IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 4, NOVEMBER 1997
269
Ground Effects on Induced Voltages
from Nearby Lightning
Hans Kr. Høidalen, Jarle Sletbak, and Thor Henriksen
Abstract— The lightning induced voltage on overhead lines
from return strokes and it’s dependency of a lossy ground is
analyzed using a new, analytical vector potential formulation.
Norton’s approximation and the surface impedance approach are
used to take loss effects into account. The surface impedance
method predicts in general induced voltages in good agreement
with Norton’s approximation, but the accuracy of the method
is dependent on the variation of the current along the lightning
channel. Norton’s method is compared with the exact Sommerfeld
solution, showing a deviation <10% even for low conducting
grounds and distances from 100–1000 m. The effect of stroke
location and line termination is also analyzed, showing that a
line terminated by it’s characteristic impedance and excited by a
return stroke at the prolongation of the line is especially sensitive
to lossy ground effects. Strokes near the mid-point of an overhead
line gives less loss effect than strokes at the end of the line.
The surface impedance approximation is derived from Norton’s
method and the necessary assumptions are outlined.
Index Terms— Calculation model, lightning-induced voltage,
lossy ground.
I. INTRODUCTION
I
NDUCED voltages due to nearby lightning is an important
reason for insulation failures in low voltage systems. An
optimal protection can only be achieved if one can predict
the expected over-voltages induced by lightning. Simultaneous
measurements of the lightning current and induced voltages
show a large deviation between measured and calculated
voltages, e.g., [1], [2]. The reasons for this are mainly:
1) an over-simplified model of the lightning channel is
used;
2) the attenuation of the electric fields due to propagation
over ground is not taken properly into account.
This paper focuses on the latter factor, but also examines the
impact of one of the lightning channel parameters.
The effect of losses on fields from remote lightning is
well known. While the vertical electric field is little affected
by ground effects, the horizontal field is strongly affected,
(decreasing from zero at the ground to a negative value when
loss is taken into account). Measurements [3], [4] have even
shown an inversion of the induced voltage for some special
configurations.
Manuscript received July 16, 1996; revised August 19, 1997.
H. Kr. Høidalen and J. Sletbak are with the Norwegian University of
Science and Technology, Trondheim N-7034, Norway.
T. Henriksen is with the Norwegian Electric Power Research Institute,
Trondheim N-7034, Norway.
Publisher Item Identifier S 0018-9375(97)08377-4.
Fig. 1. Configuration: cylindrical coordinates.
For nearby lightning ( 1000 m), the effect of losses is more
uncertain. Also the accuracy of the currently used methods
for loss calculations has been questioned, but they are in [5]
shown to predict the peak value of the induced voltage with
reasonable accuracy. Zeddam and Degauque [6] have used
and evaluated Norton’s method [7] showing better than 10%
accuracy for practical dipole heights and frequencies, even for
100 m distance. Ishii et al. [8] also used Norton’s method
to calculate the loss effects for nearby lightning and the loss
effects are shown to be important even for distances less than
1000 m.
In this paper, the induced voltages in an overhead line is
calculated by using a new vector potential formulation. The
modified transmission line model (MTL) [9] is used for the
lightning current. Norton’s method is first evaluated, also for
lower ground conductivities and other permittivities than used
in [6].
II. BASIC CONFIGURATION
The analyzes made in this paper are based on the configuration shown in Fig. 1.
The lightning channel is assumed to be vertical, as shown
in Fig. 1. This configuration is used to calculate the vector
potential caused by the current dipole
at a height .
The observation point has a horizontal distance from the
lightning channel and a height . R and R are the distances
from the dipole and it’s image, respectively. The ground has
and a permittivity
. The system is
a conductivity
axial symmetric with the -axis directed along the overhead
line and the -axis perpendicular to the line (
).
The vector potential will in this paper be used to calculate
the induced voltage in an overhead line. The advantage of
0018–9375/97$10.00  1997 IEEE
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 4, NOVEMBER 1997
this formulation is that Norton’s approximation can be used
directly. Besides, the vertical and horizontal electric fields
can both be expressed by the vector potential, making it
easier to calculate the total line voltage and to analyze it’s
dependency of the line terminations. By choosing the Lorentz
gauge [
] for the vector potential
divergence, the electric fields can be written
(7)
(8)
R
R
R
(1)
where
is the -component of the vector potential.
III. EVALUATION
OF
NORTON’S METHOD
In [6] several approximations were compared to Sommerfeld’s exact formulation of fields from a vertical dipole,
showing reasonable agreement for the Norton’s method. However, in [6] the ground parameters
and are kept constant
and a relatively high conductivity of 0.01 S/m is used. Before
using Norton’s approximation for lower ground conductivities
and other permittivities an extension of the analysis in [6] is
presented here. The exact expression for the vector potential
set up by a vertical current dipole (
) over flat, homogeneous ground is given by Sommerfeld [10], [11] in the
frequency domain
R
R
and where erfc is the complementary error function.
The first term inside the brackets in (3) is the direct wave
from the dipole to the point of observation, the second term is
the reflected wave from the ground surface and the third term
is the ground wave.
The vector potential in Norton’s formulation can also be
written as a sum of the potential in a lossless situation,
,
and a contribution from the lossy ground,
R
R
R
R
R
R
(10)
where
R
R
(9)
R
(11)
R
R
J
(2)
R
(12)
erfc
where J is the Bessel function of first kind
(13)
The accuracy of Norton’s method is further analyzed by
comparing the Norton vector potential in (3),
, with the
exact vector potential from Sommerfeld’s method (2),
.
The vector potentials are calculated at a height
10 m above
ground. The error of Norton’s approximation is defined as
The integral in (2) must be solved numerically which is
difficult and time consuming, due to the highly oscillatory
behavior of the Bessel function at large distances and the
exponential function at large dipole heights. The integrand is
also nearly singular when
.
Norton has formulated an approximation to (2), assuming a
high ground conductivity [7], [11]
R
R
R
R
R
R
R
(3)
R
where
R
(4)
erfc
R
(5)
(6)
with dipole height and frequency as variables and horizontal distance and the ground characteristic
and
as parameters. Figs. 2 and 3 show the error in Norton’s
approximation for two different radial distances, ; 100 and
1000 m. A ground conductivity of
S/m and a
relative permittivity of
is assumed. Frequencies from
0–10 MHz and dipole heights from 0–4000 m have been
investigated.
As seen from Figs. 2 and 3 the error in Norton’s method is in
general low and less than 5% in an important frequency range
(0–1 MHz). For frequencies less than 1 MHz the accuracy
of Norton’s method is better for larger distances, but above
1 MHz the error seems to increase with distance. Also, for
frequencies above 1 MHz the error is larger for lower dipole
heights. The dependency of the ground permittivity has also
been investigated. This shows that the error is reduced with
increasing permittivity and using
40 gives an error less
than 2% in the whole frequency range (0–10 MHz). A relative
permittivity of
2.5 gives errors between 10–37% for
HØIDALEN et al.: GROUND EFFECTS ON INDUCED VOLTAGES FROM NEARBY LIGHTNING
271
Fig. 4. Ratio between lossy and lossless vector potential at ground level.
Fig. 2. Error in Norton’s approximation.
r = 100 m.
1
=
A. Current Model
0:001 S/m, "r = 10,
The simple Modified Transmission Line model (MTL) [9]
has been used. In this model a current starts from the ground
and travels upward with a constant velocity , and its amplitude is exponentially decaying with increasing height with
the factor
, where is the decay constant. In our
analysis the current at ground is assumed to be a step with
amplitude . An arbitrary current shape at the ground can
be taken into account by a final convolution integral of the
calculated voltages, since the system is assumed to be linear.
In the frequency domain the current along the lightning
channel can be written
(14)
Fig. 3. Error in Norton’s approximation.
r = 1000 m.
1 = 0:001 S/m, "r = 10,
frequencies
1 MHz and dipole heights
200 m when
1000 m.
The attenuation of the vector potential can be expressed as
the ratio between the lossy and lossless ground potential,
from (9) and (10) at the ground (
).
This ratio is in Fig. 4 plotted as a function of frequency ( )
and dipole height ( ), with
0.001 S/m,
10 and
100 m. From Fig. 4 one important observation can be made:
At low frequencies the impact of the lossy ground increases
with dipole height. Above 1 MHz the loss effect is largest for
low dipole heights, however.
Norton’s method is often used to calculate the loss effects
on electrical far-fields which are dominated by the radiation
term. The attenuation of the radiation fields is similar to that
of the vector potential and this approach has been used in e.g.,
[12], [13]. The electrical close-fields show a more complex
dependency of lossy ground obtained by differentiating (10)
according to (1). The full expressions for the electrical fields
using Norton’s method are given in [14]. Calculating the
induced voltage from a vector potential formulation allows
the Norton’s method to be used directly.
IV. VECTOR POTENTIAL FROM A LIGHTNING CHANNEL
The vector potential from a current element (
) is given
by (9) and (10). To calculate the total potential, the lightning
current must be modeled.
The situation occurring when the lightning current wave
reaches the top of the lightning channel is not analyzed in
this paper. The height of the lightning channel is assumed to
be so large that this does not occur within the studied time
span. The limited height is anyhow of no practical importance
if it is large compared to the decay constant .
B. Total Vector Potential
The total vector potential equals the integral over the
lightning channel
(15)
is the total height of the lightning channel.
where
Inserting the Norton’s approximation expressions from (4)
and (5) gives
R
R
R
R
(16)
(17)
is plotted as
In Fig. 5 the absolute value of the ratio
a function of frequency and radial distance . The height
of the observation point is
10 m. The height of the
lightning channel is assumed to be
4000 m and the decay
constant in the current model is
1500 m. The ground has
0.001 S/m and a relative permittivity
a conductivity
10. Clearly the attenuation increases with frequency, ,
and distance, . Up to 400 kHz the vector potential is almost
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 4, NOVEMBER 1997
Fig. 7. Overhead line terminations.
Fig. 8. End stroke configuration.
Fig. 5. Attenuation of the vector potential from a lightning channel.
j
= 0 j.
A A
Fig. 9. Side stroke configuration.
Fig. 6. Induced voltage configuration.
unaffected by the lossy ground. At 1 MHz the vector potential
is reduced to 65% 1000 m away from the lightning channel.
in our analysis. The scattered voltage
can, however, be
decomposed in a sum of an incoming wave and a reflected
wave dependent on the termination. The incoming voltage
wave, which will be focused in our study, consists of the
time delayed, reflected wave from the other terminal plus
a contribution from the horizontal electric field, called the
horizontal field contribution
. This gives
(22)
V. INDUCED VOLTAGE CONFIGURATION
where
The configuration used when calculating the induced voltage
in an overhead line is shown in Fig. 6. The overhead line
has two terminals A and B with -coordinates
and
,
respectively (
). The length of the overhead line
is
. The observation point has coordinates
(
). All the electromagnetical fields handled in this paper
are incident fields written without superscript .
The overhead line is modeled by the Agrawal coupling
model [15]. The total line voltage is in this model a sum of
the scattered voltage
and the incident voltage
(23)
is the transmission coefficient of the overhead line (being equal
to
in a lossless situation).
When performing calculation examples, the two stroke
locations, as shown in Figs. 8 and 9, will be used.
VI. HORIZONTAL FIELD CONTRIBUTION
(
The voltage wave coming in to an overhead line terminal
) due to the horizontal electric field
is given by [5], [16]
(18)
where
(19)
and the following two equations being valid for
(20)
(24)
This voltage wave is equal to the scattered voltage
if the
line is terminated by it’s characteristic impedance at both ends.
Replacing the horizontal electric field by the vector potential
formulation in (1) gives
(21)
The terminations of the overhead line can be included as
shown in Fig. 7 when the line is terminated by impedances.
which is the solution of the wave
The scattered voltage
equations (20) and (21) is dependent on the terminations of
the line and is therefore not so useful as a key parameter
(25)
HØIDALEN et al.: GROUND EFFECTS ON INDUCED VOLTAGES FROM NEARBY LIGHTNING
273
Fig. 10. Loss effect on horizontal field contribution, far end. End stroke,
r0
100 m, 1 0.001 S/m. Solid line: MTL model ( 1500 m). Dotted
line: TL model.
Fig. 11. Loss effect on horizontal field contribution, far end. End stroke,
r0 1000 m, 1 0.001 S/m. Solid line: MTL model ( 1500 m).
Dotted line: TL model.
where the two
terms outside the integral in (25) will
dominate in most cases except for the side stroke (Fig. 9)
where the two
terms evaluated at
and
partly
cancel each other for symmetry reasons.
between two consecutive frequency points.
is set equal
to
Hz and
to 10 MHz.
,
The lossy part of the horizontal field contribution
is calculated in the time domain and shown in Figs. 10 and
. The overhead line is
11, together with the lossless part
1000 m and height
assumed lossless with length
10 m. The velocity of the lightning current is set to
m/s. The current amplitude is
1
and the height of
the lightning channel is assumed to be 4000 m. The decay
constant has the values 1500 m or 10 m corresponding
to the MTL and the transmission line TL, lightning channel
models, respectively.
Figs. 10 and 11 show how the lossy ground affects the
horizontal field contribution. For the distance
1000 m the
loss effect is large and the total horizontal field contribution
will be negative up to around 10 s, while
at
100 m,
will only have a negative initial peak.
We also observe that the lossy part of the horizontal field
is more dependent on the decay constant
contribution
than
, and thus on the variation of the current along
the lightning channel. The reason for this is that the upper
parts of the lightning channel contribute less for smaller ,
and as seen from Fig. 4 the loss effect at low frequencies is
more pronounced for the upper than for the lower parts of the
lightning channel. The peak voltage is, however, almost the
same for the two decay constants. Besides, it is important
to keep in mind that a step current has been used in the
calculations and a more realistic current shape will make the
first peak more significant. Increasing the conductivity by a
in
factor 10 reduces the horizontal field contribution
Fig. 10 by a factor 4. Another observation is that the lossless
will be higher for the MTL
horizontal field contribution
than for the TL model.
The importance of the loss effect on wave propagation in the
overhead line has also been investigated. The overhead line’s
transmission coefficient, , was calculated by Cable Constants
in ATP-EMTP [17] which uses Carson’s formulas, assuming
a 5 mm diameter Cu-line 10 m above ground. The result
shows that the overhead line losses are ignorable compared
to the attenuation of the horizontal electrical field when the
=
=
=
A. Loss Effects on the Horizontal Field Contribution
The horizontal field contribution expressed in (25) can be
split in a lossless part and a lossy ground part dependent only
on
from (17). This latter voltage contribution, called
,
is the key quantity when analyzing the loss effects on induced
voltage in an overhead line. The ground conductivity and the
importance of the decay constant in the lightning channel
model will also be studied.
Instead of differentiating the vector potential
in (25)
with respect to , the theorem of reciprocity can be utilized,
, resulting in
(26)
Inserting (26) in (25) gives a complete analytical expression
for the lossy part of the horizontal field contribution. The
function
is smooth and well behaved. It approaches
for low frequencies and
for high frequencies. Thus, the voltage in the time domain can be calculated
numerically by the inverse Fourier transform, using only ten
frequency samples per decade and the sine form of the inverse
Fourier transform
(27)
By assuming the function
to vary linearly between each
sample point, the integration can be performed analytically
=
=
=
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 4, NOVEMBER 1997
Fig. 12. Loss effect on horizontal field contribution, far end. End stroke,
r0
100 m, 1 0.001 S/m. Solid line: MTL model ( 1500 m). Dotted
line: TL model.
=
=
=
line is short. Our simulations suggest that the line losses
can be ignored for lines shorter than 1000 m whereas [5]
suggests 500 m. Similar results have also been obtained in
[18]. This result enables a very simplified calculation model
of induced voltages in short overhead lines. This model is
based on calculating the horizontal field contribution and the
incident voltage in the frequency domain, using an inverse
Fourier transform and finally keep track of reflections at the
line terminations in the time domain.
VII. SURFACE IMPEDANCE FORMULATION
A commonly used approximation for taking loss effects into
account is the surface impedance approach. This technique
is used e.g., in [5], [20]–[22]. A simplified version of this
technique (neglecting the displacement current) was also used
in [19, Eq. (95)]. The surface impedance method has been
evaluated by Cooray and compared with Norton’s method
showing less than 15% error for the considered parameter
range [5], [20].
According to the surface impedance approach the horizontal
electric field is approximated by
(28)
is the lossless electric field at line height and
where
is the lossless magnetic field at ground and where the surface
impedance is
(29)
The loss effect on the horizontal field contribution due to the
attenuation of the horizontal field can be written
Fig. 13. Loss effect on horizontal field contribution, far end. End stroke,
r0 1000 m, 1 0.001 S/m. Solid line: MTL model ( 1500 m).
Dotted line: TL model.
=
=
=
where
R
(32)
R
assuming a step lightning current with decay constant and
traveling upward with a velocity . The vector potential is
calculated at ground level so that R
.
The curves in Fig. 10 and 11 have been recalculated,
as shown in Figs. 12 and 13, using the surface impedance
approach.
Comparing Figs. 10 and 11 with Figs. 12 and 13, respectively, shows a reasonably good agreement between the
Norton’s method and the surface impedance method when
1000 m for both the MTL and TL lightning channel
models. At distance
100 m, the agreement between
the two models is best when the MTL model is used. The
surface impedance approach is less dependent on the decay
constant, than Norton’s method. This is due to the fact that
the variation of the loss effects along the lightning channel is
ignored in the surface impedance approach. From Fig. 4 we
see that the loss effect is strongest for the highest dipoles at
low frequencies. When a decay constant of
1500 m is
assumed the contribution from the upper parts of the lightning
channel is reduced. This will therefore lead to an improved
accuracy of the surface impedance method.
A. Deducing the Surface Impedance Approach
from Norton’s Method
The surface impedance approach gives the same result as
Norton’s approximation if (31) equals lossy part of (25). This
is true if
(33)
(30)
Utilizing that
, (30) can be expressed
, to
Requiring the contribution from each current dipole,
be equal in the two cases (instead of the total integral) results
in
R
(31)
R
(34)
HØIDALEN et al.: GROUND EFFECTS ON INDUCED VOLTAGES FROM NEARBY LIGHTNING
275
The hypothesis in (34) is now investigated, and the necessary
assumptions are outlined.
For small (small frequencies and/or small dipole heights),
can, by a series expansion, be expressed as
R
R
and taking the limit
Performing the differentiation of
as approaches zero gives
R
Multiplying the right hand side of (36) with
R
(35)
R
(36)
Fig. 14. Induced voltage, open ends. Dotted line: lossless. Solid line: Norton’s approximation. Dashed line: surface impedance. 1 0.001 S/m, "r
10. r0
100 m.
=
=
gives
=
R
R
(37)
Assuming R
(or
) will make the surface
impedance approach equivalent to Norton’s method. This
assumption is reasonable for large distances or short times
(
). For short times however, the high frequency
content makes the series expansion approximation in (35)
doubtful. When the lightning current is decaying along the
lightning channel the lower parts of the channel will contribute
more than the upper ones, making the surface impedance
method more accurate for lower decay constants, .
The surface impedance approach is a good approximation
if the following conditions are met:
1) low frequencies dominate ( is small);
2) the overhead line height, is small compared to the
horizontal distance ;
3) the approximation, R
, holds. This assumption is
reasonable for: Large distances , small decay constants
or short times .
formulation are compared with those obtained in [18] based
on the surface impedance method.
The incident voltage in (19) can also be expressed by the
vector potential using the formulation in (1)
(38)
A. Open Ends
If the line terminations are open, the overhead line voltage
at end
can be written
(39)
Inserting the vector potential formulation from (25) and (38)
this gives
VIII. TOTAL LINE VOLTAGE
The total voltage in an overhead line is the sum of the
scattered and incident voltage. The horizontal field contribution
, has been shown to be strongly affected by a
lossy ground in contrast to the incident voltage which is
little affected for short distances. Due to the vector potential
formulation analytical expressions for the total line voltage can
be developed and the lossy ground effect on the whole line
can be studied, including the dependency on line terminations.
Two different situations are here analyzed:
1) open end;
2) matched termination.
The lightning current is no longer assumed to be a pure step
function but is approximated by the Heidler model [23] with
the same parameters as in [18], [24] resulting in a peak current
of 12 kA and a maximum current/time derivative of 40 kA/ s.
[This current shape is taken into account by performing a
convolution integral based on the step responses (40 or 42)].
In Fig. 16 our results using the analytical vector potential
cosh
sinh
(40)
in (40) is for
The loss effect on the total line voltage
high frequencies (corresponding to the initial part in the time
domain) mainly determined by the first
term which in
since the lossless part actually
(40) is written with index
is zero at ground level (
0). For lower frequencies the
integral term contribution becomes important (typically after
the first reflection in the time domain) and this will reduce
the loss effect.
Fig. 14 shows a simulation of the induced voltage for an
end stroke at distance
100 m from a 1000 m long and
10 m high overhead line with open ends. The simulations
taking a lossy ground into account are compared with a lossless
simulation. The lightning current parameters are
1.1 10
m/s and
1500 m and
4000 m.
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 4, NOVEMBER 1997
Fig. 15. Induced voltage, matched terminations. Dotted line: lossless. Solid
0.001
line: Norton’s approximation. Dashed line: surface impedance. 1
S/m, "r
10. r0
100 m.
Fig. 16. Induced voltage, side stroke. Dotted line: lossless. Solid line: Lossy
ground (Norton). Dashed line: Lossy ground (Surf. imp.). Dots () from [18]
Fig. 12. 1
0.001 S/m, "r
10. r0
50 m.
From Fig. 14 we observe that the initial part of total line
voltage at the far end is strongly influenced by the lossy ground
and actually the polarity is reversed. After a while the lossy
ground effect is reduced, but a negative peak occurs in the
lossy ground voltage at each reflection in the line.
Fig. 16 shows the induced voltage at the end of a 1000 m
long overhead line terminated by it’s characteristic impedances
and excited by a side stroke 50 m from the center of the
line. The Norton’s method is compared with the results using
the surface impedance method in [18]. The lightning current
parameters are
1.3 10 m/s and
1700 m and
4000 m.
Fig. 16 shows that our simulations give results similar to
those in [18]. Also the surface impedance approach gives very
good agreement with Norton’s method except at the initial
negative peak and the positive peak. This is due to the high
steepness of the induced voltage.
=
=
=
B. Matched Terminations
If the line terminations are matched i.e., terminated by the
characteristic impedance, the overhead line voltage of end
becomes
(41)
Inserting the vector potential formulation from (25) and (28)
gives
(42)
The loss effect on the total line voltage
in (42) is dominated
by the first two
terms which in (42) are written with
index since the lossless part actually is zero at ground (
0). For the side stroke these terms tend to cancel each other
(
).
Fig. 15 shows a simulation of the induced voltage for an
100 m from a 1000 m long and
end stroke at distance
10 m high overhead line with matched ends. The simulation
taking a lossy ground into account is compared with a lossless
simulation. The lightning current parameters are
1.1 10
m/s and
1500 m and
4000 m. From Fig. 15 we
observe that a line terminated by it’s characteristic impedance
and excited by an end stroke is very sensitive to lossy ground
effects. However, the application of the telegraph equations
and thus Agrawal’s coupling model becomes doubtful for an
end stroke since the variation of the exciting field
, is large
along the line. This is discussed by Agrawal et al. [15].
=
=
=
IX. DISCUSSION
The performed evaluations of Norton’s method in Figs. 2
and 3, show that this method is highly accurate ( 10%
deviation) even for short distances (100 m) and low conducting
ground (
0.001 S/m). Lower relative permittivity than 10
reduces the accuracy.
Fig. 4 shows how the loss effect on the vector potential
from a vertical electric dipole is affected by dipole height
and frequency. The important observation is the increased loss
effect with dipole heights for low frequencies.
Figs. 10 and 11 show that the lossy part of the horizontal
field contribution
, after the first initial peak is strongly affected by the decay constant . Beyond 10 s the dependency
is largest for short distances. The dependency of is important
to notice and it stresses the necessity of using an adequate
current model when calculating induced voltages from nearby
lightning. The lossless part of the horizontal field contribution
, is for
100 m also very dependent on the decay
constant and is in fact larger for the MTL model than for the
TL model. Equation (43) shows the slope of the horizontal
field contribution for an end stroke at large times
R
R
(43)
HØIDALEN et al.: GROUND EFFECTS ON INDUCED VOLTAGES FROM NEARBY LIGHTNING
Calculations in Figs. 12 and 13, using the surface impedance
approach, show that this method is a reasonable approximation
for distances larger than 1000 m. The approximation is poor
for high frequencies and large dipole heights. In a practical
situation assuming a standard 1.2/50 s lightning impulse the
surface impedance method will give wrong results for very
short times (high frequencies) and for large times (large dipole
heights). Within these boundaries where often the maximum
value of the induced voltage occurs the method is accurate,
however.
Figs. 14 and 15 show how the loss effect on the total
induced voltage depends on the line terminations. When the
line terminals are open the loss effect is initially about twice as
large as when the line is matched. This is also seen from (40)
and (42) where the
term initially dominate and where
the voltage in (42) is multiplied by a factor 0.5. One exception
is the side stroke configuration where the two
terms
in (42) tend to cancel each other. A matched line is therefore
more sensitive to the stroke location.
Figs. 14–16 show that the end stroke configuration is very
sensitive to loss effects while the side stroke configuration is
less sensitive, which is in good agreement with the results in
[25], [26]. The reason for this is that the latter configuration
is less dependent on the horizontal field contribution and the
loss effect on the total voltage will actually decrease with
increasing distance until the loss effect on the incident voltage
becomes important.
Figs. 14–16 show that the surface impedance method is
accurate compared to Norton’s method for the applied current
waveform. A deviation arises when the voltage is steep like
at the far end in an end stroke configuration and for the side
stroke configuration. For times beyond the first peak value in
the voltage the surface impedance method cause the voltage
to drift off due to the increased dipole height. It is also worth
mention that the used current waveform is very steep with a
front time below 1 s. Larger front times will improve the
accuracy of the surface impedance method around the first
peak in induced voltage.
X. CONCLUSION
Norton’s approximation for calculating the loss effects on
the vector potential from a vertical current dipole has been
evaluated, showing very good agreement also for ground
conductivities as high as
0.001 S/m.
The significance of loss effects on the horizontal field
contribution has been studied. The lossy ground has significant
effect on the vector potential while the line loss in overhead
lines shorter than 1000 m long is insignificant.
The first peak of the induced voltage is independent on the
decay constant but after the first micro second the lossy part
of the horizontal field contribution is strongly dependent on
the variation of the current along the lightning channel.
The total induced voltage is strongly dependent on the line
terminations. When the line terminations are open the loss
effect is initially twice as high as when the line terminations
are matched. However, after the first reflection the voltage in
the open end configuration approaches the lossless situation.
277
The loss effect is in general large even for distances in the
range of 100 m and may lead to an inversion of the voltage.
Lightning induced voltages in a side stroke configuration is
less influenced by lossy ground effects than for an end stroke.
The surface impedance method has also been analyzed,
showing a reasonable agreement with Norton’s approximation for distances larger than 1000 m. Usage of the MTL
lightning channel model (
1–2 km) gives higher accuracy
of the surface impedance method than the TL model. The
surface impedance method has in this work been derived from
Norton’s method.
REFERENCES
[1] P. P. Barker, T. A. Short, A. R. Eybert-Berard, and J. P. Berlandis,
“Induced voltage measurements on an experimental distribution line
during nearby rocket triggered lightning flashes,” in Proc. IEEE/PES
Summer Meeting, Portland, OR, 1995.
[2] M. Rubinstein, M. Uman, P. J. Medelius, and E. M. Thomson, “Measurements of the voltage induced on an overhead power line 20 m from
triggered lightning,” IEEE Tran. Electromag. Compat., vol. 36, May
1994.
[3] N. Georgiadis, M. Rubinstein, M. Uman, P. J. Medelius, and E. M.
Thomson, “Lightning induced voltages at both ends of a 448-m powerdistribution line,” IEEE Trans. Electromag. Compat., vol. 34, Nov.
1992.
[4] V. Cooray and F. de la Rosa, “Shapes and amplitude of the initial peaks
of lightning-induced voltage in power lines over finitely conducting
earth: Theory and comparison with experiment,” IEEE Trans. Antennas
Propagat., vol. 34, Jan. 1986.
[5] V. Cooray, “Lightning-induced overvoltages in power lines: Validity of
various approximations made in overvoltage calculations,” in Proc. 22nd
Int. Conf. Lightning Protection, ICLP’94, Budapest, Hungary, 1994.
[6] A. Zeddam and P. Degauque, “Current and voltages induced on telecommunication cables by a lightning stroke,” in Lightning Electromagnetics,
R. Gardner, Ed. New York: Hemisphere, pp. 377–400.
[7] K. A. Norton, “The propagation of radio waves over the surface of the
earth in the upper atmosphere,” Proc. IRE, vol. 25, pp. 1203–1237, 1937.
[8] M. Ishii, K. Michishita, Y. Hongo, and S. Oguma, “Lightning-induced
voltage on an overhead wire dependent on ground conductivity,” IEEE
Trans. Parallel Distrib. Syst., vol. 9, Jan. 1994.
[9] C. A. Nucci, C. Mazzetti, F. Rachidi, and M. Ianoz, “On lightning
return stroke models for LEMP calculations,” in Proc. 19th Int. Conf.
Lightning Protection, Graz, Apr. 1988.
[10] A. Sommerfeld, Partial Differential Equations in Physics. New York:
Academic, 1949.
[11] T. S. M. Maclean and Z. Wu, Radiowave Propagation Over Ground, 1st
ed. London, U.K.: Chapman and Hall, 1993.
[12] C. Hopf, “Effects of propagation on the electrica fields radiated by return
strokes in distances up to 15 km,” in Proc. ICLP’94, 1994.
[13] V. Cooray and H. Pérez, “Propagation effects on first return stroke
radiation fields: Homogenous paths and mixed two section paths,” in
Proc. ICLP’94, 1994.
[14] R. King, “Electromagnetic field of a vertical dipole over an imperfectly
conducting half-space,” Radio Sci., vol. 25, pp. 149–160, Mar./Apr.
1990.
[15] A. K. Agrawal, H. J. Price, and S. Gurbaxani, “Transient response of
multi-conductor transmission line excited by a non uniform electromagnetic field,” IEEE Trans. Electromag. Compat., vol. EMC-22, pp.
110–129, 1980.
[16] T. Henriksen, “Calculation of lightning overvoltages using EMTP,” in
Proc. Int. Conf. Power Syst. Transients, Lisbon, 1995.
[17] Alternative Transients Program (ATP) Rule Book, Can/Am EMTP User
Group, Leuven EMTP Center, Belgium, 1987.
[18] F. Rachidi, C. A. Nucci, M. Ianoz, and C. Mazzetti, “Influence of a
lossy ground on lightning-induced voltages on overhead line,” IEEE
Trans. Electromag. Compat., vol. 38, Aug. 1996.
[19] S. Rusck, “Induced lightning over-voltages on power-transmission lines
with special reference to the over-voltage protection of low-voltage
networks,” Ph.D. dissertation, Royal Institute of Technology, Stockholm,
Sweden, 1957.
[20] V. Cooray, “Horizontal fields generated by return strokes,” Radio Sci.,
vol. 27, no. 4, pp. 529–537, July/Aug. 1992.
278
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 4, NOVEMBER 1997
[21] M. Rubinstein, “Voltages induced on a test power line from artificially
initiated lightning: Theory and experiment,” Ph.D. dissertation, Univ.
of Florida, Gainesville, 1991.
[22] V. F. Hermosillo, “Voltages induced on overhead power distribution
lines by lightning return strokes,” Ph.D. dissertation, Univ. of Uppsala,
Sweden.
[23] F. Heidler, “Analytische Blitzstrom-function zur LEMP-Berechnung,”
in Proc. Int. Conf. Lightning Protection, ICLP’85, Munich, Germany,
1985, paper 1.9, pp. 63–66.
[24] C. A. Nucci, F. Rachidi, M. Ianoz, and C. Mazzetti, “Lightning-induced
voltage on overhead lines,” IEEE Trans. Electromag. Compat., vol. 35,
Feb. 1993.
[25] Y. Hongo, K. Michishita, and M. Ishii, “Distribution of lightninginduced voltage along an overhead wire,” in Proc. Int. Conf. Lightning
Protection, ICLP’96, 1996.
[26] S. Guerrieri, M. Ianoz, C. Mazzetti, C. A. Nucci, and F. Rachidi,
“Lightning-induced voltages on an overhead line above a lossy ground:
A sensitivity analysis,” in Proc. Int. Conf. Lightning Protection,
ICLP’96, 1996.
Hans Kr. Høidalen was born in Draugedal, Norway, in 1967. He received
the M.Sc. degree in electrical engineering from the Norwegian Institute
of Technology, in 1990 and is currently pursuing the Ph.D. degree at the
Norwegian University of Science and Technology.
Jarle Sletbak was born in Namdalseid, Norway, in 1928. He received
the M.Sc. degree in electrical engineering from the Norwegian Institute of
Technology, in 1953.
He is presently a Professor at the Norwegian University of Science and
Technology. His main field of interest is insulating materials and their
application in high voltage equipment.
Thor Henriksen was born in Trondheim, Norway, in 1946. He received
the M.Sc. degree in electrical engineering from the Norwegian Institute of
Technology, in 1970, and the Dr.Ing. degree in 1973.
He is presently working at the Norwegian Electric Power Research Institute
as a Senior Research Scientist, mainly in the field of transient studies.