are shown in Fig. 13. It can be seen that the wave-shapes of the induced voltages predicted by the two models are quite similar, but the TCS model yields higher peak value and maximum front-steepness (see Table IV). This difference can be explained considering that the TCS model produces sharper and more intense initial field peaks than does the MTL [33]. For the same conditions, the voltages calculated using the other models (BG, TL, MULS) exhibit peak value and maximum front steepness that are practically equal (TL, MULS) or greater (BG) than those predicted by the MTL, and do not exceed those obtained using the TCS. We can conclude that peak value and maximum front steepness of the induced voltages calculated adopting other

E. Discussion ( f Other Coupling Models In the power-lightning literature, coupling equations expressed in terms of total voltage, introduced by Rusck [ 5 ] (see (B-1) and (B-2) of Appendix B), have been adopted by several authors for the determination of lightning-induced voltages on overhead lines (e.g., [6), [ 131, [ 151, [38]). These equations differ from (7) and (8). Since the two approaches are not equivalent and since the authors feel that, in this respect, a general agreement has not yet been reached, a further discussion is given in what follows. Without loosing generality, we shall consider a lossless line and a perfect conducting ground, since (B-I) and (B-2) refer to this case. It can be seen [36] that (7) and (8) (neglecting the line resistance and the line conductance since we are now considering a lossless line), which are expressed in terms of scattered voltage, are equivalent to another couple of equations given by Taylor et al. [37], which are written in terms of total line voltage (see (B-4) and (B-5) of Appendix B). These equations contain two source terms, which are due, respectively, to the incident (or inducing) magnetic flux density field [series voltage source, in (B-4)] and to the incident (inducing)vertical electric field [parallel current source, in (B-5)). Since (B-2) is equivalent to (B-5), the above allows us to conclude that in the coupling set of two equations (B-I) and (B-2), a source term is omitted, namely the contribution of the incident magnetic flux density field [compare (B-1) with (B-4)]. The result of omitting that source term can lead to a less accurate estimation of the induced voltages and, in some cases, to their underestimation. This is shown in Fig. 14, where the induced voltage calculated adopting the coupling model by Rusck [described by (B-I) and (B-2)] is compared with voltages calculated adopting the Agrawal et al. [(7) and (8)] and the Taylor et al. [(B-4) and (B-5)] models. For the calculation, we considered a 500-m-long line, matched at both ends, with a striking point along the line prolongation at 50 m from the left line termination. The difference between the calculated voltages is quite evident: the Rusck voltage has a peak value of 19 kV, while the Agrawal et a/. (and Taylor et ul.) voltage reaches a value of about 33 kV. It is worth observing that the magnetic field source term in (B-4) can be expressed in terms of both horizontal and vertical electric field components as shown by ( 1 2), which can be derived from Maxwell's equations [29]

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magnitude as those that would result from different sets of characteristic parameters of the lightning discharge. It has also been shown that a different coupling model used in the power-lightning literature by several other authors, in which a source term is omitted, may result in a less accurate estimation of the intensity of the induced voltages. Further theoretical work, as well as other experimental results, are needed in order to better estimate the influence on the induced voltages of ground characteristics and possible corona effects. 0

1

2

3

4

5

6

7 f

8

IP1

Fig. 14. Calculated voltage at the left line termination of a 500-m matched overhead line according to the Rusck (dotted line) and the Agrawal er al. and Taylor et al. (solid line) models. Striking point along the line prolongation at 50 m from left line termination.

B&(X,Z,t ) d z =

- E:(z. hi, t )

+

d

.kl

hr

APPENDIXA SPATIAL-TEMPORAL DISTRIBUTION OF LIGHTNING CURRENT ACCORDING TO THE TRAVELING CURRENTSOURCE(TCS) MODEL [19] In the TCS model, developed by Heidler ,191, a current source is assumed to travel along the channel from ground to cloud at the retum stroke velocity. The current flows into the lightning channel from the traveling source to the ground with the speed of light. The spatial-temporal distribution of the lightning current is

t ) = i ( O , t + z ' / c ) , for z' 5 vt i ( 2 . t ) = 0: for z' > lit z(z',

E 4 ( 2 . z ,t ) d z . (12)

V. CONCLUSIONS For the calculation of voltages induced by indirect lightning return strokes on overhead lines one needs 1) a lightning retum-stroke model for the calculation of the electromagnetic field; and 2) a field-to-transmission line coupling model. The MTL lightning retum-stroke model and the coupling model based on the transmission line theory, adopted in this paper, give results in reasonable agreement with existing experimental data. These two models have been applied for the analysis of the voltages induced on a 1-km-long overhead line by a typical subsequent retum stroke with striking point close to the line and equidistant from the line terminations. In addition, the results have been compared with those obtained adopting different lightning return stroke models. The study allows to conclude that: the vertical and horizontal components of the electric field for distance ranges not exceeding a few hundred meters can be calculated with reasonable approximation assuming the ground as a perfect conductor, even though the determination of the horizontal component requires a more rigorous approach for poor values of the ground conductivity; the peak value and the maximum time derivative of the channel-base current affect both the peak value and the maximum front steepness of the induced voltages. On the other hand, for the examined case, the return stroke velocity affects practically only the front steepness of the induced voltages; peak value and maximum front steepness of the induced voltages calculated using other lightning return-stroke models differ; these differences are of the same order of

(A- 1)

where c speed of light; and 'u retum stroke velocity. APPENDIXB The system of equations presented by Rusck [ 5 ] describing the coupling between lightning electromagnetic field and a lossless wire above a perfect conducting ground, in the form used by several other authors (e.g., [6], [13], [.l5], [38]) reads i,,u(z, t ) d:1.

di(z,t)

+

+ L'-d i (ats ,t ) = O

(B-1)

i,

C7(?L(.Lt) - d(.C,t) =0 dr dt where ~ ( 5t ),and i ( x ,t ) are the total line voltage and current, and uLi(lc, t ) is the so-called inducing voltage defined as (see also Section 111-A)

u ' ( L ,t ) = -

I"

E ~ ( . EZ ,. t ) d z zz -h . E ~ ( x0,, t )

(B-3)

The Taylor et al. coupling equations for a lossless wire above a perfect conducting ground, adapted to the coordinate system used in this paper and expressed in the time domain, read [37] d'lL(.C,2)

+ L d /' (atJ At ) =

-fI a

'L

Bi(z,z,t)dz (B-4)

where Bb(z.z.t ) is the component along the y axis of the incident magnetic-flux density field. While (B-2) is equivalent to (B-5), (B-1) differs from (B-4) due to the missing source term.

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ACKNOWLEDGMENT The authors are indebted to Prof. M. A. Uman and to Dr. M. Rubinstein for the provided sets of experimental results and for discussions‘ thanks are due to Prof. D. Zanobetti for profitable discussions during the development of work. REFERENCES A. C. Liew and M. Darveniza, “Lightning performance of unshielded transmission lines,” IEEE Trans. Power App. Syst.. vol. PAS-101, pp. 1478-1486, June 1982. A. J. Eriksson, M. F. Stringfellow, and D. V. Meal, “Lightning-induced overvoltages on overhead distribution lines,” IEEE Trans. Power App. Syst., vol. PAS-101, pp. 96CL969, Apr. 1982. D. E. Parrish, “Lightning-caused distribution transformer outages on a Florida distribution system,” IEEE Trans. Power Delivery, vol. 6, pp. 880-887, Apr. 1991. C. F. Wagner and G. D. McCann, “Induced voltages on transmission lines,” AIEE Trans., vol. 61, pp. 916-930, 1942. S. Rusck, “Induced lightning overvoltages on power transmission lines with special reference to the overvoltage protection of low voltage networks,” Trans. Royal Inst. Tech., no. 120, 1958. P. Chowdhuri and E. T. B. Gross, “Voltage surges induced on overhead lines by lightning strokes,” Proc. Inst. Elec. Eng., vol. 114, pp. 1899-1907, Dec. 1967. M. J. Master and M. A. Uman, “Lightning induced voltages on power lines: Theory,” IEEE Trans. Power App. Sysr., vol. PAS-103, pp. 2502-2518, Sept. 1984. G. Diendorfer, “Induced voltage on an overhead line due to nearby lightning,” IEEE Trans. Electromug. Contput., vol. 32, pp. 292-299, Nov. 1990. Y. T. Lin, M. A. Uman, J. A. Tiller, R. D. Brantley, W. H. Beasley, E. P. Krider, and C. D. Weidman, ‘Characterization of lightning return stroke electric and magnetic fields from simultaneous two-station measurements,” J . Geophysical Res., vol. 84, pp. 6307-6314, Oct. 1979. C. Leteinturier, C. Weidman, and J. Hamelin, “Current and electric field derivatives in triggered lightning return strokes,” J . Geophysical Res.. vol. 95. pp. 811-828, Jan. 1990. M. J. Master, M. A. Uman, W. Beasley, and M. DarveniLa, “Lightning induced voltages on power lines: Experiments,” IEEE Trun.c. Pobver App. Syst., vol. PAS-103, pp. 2519-2529, Sept. 1984. F. De la Rosa, R. Valdivia, H. Perez, and J. Loza, “Discussion about the inducing effects of lightning in an experimental power distribution line in Mexico,” IEEE Truns. Power Delivery, vol. 3 , pp. 1080-1089, July 1988. S. Yokohama, K. Miyake, and S. Fukui, “Advanced observations of lightning induced voltage on power distribution lines,” IEEE Truns. Power Deliveiy, vol. 4, pp. 2196-2203, Oct. 1989. M. Rubinstein, A. Y. Tzeng, M. A. Uman, P. J. Medelius, and E. W. Thomson, “An experimental test of a theory of lightning-induced voltages on an overhead wire,” IEEE Truns. Electromagn. Cumpat., vol. 31, pp. 376383, Nov. 1989. P. Chowdhuri, “Analysis of lightning-induced voltages on overhead lines,” IEEE Trans. Power Delivety, vol. 4, pp. 479492, Jan. 1989. C. Bruce and R. M. Golde, “The lightning discharge,” J. Inst. Elec. Eng., vol. 88, pp. 487-520, 1941. M. A. Uman and D. K. McLain, “Magnetic field of lightning return stroke,” J . Geophysicul Res., vol. 74, pp. 6899-6910, Dec. 1969. M. J. Master, M. A. Uman, Y. T. Lin and R. B. Standler, “Calculations of lightning return stroke electric and magnetic fields above ground,” J. Geophysical Res., vol. 86, pp. 12 127-12 132, Dec. 1981. F. Heidler, “Traveling current source model for LEMP calculation,” in Proc. 6th Inr. Symp. Tech. Exhibition o11 Electromugn. Conlptrt.. Zurich, Switzerland, Mar, 1985, pp. 157-162. C. A. Nucci, C. Mazzetti, F. Rachidi, and M. lanoz, “On lightning return stroke models for LEMP calculations,” in Proc. 19th Int. Cot$ Lightning Protection, Graz, Apr. 1988, pp. 463470. C. A. Nucci and F. Rachidi, “Experimental validation of a modification to the transmission line model for LEMP calculations,” in Proc. 8th Int. Sjmp. Tech. Exhibition on Electromagri. Comput.. Zurich, Switzerland, Mar. 1989, pp. 389-394. I221 K. Berger, R. B. Anderson, and H. Kroninger, ”Parameters of lightning flashes,” Electra. no. 41, pp. 23-37, 1975. (231 M. J. Master and M. A. Uman, “Transient electric and magnetic fields associated with establishing a finite electrostatic dipole,” Ant. J. Phy c...

no. 51, pp. 118-126, 1983. [24] A. Sommerfeld, “Uber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys., vol. 28, pp. 665, 1909. [251 A. Zeddam and P. Degauque, “Current and voltage induced on a telecommunication cable by a lightning stroke,’’ Lightning Electromgneric-, R, L, Gardner, Ed, Emisphere Publishers, 1990, pp, 377400. (261 V. Cooray and S. Lundquist, “Effects of propagation on the rise times and the initial peaks of the radiation fields from return strokes,” Radio Sci., vol. 18, pp. 409415, June 1983. [27] E. M. Thomson, P. J . Medelius, M. Rubinstein, M. A. Uman, J. Johnson, and J. W. Stone, “Horizontal electric fields from lightning return strokes,” J. Geophysical Res., vol. 93, pp. 2429-2441, Mar. 1988. 1281 V. P. Idone and R. E. Orville, “Lightning return stroke velocities in the thunderstorms research international program (TRIP),” J. Geophysical Res., vol. 87, pp. 49054915, June 1982. 1291 A. K. Agrawal, H. J. Price, and S. Gurbaxani, “Transient response of a multiconductor transmission line excited by a nonuniform electromagnetic field,” IEEE Trans. Electromagn. Compat., vol. EMC-22, pp. 119-129, May 1980. 1301 E. Vance, Coupling to Shielded Cuble.s.\quad New York, Wiley, 1978. 1311 F. Rachidi, M. Ianoz, and C. A. Nucci, “On the inclusion of loss in time-domain solutions of field-to-transmission line coupling,” in Nuclear Electromagn. Meeting 1990, Albuquerque, NM, May 1990. 1321 C. A. Nucci, C. Mazzetti, F. Rachidi, and M. Ianoz, “Analyse du champ ClectromagnCtiquedfi B une dtharge de foudre dans les domaines temporel et friquentiel”, Anrzales des n’eR’ecommunications, vol. 43, no. 11-12, pp. 625-637, 1988. 1331 C. A. Nucci, G. Diendorfer, M. A. Uman, F. Rachidi, M. Ianoz, and C. Mazzetti, “Lightning return stroke models with specified channel-base current: A review and comparison,” J . Geophysical Res., vol. 95, pp. 20 395-20 408, 1990. 1341 E. Cinieri and A. Fumi, “Sulle sovratensioni indotte dal fulmine nelle linee elettriche di energia,” L’Energiu Elettrica, vol. LV, no. 9, 1978. [35] G. Diendorfer and M. A. Uman, “An improved return stroke current model,” J. Geophysical Res., vol. 95, pp. 13 621-13 644, Aug. 1990. 1361 M. Ianoz, C. A. Nucci, and F. M. Tesche, “Transmission line theory for field-to-transmission line coupling calculations,” E1ectromagnetic.s. vol. 8, no. 2 4 , pp. 171-211, 1988. [37] C. D. Taylor, R. S. Satterwhite, and C. W. Harrison, “The response of a terminated two-wire transmission line excited by a nonuniform electromagnetic field,” IEEE Trans. Antennas and Propagat., vol. AP- 13, pp. 987-989, 1965. [38] P. Chowdhuri, “Parametric effects on the induced voltages on overhead lines by lightning strokes to nearby ground,” IEEE Trans. Power Deliven, vol. 4, pp. 1 185-1 194, Apr. 1989.

-

Carlo Alberta Nucci (M’91) was born in Bologna, Italy, on October 21, 1956. He received the degree in electrical engineering in 1981 from the University of Bologna In 1982, he joined the LJniversity of Bologna a\ a Re\earcher in the Power Electncal Engineering Imtitute, where he is now Associate Professor of Power System\ His research interests concern lightning and nuclear EMP impact on power lines, power system? simulation, and the study of power components including medium voltage capacitors dnd traction batteries He I $ author or coauthor of about 30 scientific papers presented at internatlonal conferences and publi\hed in reviewed journals

Farhad Rachidi was born in Geneva, Switzerland, in 1962. He received the M.S. degree in electrical engineering and the Ph.D. degree from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1986 and 1991, respectively. He is currently working at the Power Network Laboratory at this institute. His research interests concern lightning and EMP interactions with power lines. He is the author or coauthor of more than 15 scientific papers published in reviewed journals and presented at international conferences.

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Carlo Mazzetti (M’90) was born in Rome, Italy, Michel V. Ianoz (SM’85) was born in 1936. He in 1943. He received the Dr. Ing. degree in electrireceived the B.S. degree in electrical engineering cal engineering from the University of Rome “La from the Polytechnical School of Bucarest, RoSapienza,” Italy, in 1967. mania, and the Ph.D. degree from the Moscow In 1967 he joined the University of Rome, first as University, Moscow, U.S.S.R., in 1968. a Scientist in High-Voltage (HV) Problems and then He worked on magnetic field calculations for paras an Assistant Professor of Advanced Electrical ticle accelerators and focusing devices at the Joint Engineering. In 1974 he became Associate Professor Institute for Nuclear Research, Dubna, U.S.S.R., in High-Voltage Engineering, and since 1986 he is and at the European Center for Nuclear Research Full Professor in the same field. Since 1967 his main (CERN), Geneva. Since 1975 he joined the Power interests have been the HV transient analysis and Network Laboratorv of the Swiss Institute of Techmeasurements with particular reference to lightning effects and diagnostic tests nology, Lausanne Switzerland, where he is presently teaching EMC as a Professor of the Electrical Department and is engaged in research activities on electrical insulation. His activities include design and test experience on fluid insulations and studies on lightning electromagnetic fields. From 1986 concerning the calculation of electromagnetic fields, transient phenomena, to 1989 he was Director of the Electrical Engineering Department of the lightning, and EMP effects on power and telecommunication networks. He University of Rome “La Sapienza,” and from March 1989 Director of the HV is coauthor of a book on high-voltage engineering, editor of a book on electromagnetic compatibility, and author or coauthor of about 80 scientific Group of the Italian National Research Council. He is author and coauthor of more than 50 scientific papers. papers. Dr. Mazzetti is a member of the Italian Electrotechnical Committee (CEV81) Dr. Ianoz is chairman of the Swiss National Committee of the URSI, memand the International Electrotechnical Commission (1EC/81) on Lightning ber of the Study Committee 36 “Perturbations” of CIGRE, and of the WGlO of Protection. the TC77 (EMC) of the International Electrotechnical Commission (IEC). He ON ELECTROMAGNETIC is also an Associate Editor of the IEEE TRANSACTIONS COMPATIBILITY.

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