Journal of Electromagnetic Analysis and Applications, 2012, 4, 475-480 Published Online December 2012 (
Computer Analysis of Electromagnetic Transients in
Grounding Systems Considering Variation of Soil
Parameters with Frequency
Marco A. O. Schroeder1*, Márcio M. Afonso2, Tarcísio A. S. Oliveira2, Sandro C. Assis3
1Department of Electrical Engineering, Federal University of São João del-Rei, Laboratory of Applied Electromagnetism, São João
del-Rei, Brazil; 2Department of Electrical Engineering, Federal Center of Technological Education of Minas Gerais, Belo Horizonte,
Brazil; 3Energetic Company of Minas Gerais, Belo Horizonte, Brazil.
Email: *
Received September 15th, 2012; revised October 14th, 2012; accepted October 25th, 2012
This paper presents a mathematical model to calculate transients in grounding systems. The derived equations arise
from direct application of basic electromagnetic equations in frequency domain, whose solution is obtained by the ap-
plication of the Moment Methods. A formulation based on experimental measurements is applied to quantify the soil
parameters for each frequency. The unified approach is applied in the calculation of the grounding impedance of hori-
zontal electrodes. Results show that the inclusion of frequency dependence of the soil parameters leads to a reduction of
the values of grounding impedance, in comparison with results for soils with parameters independent of frequency.
Keywords: Grounding Electrodes; Grounding Impedance; Transient Response; Frequency Response; Electromagnetic
1. Introduction
The grounding systems are an important element for good
electrical systems performance, mainly when they are
subjected to faults. Their basic function is to disperse the
current of fault to earth without causing any potential dif-
ferences or induced voltages that might endanger people
or damage equipments. Grounding performance and de-
sign at low frequencies are well established and described in
international standards [1]. However, when energized by
lightning currents, they present a very particular behavior
and their analysis cannot be carried out with traditional
methodologies employed in low frequency occurrences [2].
The transient analysis of grounding electrodes is usually
developed based on three main different approaches: 1)
electromagnetic field theory [3]; 2) transmission line
theory [4], and 3) circuit theory [4]. The first one is con-
sidered as the most accurate since it is based on least
neglects in comparison to the methods based on trans-
mission line and circuit theory.
Further, in the grounding study, it is of major impor-
tance the adequate soil modeling [5-13]. In most works
dealing with lightning transients in grounding, the soil
electrical conductivity and permittivity are usually as-
sumed to be frequency independent (e.g. in [3,4,14]).
Typically the soil conductivity is assumed as the value
measured by conventional measuring instruments, which
employ low frequency signals [2]. In the same approach,
the soil relative permittivity is assumed to vary from 4 to
81, depending on the soil humidity [2]. However, meas-
urements of the soil electromagnetic behavior show that
both parameters are strongly frequency dependent [6-13].
Experimental data obtained in [6-12] for a large number
of soil samples indicate an increase of the soil conducti-
vity and a reduction of the relative permittivity when
frequency rises from about 100 Hz to 2 MHz. An expla-
nation physically consistent of these behaviors is de-
scribed in [13]. The soil magnetic permeability is, in
general, equal to vacuum magnetic permeability [6].
In this work a methodology based on the electromag-
netic field theory and Moment Method is proposed to
evaluate the impulse behavior of grounding. The tran-
sient is solved in the frequency domain. Proper relations
to quantify the soil electric conductivity and permittivity
for each frequency are employed. The developed me-
thodology is applied in the calculation of the grounding
impedance of horizontal electrodes.
This paper presents two main contributions in the ana-
lyses of grounding transients. First, the presented results
are based on an accurate mathematical model, which take
into account the propagation effects and electromagnetic
*Corresponding author.
Copyright © 2012 SciRes. JEMAA
Computer Analysis of Electromagnetic Transients in Grounding Systems
Considering Variation of Soil Parameters with Frequency
interaction between the grounding elements. Second, a
formulation to quantify the frequency dependence of the
soil parameters is included and an evaluation of the in-
fluence of such dependence is developed.
This article is organized as follows: presented in Sec-
tion 2 the electromagnetic grounding model; in Section 3
is described the soil modeling; the results are presented,
discussed and interpreted in Section 4 and finally in Sec-
tion 5 outlines the conclusions.
2. The Electromagnetic Grounding Model
The Hybrid Electromagnetic Model (HEM) was used to
simulate the grounding behavior. The formulation of this
model was first proposed by Visacro and Portela [15],
who developed a frequency-domain model to address the
transient response of grounding electrodes using Fourier
Transform. This model employs the scalar electric poten-
tial and the vector magnetic potential to take into account
the electromagnetic coupling between the grounding ele-
ments. Later, this formulation evolved to the so-called
HEM model, detailed by Visacro and Soares in [16], to
address the simulation of general lightning related engi-
neering problems, such as overvoltages developed by di-
rect lightning strikes and voltages induced by nearby
strikes [17,18].
The electromagnetic model is based on the fundamen-
tal idea of represent a current-carrying conductor as a
source of transversal and longitudinal currents. The trans-
versal current crosses the conductor surface and is spread
into the surrounding medium and the longitudinal current
flows along the conductor. Such idea was first explored
in [16] to model cylindrical conductors carrying lightning
currents. Here, it is applied in modeling grounding sys-
tems under transient conditions.
The grounding system is represented by a set of cylin-
drical electrodes immersed in the soil. Each electrode is
source of a transversal current IT and a longitudinal cur-
rent IL, as illustrated in Figure 1. The current IT gene-
rates a divergent electric field at a generic point that es-
tablishes a potential rise in relation to remote earth in
such point [19]. Considering each pair of electrodes, as
illustrated in Figure 2, this current yields capacitive and
conductive coupling (self and mutual ones). The current
IL generates a nonconservative electric field, which inte-
gration between two different points results in a voltage
drop along the defined path [19]. Considering each pair
of electrodes, as illustrated in Figure 2, this current yields
inductive coupling (self and mutual ones).
The electromagnetic coupling between the electrodes
can be computed by electromagnetic fields integration
along each one and leads to the following equations [16]:
where Vij refers to average potential of electrode i due to
the transversal current dispersed by electrode j; ΔVij is
Figure 1. Grounding system representation.
Figure 2. Electrodes interaction.
ijTjj i
jLL r
 ll, (1)
ijLjj i
VjI l
 
 , (2)
the voltage drop along electrode i due the longitudinal
current flowing along electrode j; σ, ε and μ are, respec-
tively, the medium electric conductivity, electric permit-
tivity and magnetic permeability;
is the propagation
constant and ω is the angular frequency.
The final solution is obtained by the application of
Moments Method (MoM) [20]. The electrode (length L)
is divided into N elements of length LN. The value
of N is determined in a way that the thin-wire approxi-
mation is valid for the longitudinal current along the
element. Also, the length of each element is sufficiently
small, so that the longitudinal current along it may be
considered uniform. In this way, the longitudinal current
is represented by a piecewise-constant distribution. In a
similar approach the current that diverges from each ele-
ment is assumed to be uniform along it. The choice of
these basis functions to represent IT and IL leads to a
straightforward physical interpretation. The longitudinal
current of an element N is exactly that of the element N-1
subtracted the transversal current dispersed to the soil
from this last one.
From the above considerations, the transversal
and the longitudinal
I current distributions along
Copyright © 2012 SciRes. JEMAA
Computer Analysis of Electromagnetic Transients in Grounding Systems
Considering Variation of Soil Parameters with Frequency
the electrode are represented by a linear combination of
N basis function, or:
1 on
0 o
Tn n
, (3)
Ln n
, (4)
the -th element; and
P (5)
In Equations (3) and (4), Tn
are unknown
coefficients and correspond, respectively, to the trans-
versal and longitudinal current of the n-th element. To
determinate such coefficients N independent linear equa-
tions are necessary. These equations are obtained by the
substitution of Equation (3) in Equation (1) and Equation
(4) in Equation (2) and by considering the self and mu-
tual interactions between the N elements. From that two
systems of linear equations can be obtained: V = ZTIT and
V = ZLIL. The terms of V and IT correspond, respectively,
to the average potential and transversal current in each
element and the terms of ZT are defined as the transversal
impedance between the elements i and j. The terms of ΔV
and IL correspond, respectively, to the voltage drop and
longitudinal current in each element and the terms of ZL
are defined as the longitudinal impedance between the
elements i and j. In ZT and ZL computation, the influence
of air-soil interface is taken into account by means of the
modified image theory [21]. In this case, the image
source is symmetrically positioned with reference to real
source and has magnitude equal to the real one multiplied
by a coefficient, which depends on the electromagnetic
characteristics of air and soil [21]. The reciprocity theo-
rem is valid and, thus, the Z-matrices are symmetrical
and only half of the elements have to be evaluated, what
permits a substantially reduction of computational time.
The two systems of linear equations may be reduced to a
unique system by means of two fundamental considera-
tions [16,19]: 1) the average potential in each element is
equal to the arithmetic media of nodal potentials and the
voltage drop in each element is equal to the difference of
nodal potentials; 2) the Kirchhoff’s current law is explic-
itly enforced over all element’s nodes. Considering these
two assumptions, an only system of linear equations AVN
= b is composed [16,19]. The matrix A is a composition
of original Z-matrices, the vector VN corresponds to nodal
potentials and b refers to external current injected into
each node (null value for nodes with no current injection).
By the system solution, all nodal potentials are deter-
mined and transversal and longitudinal currents of each
element can be immediately calculated. From these vari-
ables, other quantities may be obtained, as the potential
in node of current injection and the grounding impedance.
The described procedure provides solution for only
one frequency. The solution must be determined for a
range of frequencies, which depends of the analyzed
problem (short circuit or lightning discharge, for exam-
ple). From the frequency response, the time domain re-
sponse may be obtained by application of the inverse
Fourier transform. The main motivation to solve the tran-
sient in frequency domain is to include the frequency
dependence of the soil parameters [13].
3. Soil Modeling
The presented work considers a formulation for express-
ing the frequency dependence of the soil parameters (σ
and ε) originated from field measurements [6-9]. A very
large number of different soil and geological structures in
Brazil were tested in the frequency range of 100 Hz to 2
MHz, as described in [6-9]. After the measurements, the
soil samples were fitted to a curve assuming [6-9]:
cotang 22π10
ji j
 
 
 
 
 
, (6)
where σ0 is the electric conductivity at low frequency and
i and α are statistical parameters, which are responsible
for the frequency dependence of soil conductivity and
permittivity [6-9]. To evaluate the probability density
functions associated with parameters i and α, Weibull
approximations were considered. As discussed in [6-9],
for most purposed, it may be acceptable to consider me-
dian values for both i and α, which are, respectively,
11.71 S/m and 0.706. Only as a reference, considering
the median values, Equation (6) indicates a variation of εr
from about 3000 to 170, when frequency varies from 100
Hz to 2 MHz. In the same frequency range, a soil with σ0
= 1 mS/m has it conductivity increased to about 6 σ0. It is
important to observe that even though ε decreases with
frequency, the relation ωε still increases with the fre-
quency rise. Further details can be seen in [13].
A valuable aspect of the presented expression is that it
was developed from measurements performed in field
conditions and with the natural soil humidity in contrast
with the laboratorial experiments of the classical works
[10,11]. A detailed description of the measurement pro-
cedure is presented in [6-9], including experimental setup,
methods for soil sampling, and checking physical con-
It is worth to mentioning that such formulation has
been extensively used in the last few years to investigate
the influence of frequency dependent soil parameters in
transmission line modeling, for example, in [7-9]. Also,
has been used to investigate the influence of frequency
dependent soil parameters in electric fields of grounding
Copyright © 2012 SciRes. JEMAA
Computer Analysis of Electromagnetic Transients in Grounding Systems
Considering Variation of Soil Parameters with Frequency
electrodes in [13]. Nevertheless, the impact of the fre-
quency dependence of soil parameters in the grounding
impulse behavior is still an open issue. The next section
presents a preliminary analysis of such impact, consider-
ing the calculation of the grounding impedance of some
simple electrode configurations. In simulations, the va-
riation of soil parameters with frequency is computed ac-
cording to Equation (6).
4. Results
4.1. System under Study
The evaluated grounding configuration corresponds to ho-
rizontal electrodes, buried in a depth of 0.5 m, with ra-
dius of 1 cm and of lengths ranging from 5 to 80 m.
Two soil models were adopted, that is: Soil 1—Soil
represented by its low-frequency electric conductivity σ0
(three representative values considered in this work: 10,
2 and 1 mS/m) and relative permittivity equal to 15, both
parameters frequency independent. Soil 2—Soil with the
inclusion of frequency-dependent parameters and the
same previous low-frequency conductivity values. In si-
mulations, the effects of the soil ionization are disre-
The electrode excitation was obtained by the injection
of a typical double exponential current wave of 1 kA and
1.2/50 μs. In all simulations, the current injection was
made in the electrode termination.
The following definitions for the analyzed quantities
are adopted in this paper:
Harmonic impedance:
 
, where
are phasors of the injected cur-
rent and of the potential at the injection point, respec-
Impulse impedance:
VI, where Vp is the peak
value of the transient voltage at the injection point
and Ip is the peak value of the injected current.
4.2. Harmonic Impedance
To evaluate the influence of the soil parameters variation
with frequency on the harmonic grounding impedance, a
60-m long horizontal electrode is considered, buried in
both Soil 1 and 2, with σ0 = 10, 2 and 1 mS/m. Figure 3
illustrates the amplitude [|Z(ω)|, Figure 3(a)], and the
angle [θ(ω), Figure 3(b)], of the harmonic grounding
impedance Z(ω). The harmonic impedance is frequency
independent and equal to the low frequency ground re-
sistance R, in the Low Frequency (LF) range, for both
Soil 1 and 2. Nevertheless, as frequency rises, a different be-
havior of Z(ω) is observed depending on the soil model.
Results for Soil 1 exhibit an inductive behavior and the
amplitude of Z(ω) becomes larger than R, being this ef-
f (Hz)
Soil 1
Soil 2
2 mS / m
10 mS/m
1 mS/m
f (Hz)
) (degrees)
10 mS/m
2 mS/m
1 mS /m
Soil 1
Soil 2
Figure 3. 60 m horizontal electrode. (a) Amplitude; (b) An-
gle of the harmonic grounding impedance.
fect more relevant for less conductive soils. These results
are in perfect consonance with some classical ones, for
example, those presented by Grcev in [14]. On the other
hand, when the soil parameters dependence with fre-
quency is considered (Soil 2), the capacitive effect be-
comes relevant, especially for high-resistivity soils [θ(ω)
< 0, Figure 3(b)]. The capacitive effect plays an impor-
tant role in the grounding performance and is responsible
for the reduction of |Z(ω)|. Indeed, as may be observed in
Figure 3(a), the frequency dependence of the soil pa-
rameters leads to reduction of the amplitude values of
Z(ω) in the High Frequency (HF) range.
4.3. Impulse Impedance
Figure 4 shows the simulation results for the low fre-
quency ground resistance R and the impulse impedance
Zp of horizontal ground electrodes in a range from 5 to 80
m, for both Soil 1 and 2, with σ0 = 10, 2 and 1 mS/m.
According to Figure 4, the value of the ground resistance
is larger for less conductive soils and decreases with in-
creasing the electrode length. Similarly, the impulse im-
pedance, for both soil models, presents larger values for
less conductive soils and decrease with increasing the
Copyright © 2012 SciRes. JEMAA
Computer Analysis of Electromagnetic Transients in Grounding Systems
Considering Variation of Soil Parameters with Frequency
010 20 30 4050 60 70 80
Electrode Length (m)
R, Z p (
2 mS / m
1 mS/m
Zp, Soil 1
Zp, Soil 2
10 mS/m
Figure 4. LF ground resistance R and impulse impedance
Zp of horizontal ground electrode of 5 - 80 m, in both Soil 1
and 2, and σ0 = 10, 2 and 1 mS/m.
electrode length. Nevertheless at a certain length it be-
comes constant, while the LF resistance continues to de-
crease. Therefore, only a certain electrode length is ef-
fective in controlling the impulse impedance, which is
referred as effective length ef [14]. The effective length
decreases with soil conductivity σ0 and frequency rise.
This can be understood as both parameters are responsi-
ble for increasing ground losses, leading to an increase in
the attenuation of the current wave that propagates along
the electrode [2]. It means that as the wave is attenuated,
the electrode length that is effectively used to disperse
the lightning current is reduced.
Figure 4 also shows two very important differences
between simulated results for Soil 1 and 2. First, it may
be observed that the effective length is larger for a soil
with frequency-dependent parameters (Soil 2). This ef-
fect is more appreciable in less conductive soils and pro-
bably is related to the reduction of the attenuation con-
stant of soil in the intermediate frequency range, when its
parameters are assumed to be frequency dependent. Se-
cond, the values of impulse impedance for soil with fre-
quency variation of its parameters (Soil 2) are lower than
that obtained for soil represented by only its LF parame-
ters (Soil 1), especially for high-resistivity soils. Such
difference between the obtained values becomes larger
above the effective length for Soil 1, since the impulse
impedance continues to decrease for Soil 2. As a conclu-
sion, a significant reduction of the overvoltages, due to
the soil parameters variation with frequency, is expected
for electrodes lengths larger than the effective one for
Soil 1.
5. Conclusions
An accurate methodology to calculate transients in ground-
ing systems, which include the frequency dependence of
the soil parameters, is proposed. Results show that:
1) The inclusion of frequency dependence of the soil
parameters leads to a reduction of the values of ground-
ing impedance and a increasing of the values of effective
length, in comparison with results for soils with parame-
ters independent of frequency, especially for high-resis-
tivity soils;
2) The consideration of frequency independent soil
parameters leads to conservative values of grounding im-
pedance. So, new investigations are still necessary con-
sidering other formulations to determinate soil parame-
ters in function of frequency.
6. Acknowledgements
The authors gratefully acknowledge the financial support
provided by Energetic Company of Minas Gerais (CE-
MIG) and also the electrical engineer R. S. Alípio for
valuable discussions and contributions.
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