High frequency response of grounding electrodes: effect of soil dielectric constant

Abstract

Grounding electrodes have an important role in electric power transmission and distribution systems.
They are used to prevent excessive hazardous voltages due to ground potential rise in the case of system faults or lightning surges.
It is important that they provide a low impedance path for the current in to the ground.
The electrical properties of soil, which vary substantially with geographical location and time of year, affect the process considerably along with the properties of the grounding electrode itself, such as its dimensions.
In order to have an accurate estimation of the developed overvoltages and the backflashover rate of the transmission lines due to a lightning strike, one has to take into account the effect of the value of the soil electrical parameters, such as the electrical conductivity and dielectric constant.
This paper investigates the high frequency behavior of the grounding electrodes by solving a full-wave electromagnetic problem using the Finite Element Method (FEM).
The focus is on taking into account the effect of the variation of soil relative permittivity which has been neglected in the previous studies of the grounding systems.
This allows an evaluation of the response of grounding systems due to seasonal changes and specifically change of the water content of the soil, which would cause its electrical properties to vary significantly.
This study demonstrates the importance of considering the variation of relative permittivity of the soil especially in the modeling of electrodes buried in highly resistive soil.

Full text

IET Generation, Transmission & Distribution
Research Article
High frequency response of grounding
electrodes: effect of soil dielectric constant
ISSN 1751-8687
Received on 17th October 2019
Revised 25th March 2020
Accepted on 20th April 2020
E-First on 21st May 2020
doi: 10.1049/iet-gtd.2019.1554
www.ietdl.org
Bamdad Salarieh1,2, Jeewantha De Silva2, Behzad Kordi1
1Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada
2Manitoba Hydro International, Winnipeg, Manitoba, Canada
E-mail: behzad.kordi@umanitoba.ca
Abstract: Grounding electrodes have an important role in electric power transmission and distribution systems. They are used
to prevent excessive hazardous voltages due to ground potential rise in the case of system faults or lightning surges. The
electrical properties of soil, which vary substantially with geographical location and time of year, affect the process considerably
along with the properties of the grounding electrode itself, such as its dimensions. To have an accurate estimation of the
induced overvoltages due to lightning strike, one has to take into account the effect of the value of the soil electrical parameters,
such as the electrical conductivity and dielectric constant. This study investigates the high frequency behaviour of the grounding
electrodes by solving a full-wave electromagnetic problem using the finite element method. The focus of this paper is on the
effect of the variation of soil relative permittivity on the induced transient voltage in grounding electrodes. This allows an
evaluation of the response of grounding systems due to seasonal changes, which would cause its electrical properties to vary
significantly. This study demonstrates the importance of considering the variation of relative permittivity of the soil especially in
the modelling of electrodes buried in highly resistive soil.
1Introduction
Vertical and horizontal rods are commonly used in power systems
as a type of earth termination to provide a path for the lightning
current to flow in to the earth [1]. An effective grounding system
directs lightning intensive currents to the earth with a low potential
rise of the grounded system, which may be hazardous to personnel
or sensitive electrical equipment. Negative first strokes have been
traditionally known to produce the worst stress on the system
insulation. The subsequent negative strokes have considerably
lower peak currents but have a higher frequency content, up to a
few MHz [2]. The dynamic behaviour of grounding electrodes in
case of fast varying currents, such as lightning strokes, is different
from their low frequency response [3]. There has been a significant
number of research that aimed at high frequency modelling of
grounding electrodes.
In general, the problem of modelling grounding electrodes is
solved using (i) theoretical, (ii) numerical, and (iii) experimental
[4–6] methods. The theories are either based on the circuit [7–9] or
transmission-line formulations [10–14]. Full-wave electromagnetic
modelling, using numerical techniques are based on finite element
method (FEM) [15–18], method of moments [19–22], finite-
difference time-domain method [23–25], and partial electric
equivalent circuit [26, 27]. These methods can be considered as the
most rigorous and accurate modelling procedures over a wide
frequency range.
In lightning studies on grounding systems, the electrical
parameters of the medium in which they are buried have a high
importance and they need to be determined accurately. It has been
shown through experiments that the conductivity and dielectric
constant of soil are both very dependent upon the moisture content
of the soil which is known to vary from 4 to 30% of the total soil
weight over the greater part of the year [28]. Moreover, as the
frequency of the waves penetrating in-to the ground increases, the
dielectric constant of soil plays a more important role in
determining the effect of the earth on the wave propagation. Due to
these facts, the importance of evaluating the effects of the variation
of the electrical parameters of the soil, particularly the dielectric
constant in the whole permissible range, on the high-frequency
response of grounding electrodes has to be studied. In the
published literature, the dielectric constant of the soil is commonly
assumed equal to 10 and its variation is not considered, although
this value may vary between 3 or 4 for dry soil up to 30 for very
moist soil, depending on the nature of the soil [10, 28].
The other characteristic of soil is the frequency dependence of
its electrical parameters (resistivity and permittivity). There are
several available frequency dependent models for the soil which
are driven based on experimental data, such as, Messier [29],
Visacro and Portela [30], Portela [31], Visacro and Alipio [32]. In
it shown in such models that both the resistivity and permittivity of
the soil decrease as the frequency increases, leading to a decreased
grounding impedance [16, 33]. The frequency dependence of soil
electrical parameters is disregarded in this paper, therefore the
results are applicable for a conservative estimate of the upper
bound of the grounding impedance and this can be considered as an
assumption on the safe side. However, it is straightforward to
consider this effect in the transient analysis of grounding electrodes
using the simulation model proposed in this paper [16].
The objective of this paper is to investigate the response of
vertical and horizontal grounding electrodes in the context of
lightning currents considering a wide range of conductivity (0.1–
0.0001 S/m) and relative permittivity (3–30) of the soil in the
frequency range of 1 kHz–20 MHz. This frequency range has been
selected because the major frequency content of the first and
subsequent lightning currents have been shown to be below
10 MHz [2] and the behaviour of grounding electrodes is purely
resistive below a frequency of 10 to 100 kHz (as will be shown in
this paper). In this paper, we develop a full-wave electromagnetic
model that is solved using the FEM [34, 35]. Solving the full-wave
Maxwell's equations in the proposed model enables the
consideration of long grounding electrodes. The frequency-domain
impedance of the grounding electrodes is calculated using the
proposed model with a focus on the effect of soil dielectric
constant on the performance of grounding electrodes. Furthermore,
the numerical simulation results are compared with those obtained
using other modelling approaches. Unlike the Fourier-transform
based approaches, the frequency-domain simulation results of this
work can be directly incorporated in EMT-type simulators. Also,
time-domain electrode voltages due to both the typical first and
subsequent return strokes are calculated and the effects of the soil
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parameters on the time domain waveforms are investigated. This
will allow one to have an accurate estimation of the grounding
impedance variations in different soil conditions.
2Determination of grounding impedance
Consider a grounding electrode buried in a soil with conductivity
σ, permittivity ε, and permeability μ. In order to find the grounding
impedance, an exciting voltage source V(jω) is applied to the top
of the grounding electrode whose other terminal is connected to a
remote ground. By calculating the injected current into the
grounding rod I(jω), its input impedance in the frequency domain,
also known as harmonic impedance [3], is obtained using
Z(jω) = V(jω)
I(jω).
(1)
To determine I(jω), a commercial finite-element full-wave
electromagnetic solver [ANSYS HFSS] is employed to solve the
wave equation in the frequency domain that is given by [36]
∇ × 1
μ∇ × E(x,y,z) − k2E(x,y,z) = 0
(2)
where
k=ω ε 1 − jσ
εω
and E(x,y,z) is the electric field vector. The finite simulation space
is enclosed by perfect electric conductor (PEC) boundary condition
that also creates a return path for the current. This boundary
condition requires the tangential component of the electric field
(Et) and the normal component of the magnetic field (Hn) to be
zero as given by [36]
Et(x,y,z) = 0
(3a)
Hn(x,y,z) = 0.
(3b)
The soil region is modelled as a hemisphere of radius r1. However,
any other symmetrical geometry can be used to represent the
boundary of the ground as long as it is large enough [16]. The air
region is modelled as a finite-length conical transmission line of
length h, a lower radius r1, and an upper radius equal to the radius
of the grounding electrode. Using the conical transmission line
results in higher cut-off frequencies for the non-TEM modes [37,
38]. The cut-off frequency for higher order TE and TM modes
depends on the cone half angle and the radial distance in the
spherical coordinate system from the cone apex [37]. As a result,
these two parameters should be chosen carefully to avoid
reflections from the outer PEC boundary of the air region in the
frequency range of interest. A schematic view of the proposed
model is shown in Fig. 1. The excitation is provided by means of a
numerical port defined as a rectangle with a width equal to the
electrode's diameter and a length of h. The numerical port will
introduce a parasitic inductance. Selecting h= 10 mm results in
the parasitic impedance to be negligible compared with the
grounding impedance [16].
In (2), as the frequency increases the term σ/εω decreases, that
means the effect of soil conductivity on the propagation of
electromagnetic waves in the ground is less significant. As a result,
at high frequencies (σ/εω ≪ 1), the dielectric constant will play a
prominent role in determining the effect of the earth on the
propagation of the electromagnetic wave [10].
2.1 Size of the computational domain
Considering a relative permittivity of εr= 10 for the soil, the
dependence of skin depth on soil conductivity over a frequency
range of 100 Hz to 20 MHz is shown in Fig. 2a [10]. As the
conductivity of the soil decreases from 0.1 to 0.0001 S /m, the skin
depth increases. Knowing the skin depth at a given frequency, one
can determine the size of truncation radius of the ground. The
variation of permittivity has no influence in the low frequency
region (i.e. < 100 kHz) regardless of the soil conductivity, as
shown in Fig. 2b. However, at high frequencies (i.e. > 100 kHz)
and for soil of low conductivity, the skin depth is larger for soil
with a higher permittivity. It can be concluded from Fig. 2 that if a
specific value for the truncation radius of the ground accurately
simulates the low frequency case, it can be assured that it also
simulates the high frequency propagation with no error due to the
truncation. Fig. 3 shows the results of the FEM electrostatic
analysis, where vertical grounding electrodes are buried in a soil
with conductivity of 0.1 S/ m. As shown in this figure, for
modelling a 1 m vertical grounding electrode, a truncation radius
(r0) of 50 m is sufficient for the ground to achieve an error of less
than 1% in the value of the DC resistance (RDC) with reference to
Fig. 1 Proposed FEM model for the calculation of the grounding
impedance. The air region is represented by the blue cone of height h, the
brown hemisphere with radius r1 is representing the ground region, and a
rectangular port is defined for the excitation
Fig. 2 Skin depth in soil as a function of frequency, considering
(a) Varying conductivity and εr= 10, (b) εr= 3, 10, 30
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the case of a 600 m radius. Similarly for 3 and 5 m electrodes a
truncation radius of 55 and 70 m is sufficient, respectively.
2.2 Soil electrical parameters
2.2.1 Electrical conductivity: Soil conductivity is determined by
measuring the resistance of a sample at very low frequencies, often
DC to 20 Hz. It is shown that the soil conductivity is nearly
constant in this frequency range [32]. The value of conductivity at
higher frequencies can be estimated by knowing either its low
frequency resistivity or water content. In general, clay soils have a
high conductivity of 0.11 S/ m and above, loam and chalk soils with
an average value of about 0.1 S/ m, while soil of a sandy or gritty
nature gives a much lower conductivity value. The lowest values
were obtained on solid granite or slate subsoils with conductivity
of the order of 0.0001 S/ m [28].
2.2.2 Dielectric constant: Relative dielectric constant (εr)
expresses the ability of a material to polarise under an electric
field. To measure this quantity, the material is placed in an
alternating electromagnetic field, and the time it takes for the wave
to travel through the material is measured [39–41]. Measurements
have shown that the variation of dielectric constant with moisture
content depends on soil types [42]. The dielectric constant
increases slowly with soil's water content up to a transition point,
beyond which a rapid increase occurs. It was also observed that the
dielectric constant of soils with different water contents (from dry
soil to 30% of moisture content) ranges between 3 to near 30, with
its trend being dependent on the soil type or texture [28, 40]. In
another measurement, where precautions were taken to remove all
the moisture from a sample of soil, the minimum observed
dielectric constant of 2 and conductivity of 5.5 × 10−5 S/m were
measured [43]. As reported in [44], there have been several soil
samples of highly resistive soils (σ= 0.00008 to 0.0005 S /m)
which had a high low frequency ( 10 kHz) permittivity of 20 to 30.
These soil samples had a volumetric water content of 0.2 up to
35%. From these measured data, it can be seen that the variation of
conductivity and dielectric constant in a frequency range are
related to each other [40], however, at low frequencies such
correlation cannot be easily validated from the measurements.
2.2.3 Magnetic permeability: Magnetic permeability is
determined by measuring magnetic susceptibility of soil samples
under a weak magnetic field [41, 45]. In the study of
electromagnetic pulse propagation in soil, the relative magnetic
permeability of rock and soil is less important than its conductivity
and dielectric constant. For most earth materials it is only slightly
greater than unity (between 1.0006 and 1.001) [46]. Due to this
fact, the permeability of soil is considered equal to 1 in all studies
involving the ground.
3Numerical results
In this section, the normalised magnitude of the impedance,
Z(jω) /ZDC, of vertical and horizontal grounding electrodes of
lengths 1, 3, and 5 m in the frequency region of 1 kHz to 20 MHz
are calculated. The electrodes have a radius of 12.5 mm. The
conductivity of the soil is assumed to be in the range of 0.0001–
0.1 S/ m, and the values considered for soil relative permittivity are
3, 5, 10, 20, and 30.
3.1 Frequency response of grounding electrodes
The results determined by the full-wave electromagnetic approach
(‘FEM’) are compared with those obtained using the circuit theory
(‘RLC’) [10] and transmission-line theory (‘TLine’) [47, 48]. The
harmonic impedance of 1, and 5 m vertical grounding electrodes
with εr= 10 are shown in Figs. 4 and 5.
Any termination to ground presents resistive, inductive, and
capacitive effects. The current that is injected into a grounding
electrode has two components: a longitudinal current (IL)
transferred along the length of the electrode and a leakage
transversal current (IT) dispersed into the soil [10], as shown in
Fig. 6. The response of grounding electrodes is practically constant
up to a certain frequency, which is called the characteristic or the
break frequency fc [49]. This is due to the fact that at low
frequencies, the voltage drop along the length of the electrode
caused by the longitudinal current ( jωL1≃jωL2≃ 0) and the
capacitive current dispersed into the soil are negligible ( jωC1≃ 0).
In such a condition, the behaviour of the electrode is governed by
the value of conductance G1. Note that R1 and R2 are very small,
because the grounding electrodes are made of highly conductive
materials to have a better dispersion of excessive currents into the
soil. Above the characteristic frequency, the electrode has either an
Fig. 3 Electrostatic analysis of a vertical electrode buried in a
homogeneous soil with conductivity of σ= 0.1 S/ m as a function of ground
radius obtained with FEM, and analytical formulae [10]
Fig. 4 Grounding impedance of 1 m vertical electrode for εr= 10
(a) Normalised magnitude, (b) Phase angle
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inductive or a capacitive response based on the length of the
electrode and the soil conductivity. To justify the high frequency
behaviour, one should consider the affecting parameters on the
leakage transversal current, also known as displacement current.
The ratio of leakage current to the longitudinal current (IT/IL)
increases as the frequency, the dielectric constant, or the earth
resistivity increases. As the conductivity increases, the effect of
ground displacement current becomes less important and an
inductive behaviour is seen in the high frequency region, as shown
in Figs. 4 and 5. This can also be quantitatively described in terms
of the ratio σ/ωε. Another influencing parameter is the length of
the electrode which causes the resonant region to start at a lower
frequency for longer electrodes. The vertical and horizontal
grounding electrodes of the same length have almost a similar high
frequency response (results for the horizontal grounding impedance
are not shown here).
Comparing the results obtained by the three approaches, one
can see that the theoretical models lead to very small errors up to
the MHz frequencies, especially in the case of the highly resistive
earth and shorter electrodes. When the injected current has only
low frequency components, the electrode can be approximated by a
conductance (G1). This way, the electrical potential remains the
same along the length of the electrode and the theoretical
approaches that assume a constant potential are valid.
Nevertheless, the highest amount of variation between the
theoretical modelling approaches and numerical simulation results
is seen in the regions of resonant behaviour, where they predict
resonances of much higher peaks.
Regarding the effect of soil electrical permittivity on the
harmonic impedance of the grounding electrodes, Figs. 7a and b
show the variation of high frequency impedance of a vertical
grounding electrode of length 1 m as the soil relative permittivity is
changed from εr= 3 to 30. The variation of soil electrical
permittivity (i.e. its water content) has no significant effect on the
the impedance of the vertical grounding electrodes in a low
resistivity soil (10 and 100 Ωm). Mathematically, the ratio of σ/εω
in (2) determines whether the effect of soil permittivity is
significant or not. In a highly resistive soil (1000 and 10000 Ωm)
and low frequency range, the performance of the grounding
electrode does not change significantly as the permittivity varies.
However, beyond a threshold frequency of around 10 kHz,
increasing the relative permittivity from 3 to 30 results in a
significant reduction of at least 50% in the magnitude of impedance
due to the capacitive effect. This decrease continues up to a
frequency of 10 MHz after which an oscillatory behaviour is
observed in soil of very high permittivity.
3.2 Characteristic frequency
The frequency limit for the resistive behaviour of grounding
electrodes was termed as the characteristic frequency fc by Gary
[49]. In this paper, the grounding impedance is considered to be
resistive if its phase angle is in the range of ±5°. The range of
resistive behaviour depends on the value of the inductance (L1, L2)
and capacitance (C1) of the electrode. As we decrease the length of
the electrode, both the capacitance to remote ground and self-
inductance of the electrode decrease. Also decreasing the soil
conductivity makes the capacitance smaller in value, and it makes
no change to the electrode's self-inductance. For this reason,
decreasing the length of the electrode or the earth conductivity
increases the leakage current, the frequency band of resistive
behaviour, and the characteristic frequency. This effect is
demonstrated in Figs. 4 and 5.
Fig. 7 shows that the harmonic impedance of grounding
electrodes buried in a more conductive earth (i.e. σ= 0.1 S/ m) is
Fig. 5 Grounding impedance of 5 m vertical electrode for εr= 10
(a) Normalised magnitude, (b) Phase angle
Fig. 6 Equivalent circuit for a grounding electrode showing the
transversal and longitudinal current components
Fig. 7 Normalised magnitude and phase angle of harmonic impedance of
a 1m vertical grounding electrode for ground conductivity σ= 0.1 and
0.001 S/ m and relative permittivity εr= 3,5, 10, 20, 30
(a) Normalised magnitude, (b) Phase angle
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inductive and not much dependent on the earth permittivity.
However, a grounding electrode in highly resistive soil (i.e.
σ= 0.001 S/m) shows a capacitive high frequency response
(∠Z(jω) < 0), and increasing the soil permittivity makes its
capacitive behaviour more pronounced. As such, in this case,
increasing the permittivity of the ground will result in a reduced
characteristic frequency. In contrast, increasing the soil permittivity
of soil with a small resistivity increases the characteristic
frequency. The dependence of the characteristic frequency for
vertical 1 and 3 m − long electrodes on the ground permittivity for
various values of earth conductivity is plotted in Fig. 8. As it can
be seen, the characteristic frequency is more affected by the soil
permittivity in highly resistive grounds. Furthermore, the two
lengths of the electrode have a similar variation of characteristic
frequency as the permittivity is increased in a highly resistive soil,
while its different in a lower resistive soil.
4Time domain analysis
In this section, the influence of soil permittivity on the potential
rise of grounding electrodes is analysed in the time domain. A
lightning surge current pulse is applied to the electrode and the
potential rise with reference to remote earth is determined. The
lightning current waveforms of first and subsequent strokes
employed in this work are approximated by Heidler's formulation
[2, 50, 51] and plotted in Fig. 9. They have a rise time of 4.61 μs
and 0.49 μs, respectively, as defined in [52]. To determine the
potential rise of grounding electrodes, v(t), directly in the time
domain, Vector Fitting [53] is employed to approximate the
impedance of grounding electrodes with rational functions.
Recursive convolution method [54] is then used to obtain the time
domain potential rise of grounding electrodes, v(t). In Fig. 10, the
voltage of a vertical electrode of length 1 m buried in soil with a
conductivity of σ= 0.001 S/ m and a varying relative permittivity
is shown. The important characteristics of the induced overvoltage
in the grounding electrode v(t) include the peak and the rise time
(or the front time) [52]. The rise time influences the withstand
capability of an insulator. The initial change in the voltage is called
the surge region and corresponds to the high frequency response of
the electrode and the lightning current. Once the peak of the
voltage is passed, the stationary period is reached where the
behaviour of the grounding electrode can be estimated by its low
frequency resistance. As such, the difference between the
magnitude of the voltage in the stationary region is almost the same
for all values of relative permittivity. Nevertheless, one can see for
a conductivity of σ= 0.001 S/ m the dependence of the peak and
rise time on the relative permittivity is not significant. Fig. 11
shows the potential rise of grounding electrode for a less
conductive ground (σ= 0.0001 S/m ) where the peak and rise time
show a very strong dependence on the relative permittivity of the
ground. As the conductivity is reduced, there are two major effects
on the voltages with varying soil permittivity: first, the peak of the
voltage is considerably reduced as the relative permittivity varies
from εr= 3 to εr= 30, especially in the case of the subsequent
stroke. The percentage of variation in the peak is 3.5 and 24.9% for
the first (Fig. 11a) and subsequent (Fig. 11b) strokes, respectively.
Fig. 8 Dependence of the characteristic frequency (fc) on the dielectric
constant and frequency for a vertical electrode of length 1 and 3m
Fig. 9 First and subsequent return stroke current waveforms
Fig. 10 Grounding electrode potential rise of a 1 m vertical electrode in a
soil with a conductivity of σ= 0.001 S/ m and relative permittivity of εr= 3,
5, 10, 20, and 30 due to the first and subsequent stroke currents
Fig. 11 Grounding electrode potential rise of a 1 m vertical electrode in a
soil with a conductivity of σ= 0.0001 S/m and relative permittivity of
εr= 3, 5, 10, 20, and 30 due to
(a) First stroke current, (b) Subsequent stroke current
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Similar variation for the case of σ= 0.001 S /m is 2 and 6% only.
This is in agreement with Fig. 7, where it was shown that the effect
of soil permittivity is more pronounced at higher frequencies and
more resistive soil. The second important characteristic of the
potential rise of grounding electrode is the rise time. As the
permittivity is increased, the rise time becomes longer, which is
again more affected in the case of a subsequent stroke. The rise
time of the voltage in Fig. 11a is 6.05 μs for εr= 3 and 9.27 μs for
εr= 30. For the potential rise of grounding electrode corresponding
to the subsequent stroke, the rise time varies from 0.97 to 4.16 μs.
The other parameter that has an influence on the grounding
impedance is the length of the grounding electrode. As shown in
Fig. 12, increasing the length of the electrode to 5 m results in a
considerable decrease of the voltage peak. Furthermore, increasing
the length of electrode results in a bigger rise time. For a 5m
electrode, the variation of the peak of potential rise of the
grounding electrode due to the first and subsequent stroke currents,
when ε varies from 3 to 30, is 2 and 16.86%, respectively. For the
case of the first stroke, the rise time of the grounding electrode
potential in a soil of εr= 3 and εr= 30 is 5.75 and 8.11 μs,
respectively. For the case of a subsequent stroke, the rise time
varies from 0.81 to 3.93 μs. It can be concluded that as the length
of the electrode increases, the effect of soil permittivity is less, but
it is still significant, especially for the subsequent lightning strokes.
This can be due to the fact that increasing the length of the
electrode makes the inductive behaviour more pronounced and the
variation of soil permittivity and capacitance of the electrode less
important.
5Conclusions
This paper presented a full-wave electromagnetic simulation model
to determine the potential rise of grounding electrodes due to
lightning return stroke and study the influence of ground
conductivity and permittivity on the peak and rise time of the
grounding electrode potential. Along with the length of the
electrode and the soil resistivity, the value of the soil dielectric
constant was shown to highly affect the grounding impedance of
rods. This effect is more pronounced as the frequency or the soil
resistivity are increased, where the changes in the grounding
impedance can be as high as 95% reduction for a soil of resistivity
σ= 0.0001 S/m and frequencies beyond 5 MHz when the relative
permittivity of the soil is changed from εr= 3 to 30. The changing
trend of the characteristic frequency (the frequency above which
the impedance of a grounding electrode is not resistive anymore)
with the increase of the dielectric constant depends on the soil
resistivity. It was shown that the characteristic frequency reduces
for highly resistive soils. In the time-domain calculations, two
major effects were observed as the electrical permittivity of the soil
was varied. First, the peak of the grounding electrode potential
decreases by up to 25% for highly resistive soil as the permittivity
is changed from 3 to 30. Secondly, the rise time of the grounding
electrode potential was increased by a factor of up to 4 for a
vertical electrode of length 1 m. Both effects were more noticeable
for subsequent lightning strokes. Although increasing the electrode
length reduced the dependability of the developed overvoltages on
the variation of the soil permittivity, it was still highly affecting the
peak and rise time of the grounding electrode voltage. The
proposed simulation model in this paper can be applied to the
analysis of other type of grounding electrodes and its accuracy can
be compared to field measurements in future studies. The high cost
and complexity of performing field measurements highlights the
importance of numerical simulation models, such as the one
developed in this paper.
6Acknowledgments
The authors are thankful to Canadian Foundation for Innovation
(CFI) and Prof. Ian Jeffrey for providing the computational facility.
Financial support from Manitoba Hydro International (MHI) and
Mitacs is acknowledged.
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