Voltage surges induced on overhead linesby lightning strokesP. Chowdhuri, D.Eng., C.Eng., M.I.E.E., and Eric T. B. Gross, D.Sc, C.Eng., F.I.E.E.

Synopsis

The paper analyses the voltage induced on an overhead line by the electromagnetic effects of the returnstroke of lightning. The effects of a rectangular return-stroke current as well as currents having linearlyrising front are analysed, assuming uniform charge density along the stroke channel and return-strokevelocity as a constant parameter. The analysis, which corroborates recent field data, shows that the inducedvoltage is not entirely a travelling-wave phenomenon. Contrary to the previous studies, this study showsthat the waveshape of the induced voltage can be bipolar. Furthermore, the magnitude as well as thefront time of the return-stroke current may be predicted approximately from the oscillogram of an inducedvoltage. The magnitudes of voltages induced by indirect strokes can exceed the basic impulse insulationlevel of high-voltage systems.

List of symbolsA = vector potentialc = velocity of electromagnetic waves in free space

Ei = inducing electric-field vectorG(x; x') = Green's function

h = height of overhead line above groundhc = height of cloud above ground

/ = current/0 = peak currentJ = current density vectorL = inductance per unit length of overhead line

££ = symbol of Laplace transformr = distance of field point from origin

r' = distance of source point from origins = operator of Laplace transform/ = time

tQ = time needed for the effect of return stroke to reacha point on the line

tj — front duration of return-stroke currentu(t) = unit step function

v = velocity of return strokev' = volume of source

v = Laplace transform of induced voltageV — induced voltageVj = inducing voltagex = distance of field point on the line from the point

of least distance from strokex' = distance of source point on the line from the point

of least distance from strokey0 = distance of point of least distance on line from

, strokez — distance in the vertical direction under line from

groundz — distance inside return stroke from point of strikea — rate of rise of return-stroke currentj8 = ratio of velocity of return stroke to velocity of

light in free space8(t) — Dirac delta function

V = gradient€ = permittivity of a material

e0 == permittivity of free spacefju = permeability of a material

[x0 = permeability of free spacep = volume density of electric charge<f) ~ scalar potential

1 IntroductionTransient high voltages caused by lightning are con-

sidered to be the chief source of disturbance to overheadpower lines. Such transient voltages can appear on an over-

Paper 5422 P, first received 26th May and in revised form lOth August1967Dr. Chowdhuri is with GE Co., Erie, Pa., USA, and was formerly withRensselaer Polytechnic Institute, Troy, NY, USA. Dr. Gross is PhilipSporn Professor at Rensselaer Polytechnic Institute, Troy, NY, USA

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967

head line either by direct hit or by induction from a nearbylightning stroke.

Much work has been done on the theories of protection oftransmission lines against direct strokes. In contrast, the studyof the mechanism of induced strokes has generally beenneglected; although it has been apprehended that inducedstrokes could be dangerous to transmission systems workingat 50kV and below. Recent field studies in Great Britain,1

Japan2 and the Scandinavian countries3"5 have generatedrenewed interest in the induced strokes. Apart from theimportance of protection against induced strokes, the studyof the mechanism of induced strokes is appropriate from anentirely different point of view.

Although the mechanism of lightning has been studiedfrom the early days by the Greeks, revived by Franklin andbrought to age by Schonland, much vital information con-cerning the lightning stroke is still unknown. During the lastforty years, many excellent instruments have been developed,and many field studies have been instituted, to study thecharacteristics of lightning strokes; but lightning rarelystrikes where the researcher wants it to. Consequently, ittakes a long time to obtain fruitful field data.

Transmission lines pass through many miles of territoryin which there is great likelihood of lightning activities. Alightning stroke, even at a distance of 5-IOkm from the line,would induce a voltage along the line large enough to berecorded by appropriate instruments, such as cathode-rayoscilloscopes, connected to the line. Moreover, as the trans-mission line itself acts as an antenna, the instrument wouldrecord lightning activities far removed from the instrumentitself but in the vicinity of the line. However, it would requireknowledge of the mechanism of induced strokes to interpretthe recordings in order to study the characteristics of thelightning stroke itself. Therefore, it is appropriate to studyanalytically the mechanism of induced strokes from thispoint of view, apart from the problem of protection againstsystem outages.

Accordingly, this study is entirely analytical in nature, thepurpose being to bring to light some characteristics of thelightning stroke as well as to estimate the magnitude of thevoltage on a power line induced by a nearby stroke of knownparameters.

2 Critique of previous studiesPrevious studies on induced voltages may broadly be

classified as analyses based on

(a) electrostatic effect of the cloud6"8

(Jb) electrostatic effect of the stepped leader9

(c) magnetic effect of the return stroke10

(d) electromagnetic effects of the return stroke.""16

However, it has been shown that the electromagnetic effectof only the return stroke is of any practical significance.1' >l3~15

It is difficult to compare the various studies made on theelectromagnetic effects of the return stroke, because the

1899

assumed models of the return-stroke structure were different.Wagner and McCann" assumed a uniform distribution of

• charge density along the channel and a constant velocity ofthe return stroke; Lundholm13 and Rusck14 also assumed auniform distribution of charge but a return-stroke velocitywhich is dependent upon the magnitude of the return-strokecurrent; Papet-Lepine15 assumed densities of charge andcurrent to be varying with the height of the channel; Ohwa's16

assumptions were similar to Rusck's except that he furtherassumed an excessive charge concentration at the tip of thestepped leader.19

fn spite of the divergence in assumptions, several commonconclusions can be drawn from these studies: (i) the magneticeffect of the return-stroke current is considerably less thanthe electrostatic effect of the charge in the return stroke, andtherefore the net effect of the return stroke is still electrostatic;(ii) the voltage on an overhead line induced by lightning isentirely a travelling wave in character.

As recent field data2'32 cast doubt on the conclusion thatthe induced voltage is entirely a travelling-wave phenomenon,renewed interest in such a study is worthwhile.

3.3 Mathematical formulations

A rectangular system of co-ordinates in space waschosen so that the origin of the system is the point where thelightning strikes the surface of the earth. The line conductoris assumed to be located at a distance y0 metres from theorigin, having a mean height above ground of h metres andrunning along the x direction (Fig. 1).

3 Method of analysis3.1 Nature of induced voltage

The voltage induced on an overhead line has fourcomponents.

(a) The charged cloud above the line induces bound chargeson the line while the line itself is held electrostatically atground potential by the neutral grounding of connectedtransformers and by leakage over the insulators. When thecloud is partially or fully discharged, these bound charges arereleased and travel in both directions on the line, giving riseto travelling voltage and current waves.

(b) The charges lowered by the. stepped leader furtherinduce charges on. the line. When the stepped leader isneutralised by the return stroke, the bound charges on theline are released and thus produce travelling waves similarto those caused by the cloud.

(c) The charges in the return stroke induce an electrostaticfield in the vicinity of the line and hence an induced voltageon it.

id) The rate of change of current in the return strokeproduces a magnetically induced voltage on the line.

If the lightning has subsequent strokes, the subsequentcomponents of the induced voltage will be similar to one orthe other of the four components discussed above.

i

3.2 Basic assumptions

(a) The components of the induced voltage which aregenerated by the release of the bound charges were neg-lected,1 l>l3~15 and only the electrostatic and the magneticcomponents induced by the return stroke were considered.

(b) The charge distribution along the leader stroke wasassumed to be uniform.21 Although many other types18"20

of distribution have been proposed from time to time, anyspecific form of distribution is hardly plausible.

(c) The shape of the return-stroke current was assumed tobe rectangular. The result with rectangular current wave canbe transformed to that with currents of any other waveshapeby the convolution integral (Duhamel's theorem).2829 Thecomputations were extended to current waves having linearlyrising front, in order to study the.effect of the wavefront ofthe current on the induced voltage wave.

(d) The velocity of the return stroke was assumed to be aconstant parameter. Although a few previous studies13'14'22

suggested that the velocity of the return stroke is a functionof the return-stroke current, such an a priori assumption wasconsidered to be a limitation to study the general charac-teristics of lightning. Since one purpose of this study was toanalyse the structure of the lightning channel, it was thoughtto be of advantage to be able to take the velocity of thereturn stroke as a parameter and to study its effects on theinduced voltage.

(e) It was assumed that the conductor is loss-free and thatthe earth is perfectly conducting.1900

stroke

line conductor

.81-

Fig. 1Co-ordinate system of line conductor and stroke

The electromagnetic field associated with the charge andthe current in the lightning stroke at any point in space is23

( i )

where <j> is the inducing scalar potential created by the chargeof the lightning stroke and A is the inducing vector potentialcreated by the current of the lightning stroke.

The potential difference between two points in space is theline integral of this electromagnetic field between the twopoints in question. As this electromagnetic field is not curlfree, the line integral of this field between two points isdependent on the path taken from one point to the other.Fn this analysis, such a path was taken to be the verticaldistance between the point in question on the line conductorand the ground plane.

One would indeed measure the inducing voltage only in theabsence of any conductor. It will be shown later that theinducing voltage at different distances along the length of apower line will be different. With the power line being agood conductor of electricity, these differences of voltageswill tend to be equalised by the flow of currents. Therefore,the actual voltage measured between a point on a power lineand the point on ground vertically below it will be differentfrom the inducing voltage. This voltage which is actuallymeasured on the power line is defined as the induced voltage.The calculation of this voltage is the primary objective ofour analysis. However, the importance of the inducing voltageshould not be minimised, because it is the inducing voltagewhich is the force function or the source of the inducedvoltage. Therefore, the analysis may be broadly divided intotwo parts: the evaluation of the inducing voltage and theevaluation of the induced voltage. Throughout the analysisthe MKS system of units has been employed.

PROC. /EE, Vol. 114, No. 12, DECEMBER 1967

3.3.1 Evaluation of inducing voltage

The electromagnetic inducing potentials of eqn. 1 canbe deduced from Maxwell's field equations

t -

dv' (3)

where r, r' = field and source points, respectively,p = charge density/ = current density

and the integration is taken over a volume enclosing thesource.

The equations for the electromagnetic potentials (knownas the retarded potentials) indicate that at a given field point rand a given time /, the potentials are determined by the chargeand current which existed at the source point r' at an earliertime; the difference in time is the time required to travel thedistance between the source point and the field point with avelocity v = l/\/(€/u,).

Fn our problem, the source is the lightning stroke, and thefield point is any point along the power line. The power linebeing in air, the effect of the lightning stroke travels towardsthe line with a velocity c = l/v^o/^o) - 3 x 108m/s. Fromthe distribution of charge and current in the lightning stroke,the retarded electromagnetic potentials (f> and A can bededuced. Knowing <f> and A, in turn, the inducing electricfield Ej can be computed from eqn. 1. The line integral ofEj will then provide the inducing voltage at the power line

= - \ E, dz (4)

The inducing voltage will act on each point along the lengthof the power line.

3.3.2 Evaluation of induced voltage

Neglecting losses, a power line may be represented asconsisting of distributed series inductance and distributedshunt capacitance. The effect of the inducing voltage will thenbe equivalent to connecting a voltage source (generator)along 6ach point of the line (Fig. 2). The partial differentialequations for such a configuration will be

-}Z*X = L}' (5)

— — ox = C — (V — V)ox (6)^x lit '

Eliminating /, the voltage equation can be written as~b2V 1 ~b2 V 1 <)2 V-

. i — pfa f\ f-j\

where

and

which is a known function of x and / from eqn. 4.

•7-7- / / / / ' / / / / ' — y

-. Sx •Fig. 2Equivalent circuit of transmission line with inducing voltage

Eqn. 7 is an inhomogeneous wave equation in one dimen-sion for the induced voltage along the power line. Its solutioncan be obtained by assuming F(x, /) to be a superposition ofimpulses which involves the definition of Green's function.24-25

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967

To obtain the voltage caused by a distributed source F(JC),we calculate the effects of each elementary portion of sourceand add them all. If G(x; x') is the voltage at a point x alongthe power line caused by a unit impulse source at the sourcepoint x', the voltage at x caused by a source distributionF(x') is the integral of G(x; x')F(x') over the whole domainof x' occupied by the source, provided that F(x') is a piecewisecontinuous function in the domain a < x < b\

i.e. V(x) = \ G{x; x')F(x')dx' • • • (8)

The function G(JC; X'), called Green's function, is thereforea solution for a case which is homogeneous everywhereexcept at one point. Green's function G(x; x') of eqn. 7 hasthe following properties:

(a) G(x; x' + 0) - G(x; x' - 0) = 0

x,+0 \dxJx,_0

(c) G{x;x') satisfies the homogeneous equation everywherein the domain, except at the point x = x'

(d) G(x; x') satisfies the prescribed homogeneous boundaryconditions.

4 Doubly infinite single-conductor lineA doubly infinite single-conductor line can be repre-

sented in practice by a line whose terminals are sufficientlydistant from the point of strike of lightning that reflectionsof the induced voltage at the terminals can be neglected. Thissystem is relatively simple to analyse, and, furthermore, itgives an insight into the physical nature of the phenomena.

4.1 Inducing voltage: rectangular current in returnstrokeThe assumption of rectangular waveshape for the

current in the return stroke keeps the analysis general, sothat for any particular waveshape of current the analysis canbe transformed to the desired case by Duhamel's theorem.Under the assumption of rectangular waveshape, the chargealong the leader stroke is uniformly distributed as is also thecurrent in the return stroke.

As the height h of the line conductor is small compared withthe height of the cloud (i.e. the length of the lightning stroke),the inducing electric field below the line conductor can beassumed to be constant and equal to that on the groundsurface.

Then, from eqns. 1 and 4,

(9)

where <f> and A are obtained from eqns. 2 and 3.Because of the retardation effect, the electromagnetic

potentials <j> and A at the field point r and at time / mustbe originated at the source point r' at an earlier time

t' •= t — It is convenient to choose the instant at

which the return stroke starts from the ground as the originof time, i.e. t = 0.

ff the velocity v of the return stroke is a fraction of thevelocity of light in free space (v = jSc), the disturbance of anelement of charge or current at a height z' above the groundand along the return stroke will be felt at a field point r onthe ground at time

/ = - +v (10)

Therefore, the earliest time at which the disturbance fromthe return stroke will reach a field point would be

to = (11)c

In other words, the inducing voltage at a field point remainszero till / = /0. Hence the inducing voltage is a sectionedfunction

Vt, = fax, t)u(t - t0)where u(t — t0) = shifted unit step function.

(12)

1901

The continuous function IJJ(X, /) can be evaluated from theassumed structure of the lightning stroke14-17 and is given by

> 0 z a . /{D2J1.2 , / . - P2>2}. . . . (13)

where IQ = return-stroke currentjS = ratio of return-stroke velocity and velocity of

light in free spacer — distance of point x on line from point of strike.

4.2 Induced voltage: rectangular current in returnstrokeThe partial differential equation for the induced voltage

on the line conductor is

7)x2 c2 o/2 c2 ot2

whose initial and boundary conditions are

(a) V is bounded for x -» ± oo(b) V(x, 0) = V'(x, 0) = 0

From eqn. 12

= <//'(*, t)u(t - /0) + 20'(JC, t)B(t - /

S ' ( ' - ' o ) • • • • (14)

where the primes represent differentiation with respect to t,and S(/ — /0) is the Dirac delta function.

Taking the Laplace transform of eqn. 7 and considering theinitial conditions we get

s2

—s2

where f(x, s) = —=-.cL , t + >0)} •

(15)

(16)

To solve this equation, we find Green's function G(x; x', s) of

:;x', s)ox2 tn2G(x; x , s) = 0 where m = - (17)

c

From the boundary conditions of the problem and fromthe properties of Green's function, we solve for G(x; x', s) as

cexp/ sjx' - x)\

Giix;x',s) = - I J

cexpG2(x; x', s) = —

Is

s{x' -

for x <

for x > x'

(18)

' A

3 _ \

Fig. 3Analytical representation of linearly rising return-stroke currenta Current with constant tail b Current with dropping tail/0 = at/

1902

By applying eqn. 8

Cfx(s 1vix, s) = - — exp < - ix' - x) >/ix', s)dx'

2S J - oo {C J

- ! . f ° ° / - ^ r ' - \ \ f f ' w ' _ -2 ^ J x y c )

. . . . (19)

Putting the value of fix, s) from eqn. 16, taking the inverseLaplace transform and considering the properties of the unitstep function as well as the Dirac delta function,

Vix, t) = iVn + Vi2 + V2l + V22)uit - ,0) . (20)

where30/0/t(l - ft2)

ft2(c/ - x)2 + y2 j8(c/ - x)

{ct - x)x - y2

1/2 • (21)

^12 =— X

• • • (22)

30/oW - jS2) P(ct + x)

l/2 • (23)

- 3 0 / Q A

c/ 4-• • (24)

(25)

4.3 Return-stroke current of linearly rising frontOnce the induced voltage on an overhead line caused

by a rectangular current wave in the return stroke is known,

-160

-140

-120

> "100

2-80o

-20

• 20

.tr

O 5 10 15. 20 25 30 35 40time, ps;:

Fig. 4Induced voltage on doubly infinite single-conductor line by linearlyrising return-strokeIf = 0 - 5 n s , 0 = O l , y 0 = 100m, /0 = lOkA, h = 10m, hr. = 3km

* = (00mx = 1000mx = 5000m

PKOC. /E£, Vol. 114, No. 12, DECEMBER 1967

the corresponding induced voltage caused by any other formof current in the return stroke can be computed by theapplication of Duhamel's theorem,28-29 which can be statedas follows: If the current in the return stroke /(/) is ofexponential order and is a continuous function of t, and ifits first derivative with respect to t is sectionally continuous,the induced voltage caused by this current is

V(t) = 1(0) V^t) -jHt - r)V0{r)dr (26)

-152

-V36

-1-20

-K>4

CT-0682

-O56

-040

-024

-0-08

0+OO8

1 115 20 25 30 35 40 45 50 55

time, >JS

Fig. 5Induced voltage on doubly infinite single-conductor line by linearlyrising return-stroke current// = 0-5ns, P = O ' l j j = 5000m,/0 = lOkA, h = 10m, he = 3km

x = 100mx = 1000mx = 5000m

-120

-100

- 80

en5-60

- 4 0

- 20

+ 20

403010 2 0

time, MSFig. 6Induced voltage on doubly infinite single-conductor line by linearlyrising return-stroke currenttf = 0-5|XS, P = 0-3, y0 = 100m, /0 = lOkA, h = 10m, hc = 3km

x = 100mx •= 1000mx = 5000m

PROC. IEE, Vol. 11:4, No. 12, DECEMBER 1967

where Ko(/) is the induced voltage caused by the unit-step-function return-stroke current.

Oscillographic measurements of current in lightningstrokes30*31 have indicated innumerable forms which canhardly be fitted into simple mathematical expressions. How-ever, any waveshape can be represented by elemental straightlines, and thus a linearly rising current wave is not withoutpractical significance.

A linearly rising current wave is the simplest type of wavewith which to study the effect of the front of current waveon the induced voltages. Such a wave can be analyticallyconstructed by algebraically adding two linearly rising func-tions, one being shifted by a time equal to the front time ofthe desired wave (Fig. 3). The two functions are analysedseparately, and the solutions are superposed to get theanalytical expression for the induced voltage. The systembeing assumed linear, such procedure is permissible. In otherwords

= at - *{t - tf)u(t - tf)

= /.(') + hit)(27)

If K,(/) is the component of the induced voltage due to/,(/), and if V2(t) is that due to I2(t),

V(t) = (28)

-200

-120

-40

0>.. +40

> +120

iE +200

+280

+360

+44015 25 30 35

time, .us40 4 5 50 55

Fig. 7Induced voltage on doubly infinite single-conductor line by linearlyrising return-stroke currenttf = 0-5|xs, P = 0-3, y0 = 5000m, l0 = lOkA, h = 10m, /)„ = 3km

x = 100mx = 1000m

+ 3015 20 25 30 35 40time JJS

Fig. 8Induced voltage on doubly infinite single-conductor line by linearlyrising return-stroke current

100m, /o = lOkA, h = 10m, hc = 3kmtf = 50ns , P = 0-1,x = IOOmx = 1000mx = 5000m

1903

VQ(O is obtained by putting 70 = 1 in the expressions for thecomponent voltages of eqn. 20, so that

'o2{ + V22)u(t - t0) . (29)

Where Vu, Vx2, V2l and V21 are given by eqns. 21-24.

Applying eqn. 26 to /j(0

*o JoK21(r) + V22(r)}u(r - to)dr

. . . . (30)

K2t(r) + V22(r)}dr . (31)

After integrating and simplifying

= 0 for t < t0

Similarly,

'/j8cl -

. . . . . (32)

j82)

2j32c(/ -O- jS 2 ) 2 , 2

- tf)2 + (1 - j

. (33)

= 0 for / < t0 +

Finally, the induced voltage V(t) is obtained by addingeqns. 32 and 33 as shown in eqn. 28.

The variation of the induced voltage V with time alongthe power line is shown in Figs. 4-11. It should be borne inmind that the origin of the time co-ordinate is at the instant

-1.44

-1-28

-1-12

-0-96

-060

-064

-O48

-032

-0-16

15 25 35time.>is

45 55

Fig. 9Induced voltage on doubly infinite single-conductor line by linearlyrising return-stroke currenttf - 5-0us, 3 -= 0-1, y0 = 5000m, /0 = lOkA, h = 10m, hc = 3km

x = 100mx =• 1000mx = 5000m

1904

when the return stroke starts from the ground. This producesa delay, equal to the time of travel, in the time taken for theelectromagnetic effects of the return stroke to reach a pointon the power line. This is shown in the curves by a shiftin the time axis, which is different for different field points.Calculations were made for a return-stroke current /0 oflOkA, and for a height of line above ground h of 10m. As

-16

-12>

oio<->

•oV

I o+4

48

+12

4-16

\

i l !M-1/

/

V10 25 30 35 4 015 2O

time,jJSFig. 10Induced voltage on doubly infinite single-conductor line by linearlyrising return-stroke currenttt = 5 0txs, 3 = 0-3, y0 = 100m, /0 = lOkA, It = 10m, hc = 3km

x = 100mx = 1000mx = 5000m

the induced voltage is directly proportional to /0 and h, itcan be calculated easily for other values of /0 and h from thegiven curves. The velocity of the return stroke v(= jSc), thefront time tf of the return-stroke current, the distance of thestroke from the line y0 and the field point along the line xwere taken as parameters. A cloud height hc of 3km was alsoassumed.

-150

-100

-50

o

•50

> +150T34)(J

^ + 200

+ 250

+300

•350

' +40015 20 25 30 35 40 45 50 55

time.ps

Fig. 11Induced voltage on doubly infinite single-conductor line by linearlyrising return-stroke currenttf = 50p is , 3 = 0-3, y0 = 5000m, /0 = lOkA, h -= 10m, hr = 3km

x = 100mx = 1000mx = 5000m

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967

5 Discussion5.1 Analysis of waveshape and comparison with

field-test dataA study of the curves for the induced voltage would

show that most of the voltages start with a spike; then theyreverse in polarity, rise slowly to the reversed, peak valueand finally decay at a slower rate. The voltage spikes aremore pronounced for lower return-stroke velocities andnearer field-point locations. In other words, voltages arebipolar with a few exceptions. This is the cardinal differencefrom the previous analytical studies. The first peak of voltageis caused by the rate of change of the return-stroke current.After the front effect of the current has subsided, the effectof the charge in the return stroke is evidenced by the reversingof polarity of the induced-voltage wave. The induced voltagebecomes unipolar when either the electrostatic effect or themagnetic effect solely predominates. The voltages of Figs. 5and 9 are affected primarily by the return-stroke current,whereas the voltages at the field point x = 5000m of Figs. 7and 11 are predominantly affected by the charge in the returnstroke. Whether the effect of the charge or the current inthe return stroke is predominant can be estimated from thefield components caused by the inducing scalar and vectorpotentials.

-16-4

o20 40

time, ps60 80.

+ 32-8

+ 49 2

Fig. 12Measured induced voltage on 500 k V experimental line: example I

Several field studies have been made in the past to recordinduced voltages on overhead lines by lightning.33"38 Mostof these data were taken on long time scales in order torecord voltages induced by subsequent strokes; as a result,the details of a single stroke were lost. Those data37-38 whichwere taken on shorter time scales show the prevalence ofbipolar characteristics of the induced voltages as predictedby the analysis.

A few oscillograms of induced voltages recorded on a500kV experimental line are shown in Figs. 12—15 for aqualitative comparison with the analytical curves. These arefour of many oscillograms recorded during the lightning

time, its40 60 80

- 3 2 6

-16-4>•* 0

&2 +16-45

+32-8

Fig.13Measured induced voltage on 500k V experimental line: example 2

Fig. 14Measured induced voltage on 500k V experimental line: example 3

PROC. IEE, Vol. 114, No. 12, DECEMBER 196790 P36

seasons of 1960-1962 on a I3mile-long, 2-bundle, 3-phase,single-circuit line with two ground wires. This line wasequipped with automatically operated cathode-ray oscillo-scopes at about 5000ft from one end of the line. The oscillo-grams were taken when the line was de-energised andearthed at both ends. A comparison of these oscillogramswith the analytical curves of Figs. 4-11 shows fairly reasonablesimilarities between the waveshapes. For instance, Fig. 12resembles the voltage at x = 5000m of Fig. 6; Fig. 13resembles the voltage at x — 1000m of Fig. 8; Fig. 14resembles the voltages in Figs. 5 and 9; Fig. 15 resemblesthe voltage at x = 5000m of Fig. 8, although a better fitwas found for x — 5000m, y0 = 1000m, j8 = 0-3, tf = 5/JLS.A quantitative comparison is not possible because of theunknown parameters as well as the difference in lineconfigurations.

timers40 60

-16-4,

O

+ 16-4

«T +32-8oi

1+49-2>+ 65-6

Fig.15Measured induced voltage on 500k V experimental line: example 4

5.2 Dual nature of induced voltageA close study of the analytical curves of induced volt-

ages would show that the induced voltage does not propagatealong the power line unattenuated and undistorted as wouldhave been expected for a travelling wave on a lossless lineabove a perfect earth. A closer look at these curves wouldreveal that sometimes the amplitude of the voltage at firstincreases with x along the line before it diminishes again.This is better displayed in Tables 1-3. It is interesting tonote that on several occasions apparatus connected to pointson power lines nearer a lightning stroke remained undamaged,whereas those connected to more distant points weredamaged.2

This increase in the induced voltage with distance alongthe line was first noticed by Szpor for the electrostatic com-ponent.12 However, when the electrostatic and magneticcomponents of Szpor's induced voltage are added together,an induced voltage results which diminishes with distance.Lundholm also found that the peak value of the inducedvoltage was higher at distant points than at the point closestto the stroke.13 However, he justified his results by statingthat, when losses are taken into account, the voltages atgreater distances will have more attenuation than those nearthe point x = 0, and therefore the overall attenuation will beprogressive along the line.

Table 1PEAK INDUCED VOLTAGE FOR j8 = O ' l

metres

100

500

1000

5000

'/|J.S

0-51050

0-51050

0-51050

0-5105 0

0

-346-30-256-95— 84•13

-86-47-77-25- 4 4 1 2

-37-62-35-27-24-26

-1 -56— 1-54-1-37

100

-375-43-266-68-83-24

-87-29-77-72- 4 4 0 7

-37-69— 35 31-24-24

— 1-56- 1 - 5 4-1-37

x metres

500

-475-96- 2 9 6 - II

- 7 4 - 7 1

-100-51- 8 4 - 6 4- 4 2 - 7 0

— 39-16- 3 6 1 7- 2 3 - 7 6

— 1-56— 1 - 53— 1-36

1000

-467 -40-277-43- 6 1 1 5

-113-70— 89-17- 3 8 - 7 3

- 4 1 - 5 9- 3 7 - 2 2- 2 2 - 2 6

— 1-54- 1 - 5 2— 1-34

5000

-128-91-67-46+ 3-55

— 51-40-33-76-9-47

-23-96-18-14

- 7 0 4

- 1 1 4-1 -11-0-90

Voltages are expressed in kilovolts at (OkA, h = tOm, hr. = 3km

1905

Table 2

PEAK. I N D U C E D V O L T A G E FOR / 3 - 0 * 2

metres

100

500

1000

5000

'/(AS

0-5105 0

0-5105 0

0-5105 0

0-5105 0

0

-161-98- 114-97

-31-48

-41-41-36-72-19-36

-17-91-16-72- 1 1 0 1

-0-60-0-59-0-50

. , 100

-175-90-119-34

-3.0-89

-41-80-36-94—19-33

-17-94-16-74-11 00

-0-60-0-59-0-50

x metres

500

-223-23-132-27• 25-82

-4802-40-14-18-55

-18-60-1711

-10-74

-0-60-0-59-0-50

1000

-217-15-121-98-18-64

-53-93• -41-99

-16-44

-19-62-17-49-9-95

-0-59-0-57-0-49

5000

-43-30•1-25-22+ 14-36

—19-56-12-48— 1 39

- 9 1 1— 6-81+ 2-39

+0-41+0-40+0-39

Voltages are expressed in kilovolts at l0 = lOkA, h = 10m, hr. = 3km

Table 3

PEAK INDUCED VOLTAGE FOR j6 = 0 ' 3

metres

100

500

1000

5000

|i.S

0-5105 0

0-5105 0

0-5105 0

0-5105 0

0

-97-07-65-62-16-23

— 25-61-22-44-10-70

-10-94- 1 0 1 5— 6-25

+0-31+0-31+0-29

100

-105-53-67-95-15-78

-25-84-22-57-10-67

-10-96- 1 0 1 6

-6-24

+ 0-31+0-31+0-29

x metres

500

-133-62-74-34-11-99

-29-55-24-42- 1 0 0 4

-11-32-10-34- 6 0 4

-1-0-31+ 0-31+0-29

1000

— 127-50-66-33-6-89

-32-74-25-17-8-47

-11-79-10-43-5-46

+0-32+ 0-32+0-30

5000

+ 25-32+ 24-39+ 15-78

-6-24-3-48+ 2-07

-2-88+ 2-36+ 2-19

+0-41-1-0-41+0-40

Voltages are expressed in kilovolts at /() — 10kA, h = 10m, /;,. •= 3km

Our study shows that, under certain conditions, the overallattenuation of the induced voltage along the line is notmonotonically progressive. It is proposed that the mechanismof induced voltage on a power line is not purely a travelling-wave phenomenon. A simple physical picture of the phe-nomenon can be obtained by visualising the line to be con-nected at each point along its length to a voltage sourcethrough a switch. The first switch to be closed is at the pointnearest to the stroke (x — 0). The switches on either side ofthis point are then closed progressively. This would tend toproduce a standing-wave pattern along the line, especiallyfor small distances along the line from the point x = 0. Wecall this the near zone. At a distance (far zone) along the linewhere the influence of the inducing field is negligible, thephenomenon resembles a travelling-wave pattern. The region(intermediate zone) between the near and the far zones willshow the dual nature of the phenomenon. Because of thestanding-wave pattern in the near zone, maxima and minimaof induced voltages may be expected.

5.3 Effect of front time of return-stroke current

The front of the return-stroke current would determinethe influence of the magnetic component of the inducing fieldon the induced voltage; the steeper the front, the greaterwill this influence be. If the front is very steep, the magneticcomponent may be the deciding factor on the inducedvoltage. After a time period equal to the front duration ofthe current, the effect of the magnetic component will beconsiderably diminished if the tail of the current wave iseither constant or slowly falling. At the last stage of thereturn stroke, when the charge in the lightning channel has

1906

been considerably neutralised, the magnetic component againmay outweigh the electrostatic component. Many previousstudies have estimated the magnetic component to be lessthan 20% of the electrostatic component. The magneticcomponent of the inducing field being in the opposite directionto that of the electrostatic component for the greater part ofthe return-stroke duration, the combined effect of the returnstroke is still electrostatic, according to these studies, butdiminished in magnitude by the magnetic component. Thisassertion may be true for the intermediate part of the return-stroke duration; during the front time of the return-strokecurrent this assertion may lead to serious errors, especially ifthe front duration is short.

The computed curves of induced voltages as well asTables 1-3 show that the peak value of the negative loopdiminishes with increasing front time of the return-strokecurrent. The Tables also show that the effect of the magneticcomponent is predominant at points which are nearer to thepoint of strike and that the electrostatic component is moreeffective at large distances from the point of strike.

5.4 Prediction of return-stroke characteristics

Some of the characteristics of the return stroke maybe predicted from oscillograms of the induced voltage. Thisis especially true if the induced voltage is bipolar, so that theeffects of the charge and the current in the return stroke canbe separately studied.

For instance, our study shows that the front duration ofthe first loop is approximately the same as the front durationof the return-stroke current, especially if the front durationis not long. For the 5/xs front, several of the curves showedthat the peak occurred before this time. However, by notingthe sudden drop in the voltage, the front time can beestimated.

This method of prediction of front time failed in oneinstance (Fig. lOatx - 5000m). The duration of the first loopof the induced voltage was much smaller than the fronttime of the return-stroke current. Because of the longer fronttime, the electrostatic effect became predominant even beforethe front of the return-stroke was reached. For such cases,the front time can be predicted from oscillograms obtainedat shorter distances along the line.

As the peak of the second loop is not very sensitive to j3,the peak return-stroke current may be approximately pre-dicted by substituting / /( from the first loop), x, y0, hc and //in eqns. 32 and 33. It should be emphasised that /0 and t,-thus found are only approximate values, although it isbelieved that this would still give us much needed data onlightning.

5.5 Possibility of line flashover from indirect strokes

The curves of induced voltages in this study werecomputed on the basis of 10kA for the peak of the return-stroke current / 0 and for a height of conductor above groundh of 10m. Since the magnitude of the induced voltage isdirectly proportional to /0 and h, the basic impulse insulationlevel of many high-voltage systems may be exceeded forhigher /0 and /;, especially if the front time tf of the returnstroke is short. This is indicated by Figs. 4 and 6.

5.6 Multiconductor systems and imperfect earth

The present analysis was made with the assumptionsof a single-conductor line above a perfect earth. For multi-conductor systems, such as 3-phase single- or double-circuitlines, the mutual influence of the conductors has to be takeninto consideration. In general, the induced voltage on a con-ductor of a multiconductor system will be higher than thaton a single-conductor line of the same height. The effect ofground wires will be to reduce the magnitude of the inducedvoltage to some extent. The waveshapes of the inducedvoltages on a multiconductor system will be similar to thosefor the single-conductor line if the earth is perfect and if y0

is large compared with the separation distances between theconductors.

An imperfect earth would introduce both attenuation anddistortion of the induced voltages on a single-conductorsystem. Further distortion of the induced voltage will be

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967

introduced on a multiconductor system with imperfect earth.Some of these effects have been studied,17 and it is hopedthat the results will be presented in the future.

6 ConclusionAn analytical method is presented to predict not only

the voltage induced on an overhead line by lightning, butalso to examine -the characteristics of the lightning strokeitself. The analysis shows that(a) the induced voltage is not entirely a travelling-wave

phenomenon(b) the waveshape of the induced voltage can be bipolar(c) the magnitude and the front time of the return-stroke

current can be predicted approximately from the oscillo-gram of an induced voltage

(cl) high-voltage systems may not be immune to indirectstrokes.

7 AcknowledgmentsA study of voltages induced by lightning on an over-

head line was suggested by J. G. Anderson, Technical Directorof Project EHV, GE Co., Pittsfield, Mass. The authorsexpress their thanks to both J. G. Anderson and P. L.Lumnitzer, Assistant Vice President, Pennsylvania ElectricCo., for permission to publish the oscillograms taken fromtheir field study.

8 Referencest COLDE, K. H.: 'Lightning performance of British high-voltage

distribution systems', Proc. IEE, 1966, 113, (4), pp. 601-6102 UEIIARA, K., and OHWA, G.: 'Investigation of lightning damages on

distribution lines', IEEE Winter Power Meeting, 1967, paper31PP67-94

3 GREVE, A. G.: 'Overvoltage protection in low-voltage networks',AseaJ., 1961, 34, (3), pp. 34-38

4 GRUNNER-NIELSON, B., HYLTEN-CAVALLIUS, N., KONGSTAND, E.,NIELSEN-IIOJSLER, M. K., and RUSCK, s.: 'Lightning arresters in lowvoltage networks', The Danish Association of Electricity SupplyUndertakings and the Danish Association of Mutual Fire InsuranceCompanies, Copenhagen, 1957

5 RUSCK, s.: 'Lightning over-voltages and over-voltage protection inlow-voltage networks', AseaJ., 1958, 31, (6), pp. 75-81

6 WACNER, K. w.: 'Elektromagnetische Ausgleichsvorgiinge inFreileitungen und Kabeln\ Elektrotech. Z., 1911, 32, pp. 899-903,928-931,947-951

7 BEWLEV, L. v.: 'Traveling waves-due to lightning', Trans. Amer.hist. Elect. Engrs., 1929, 48, pp. 1050-1064

8 NORINDER, H.: 'Some recent tests on the investigation of inducedsurges', CIGRE\ Paris, Paper 303, 1939

9 GOI.DE, R. H.: 'Lightning surges on overhead distribution linescaused by indirect and direct lightning strokes', Trans. Amer.Inst. Elect. Engrs., 1954, 73, (Pt. Ill), pp. 437-446

10 AIGNER, v.: 'Induzierte Blitziiberspannungen und ihre Beziehungzum riickwartigen Oberschlag', Elektrotech. Z., 1935, 56, (18),pp.497-500

11 WAGNER, c. F., and MCCANN, c. D. : 'Induced voltage on transmissionlines', Trans. Amer. Inst. Elect. Engrs., 1942, 61, pp. 916-930

12 SZPOR, s.: 'A new theory of the induced overvoltages', CIGR£Paris, Paper 308, 1948

13 LUNDHOLM, R. : 'Induced overvoltage-surges on transmission linesand their bearing on the lightning performance at medium voltagenetworks', Chalmers Tek. Hogsk. Hamll., 1957, (188)

14 RUSCK, s.: 'Induced lightning overvoltages on power transmissionlines with special reference to the overvoltage protection of low-voltage networks', K. Tekn. Hogsk. Hamll., 1958, (120)

15 PAPET-LEPINE, J. : 'Contribution au calcul des surtensions induitespar la fondre dans les lignes de transport d'energie electrique',Rev. Gen. Elect., 1960, 69, pp. 591-600

16 OHWA, c : 'Study of induced lightning surges and their frequencyof occurrence', J. Inst. Elect. Engrs. Japan, 1964, 84, pp. 44-55

17 CHOWDHURI, P.: 'Voltage surges induced on overhead lines by•lightning strokes', D.Eng. dissertation, Rensselaer PolytechnicInstitute, June 1966

18 BRUCE, c. E. R., and GOLDE, R. H.: 'The lightning discharge',/. IEE,1941, 88, Pt. II, pp. 487-505

19 GRISCOM, s. B. : 'The prestrike theory and other effects in thelightning stroke', Trans. Amer. Inst. Elect. Engrs., 1958,77, Pt. Ill,pp. 919-933

20 WAGNER, c. F., and HILEMAN, A. R.: 'The lightning strokes', ibid.,1958, 77, Pt. I l l , pp. 229-240

21 SCHONLAND, B. F. j . , HODGES, D. B., and COLLENS, H.: 'Progressivelightning V-a comparison of photographic and electrical studiesof the discharge process', Proc. Roy. Soc, 1938, [A], 166, pp. 56-75

22 WAGNER, c. F.: 'The relation between stroke current and thevelocity of the return stroke', IEEE Trans. Power Apparatus Syst.,1963, 82, pp. 609-617

23 STRATTON, J. A.: 'Electromagnetic theory' (McGraw-Hill, 1941)24 COURANT, R., and HILBERT, D. : 'Methods of mathematical physics'.

Vol. 1 (Interscience, 1953)25 MORSE, p. M., and FESHBACH, H.: 'Methods of theoretical physics',

Vol. I (McGraw-Hill, 1953)26 DIRAC, P. A. M.: 'Principles of quantum mechanics' (Oxford

University Press, 1947)27 JACKSON, j . D. : 'Classical electrodynamics' (Wiley, 1963)28 CARSLOW, H. s., and JAEGER, J. c.: 'Operational methods in applied

mathematics' (Oxford University Press, 1948)29 GARDNER, M. F., and BARNES, J. L. : 'Transients in linear systems'.

Vol. I (Wiley, 1950)30 BERGER, K.: 'Resultate der Blitzmessungen der Jahre 1947 bis 1954

auf dem Monte San Salvatore', Bull. Schweiz. Elektrotech. Ver.,1955, 46, pp. 405-424

31 BERGER, K.: 'Front duration and current steepness of lightningstrokes to earth', //; j . s. FOR REST, P. R. HOWARD and D. J. HITLER(Eds.): 'Gas discharges and the electricity supply industry' (Butter-worths, 1962), paper 27

32 OHWA, G.: 'Field investigation of induced lightning surges', reportof Central Research Institute of Electric Power Industry, 1966,Japan

33 BERGER, K.: 'Ueber den Verlauf der von Gewittern auf zweiMittelspannungsleitungen erzeugten elektrischen Spannungennach Beobachtungen in Sommer 1928', Doctoral dissertation 566,Eidgenossische Technische Hochschule, Zurich, Switzerland, 1930

34 BERGER, K.: 'Die Gewittermcssungen der Jahre 1932 und 1933 inder Schweiz', Bull. $chweiz. Elektrotech. Ver., 1934, 25, (9),pp. 213-229

35 BERGER, K.: 'Resultate der Gewittermessungen in den Juhren1934/35', ibid., 1936, 27, pp. 145-163NORINDER, H.: 'Some recent tests on the investigation of induced36

37

38

surges', CIGRE\ Paris, Paper 303, 1939PERRY, F. R.: 'The measurement of lightning voltages and currentsin South Africa and Nigeria, 1935 to 1937', J. IEE, 1941, 88,Pt. II, pp. 69-87PERRY, F. R., WEBSTER, G. H., and BAGUI.EY, p. w. : 'The measurementof lightning voltages and currents in Nigeria (Part 2, 1938-1939)',ibid., 1942, 89, Pt. II, pp. 185-203

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967 1907