High Frequency and Transient Coupling to Pasive Conductors Near Grounding Systems

in Layered Soil

Vesna Arnautovski-Toseva (1), Leonid Grcev (1), Spase Petkoski (1) and Khalil El Khamlichi Drissi (2)

(1) Ss Cyril and Methodius University, Faculty of Electrical Engineering and Information Technologies

Skopje, Macedonia, Email: (atvesna,lgrcev}@feit.ukim.edu.mk (2) University Blaise Pascal, LASMEA

Aubière, France, Email: drissi@ lasmea.univ-bpclermont.fr

Abstract—This paper presents the ongoing research results of the high frequency and transient grounding system analysis in two-layer soil structure. On the basis of rigorous electromagnetic field theory, which involves Sommerfeld’s integrals, the mathematical model is formulated by the mixed potential integral equation (MPIE). Detailed analysis of a high frequency performance of a typical horizontal grounding conductor placed in the upper or in the bottom layer is given. It is shown that because of the presence of two distinct layers the current distribution and the impedance to ground are highly affected by the parameters of both soil layers. Also, the current distribution in near-by passive horizontal conductor is analyzed. At the end of the paper some observations about the effects of two-layer soil structure at high frequencies are given.

Keywords-grounding; electromagnetic model;

I. INTRODUCTION The practical studies and experimental investigations

indicate that uniform soil parameters throughout the entire area and to considerable depth of interest are seldom found. Considering the realistic soil environment, it is shown that non-homogenous soil could be adequately represented by an equivalent multi-layer soil structure, by the presence of several horizontal layers. In practice it is assumed that the two-layer model of soil stratification is reasonably valid for non-uniform soil representation. The survey of the published work in this area implies that traditional analysis of grounding systems in two-layer soil is performed using static and quasi-static theory concepts of images [1], and modified images [2]. The electromagnetic model, which is developed on the rigorous electromagnetic approach [3], considers homogenous soil only. However, a comparison study of exact and quasi-static model for uniform soil, has suggested that quasi-static model is not valid at high frequencies (HF) [4]. Our research in this field resulted in development of a new electromagnetic model for HF analysis of grounding systems placed within the upper layer of two-layer soil [7]. The validation of this model is

achieved by comparison with the quasi-static model of images [8], and the electromagnetic model developed previously for uniform soil [9].

In this paper, we will present more general analysis of high frequency behavior of simple horizontal grounding conductor placed within two-layer soil, where the position of the conductor is assumed to be within the upper or within the bottom layer. Also, of interested in this analysis is the frequency domain behavior of a near-by horizontal passive conductor placed parallel to the grounding conductor in the same layer. The main objective of this work is to give some general observation of the high frequency behavior of grounding systems within two-layer soil in general, as well as to investigate the behavior of near-by passive conductor with respect to frequency. The mathematical model will be presented in section 2. By comparing the results obtained by using the two-layer soil model and the homogeneous soil model, in section 3, we will give some observations about the adequacy of the application of the two-layer soil model in practical lightning studies

II. MATHEMATICAL MODEL

A. Model of the Grounding Conductor and the Soil The electromagnetic model developed for high

frequency and transient analysis of grounding systems is based on rigorous formulations derived from the full set of the Maxwell’s equations, on the theoretical background of antenna analysis [5]. It is based on the integral equation for the electric field due to current and charges along grounding wires in terms of the vector and scalar potentials, so called: mixed potential integral equation. The exact Green’s functions involve Sommerfeld type integrals for the layered media and their direct numerical integration. For the excitation of the grounding conductor by injection of current in one of its end, this solution gives current distribution along the grounding conductor, impedance to ground, electric field and potentials in arbitrary points. The detailed description

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of the model which is developed for uniform soil is given in [3]. Later, this model is extended for high frequency analysis of grounding systems within the upper layer of two-layer soil [7]. For the purpose of this paper, the brief description of the electromagnetic model in general is given within this section.

We consider a single x-directed horizontal electrode of length L placed at depth H within the upper or within the bottom layer of a two-layer soil. The passive wire may be placed in the same or in opposite layer. Following the thin-wire approximation, the physical model of an electrode is based on the fictitious wire segmentation into straight tubular segments. The harmonic current excitation is assumed, which is simulated by an ideal time-harmonic current generator with one terminal connected to the grounding system and the other terminal to infinity, which yields frequency-domain response of the system.

To solve current distribution the method of moments is applied using Galerkin formulation with triangular basis and test functions [8]. The wires are segmented in fictitious segments and the current distribution is approximated by overlapped triangular expansion functions (“triangular dipoles”). Excitation is approximated by an additional triangular monopole. Following matrix equation yields current distribution:

][]][[ SS IZIZ −= (1)

Here, the column matrix [I] represents the unknown current samples, [Z] is generalized impedance matrix related to mutual impedances between triangular dipoles, [-ZSIS] is excitation matrix where IS is current injected at a point in the grounding conductors and ZS is column matrix related to mutual impedances between each of the triangle dipoles and the injection triangle monopole. [Z] matrix is symmetrical and only half of the elements have to be evaluated. For the case of one linear wire [Z] is Toeplitz matrix and only one row have to be computed.

0 – AIR

Ig L

1 – UPPER LAYER

2 – BOTTOM LAYER

z yx

H

Grounding conductor

Figure 1. General view of the end-driven single grounding electrode

and a near-by passive condutor in the same layer.

The soil and the air are modeled as linear medium characterized by corresponding permittivity, permeability and conductivity. Figure 1 shows a single grounding conductor in two-layer soil and near-by passive conductor in the same or in the opposite later. It is assumed that the two-layer soil model consists of an upper layer of finite height d1 and characterized by relative permittivity εr1, permeability μ0 and resistivity ρ1; and a bottom layer which is characterized by relative

permittivity εr2, permeability μ0 and resistivity ρ2. Air is characterized by permittivity ε0 and permeability μ0.

B. Calculation of the impedance matrix To determine the impedance matrix [Z], each element

of the matrix is determined as self or mutual impedance zmn between two wire segments (m-observation) and (n-source) carrying current In:

mn n mmn

n n

uz

I I⋅

= =E l

(2)

Here, En is the tangential electric field at the surface of the observation segment lm due to current In along the source segment.

( )n nj nω φ= − + ∇E A (3)

Here, the vector and scalar potentials are represented by their integral forms, involving current density In and charge density qn in the source segment of length ln:

∫ ⋅=nl

nnAn dI lGA , ∫=nl

nnn dlqGφφ 1 nn

n

dIq

j dlω−

= (4)

Here, AG is the dyadic Green's function for the magnetic vector potential, and Gφ is the scalar potential Green's function due to elementary horizontal electric dipole (HED) in layered medium. They are first obtained in spectral domain and take into account the reflection from the boundaries. The spectral domain Green’s functions of vector potential and scalar potential are related to the longitudinal components of the spectral domain fields, which are proportional to the spectral domain transmission coefficients [6]:

0

2xxA

z

Gj kμ

= TET (6)

2

2

1 12

TMTE TEz

qr z z

k TG T Tj k jk zkρε

⎡ ⎤⎛ ⎞∂= + ±⎢ ⎥⎜ ⎟∂⎢ ⎥⎝ ⎠⎣ ⎦

(7) Passive conductor

Here, TTE and TTM are plane wave transmission coefficients from the plane of the source to the plane of the observation; kz is vertical wave number for the layer of the source and 2

0020

22 )( kjkkk rrz ωεσεερ −==+ . The spatial domain Green's functions are obtained by numerical solving the Sommerfeld-type integrals

∫∞

=0

0,, )()(~21

ρρρρ ρπ

dkkkJkGG qAqA . (8)

C. Calculation of the impedance to ground Once the currents in segments of the grounding

conductors are computed, the electric field can be

computed by summing their contributions. The voltage between the feed point and a remote ground VG, might be computed by integrating the electric field vector along a path to infinity perpendicularly to the electrode and starting from the surface of the conductor. However, this is equivalent to the scalar potential at the feed point. Therefore, impedance to ground ZG is:

][][][][][1 1 ZZZZIII

VZ SSS

GG ′⋅⋅=′⋅== − . (9)

Here [Z'] is a row matrix with scalar potentials at the excitation segment due to currents in all segments.

III. NUMERICAL RESULTS

A. Analysis of a single grounding conductor In this section some of the results that show the high

frequency behavior of a single horizontal grounding conductor within two layer soil are presented.

The studied grounding conductor is typical end-driven linear horizontal wire: short (10m) and long (50m) with radius of 0.005m placed at depth H=-0.75m within the upper or the bottom layer. The conductor is energized by a time-harmonic current generator with amplitude of 1A in frequency range from 1kHz to 10MHz. Since there is huge number of combination of parameters of the two-layer soil model we are restricting our analysis to the cases 1 and 2 given in the table 1.

Figure 2. Magnitude and phase of a longitudinal current along 10m

short grounding conductor within two-layer soil at frequencies 0.1MHz, 1MHz and 10MHz.

TABLE I. PARAMETERS OF BOTH LAYERS

Upper layer depth d1

(m)

Upper layer ρ 1 (Ωm)

Bottom layer

ρ 2 (Ωm) Case 1

Conductor in the upper

layer

1.0

100

a) 1000 b) 10

Case 2 Conductor in

the bottom layer

0.5

a) 1000

b) 10

100

The layer containing the grounding conductor (upper or bottom) is assumed by resistivity fixed to 100Ωm. It is assumed that the relative permittivity of both layers is 10. The resistivity of the neighbor layer is assumed by: a) 1000Ωm and b) 10Ωm.

The depth of the upper layer is considered as: Case 1) 1.0m - when the grounding conductor is placed within the upper layer and Case 2) 0.5m - when the grounding conductor is placed within the bottom layer. In order to compare the results obtained for homogeneous soil and two-layer soil, it is assumed that homogeneous soil model is characterized by resistivity of 100Ωm.

Figure 2 shows the magnitude and phase of the current distribution along the short 10m horizontal conductor. The studied frequencies are: 0.1MHz (solid), 1MHz (dash) and 10MHz (dot).

Figure 3. Magnitude and phase of a longitudinal current along 50m

long grounding conductor within two-layer soil at frequencies 0.1MHz, 1MHz and 10MHz.

The results are obtained for two-layer soil with parameters given in table I: Case 1 (blue-green), and for Case 2 (red-magenta). For comparison, the results obtained for homogeneous soil are represented in grey.

It is obvious that in case of a short horizontal conductor the influence up to 0.1MHz the current distribution profiles correspond to the typical quasi-static behavior. The current discharged into the ground is almost constant except at the conductor extremities, which is in agreement with [1]. As the frequency increases, it is observed that the longitudinal current decreases very fast. For 1MHz, the wavelength in the layer containing the conductor is about 22m that is comparably to the length of the conductor. At this frequency the current distribution profiles show significant differences. For 10MHz, the current distribution profiles become closer and almost overlap, which lead to the conclusion that at very high frequencies the short horizontal conductor placed within two-layer soil is practically behaves like placed in homogeneous soil.

In comparison with the previous results, the performance of a long 50m conductor placed within two-layer soil is much more function of the frequency, and is also significantly influenced by the properties of the two-layer soil model. Figure 3 shows magnitude and phase of the current distribution along the 50m long horizontal conductor at frequencies: 0.01MHz (solid), 0.1MHz (dash) and 1.0MHz (dot). The results show that up to 0.01MHz the longitudinal current distribution profiles correspond to the typical quasi-static profiles, which correspond to [1]. Significant differences are observed for frequencies in range of 0.1MHz (the wavelength in the layer containing the conductor is 70m that is close to the length of the conductor). As the frequency increases up to 1MHz, it is observed that a large partition of the injected current is discharged through the small section of about 25% of the conductor length, especially in case when next layer is slightly resistive.

B. Analysis of the passive near-by parallel conductor In this section we present some of the results that

show the longitudinal current along passive near-by conductor placed parallel to 10m (short) grounding conductor in the same layer. The length of the passive conductor is assumed to be: 10m and 20m. The passive conductor is positioned at the same depth (-0.75m) at 0.5m parallel to the grounding conductor. Figures 4 and 5 show the current along 10m and 20m passive parallel conductor at frequencies: 0.01MHz (solid), 0.1MHz (dash) and 1MHz (dot).

IV. CONCLUSION This paper presents the high frequency

electromagnetic analysis of horizontal grounding conductor within two-layer soil. The results lead to the following conclusions:

Figure 4. Longitudinal current along 10m passive near-by conductor

parallel to 10m grounding conductor within two-layer soil at 0.01MHz, 0.1MHz and 1MHz.

Figure 5. Longitudinal current along 20m passive near-by conductor

parallel to 10m grounding conductor within two-layer soil at 0.01MHz, 0.1MHz and 1MHz.

• The current distribution along horizontal grounding conductor within two-layer soil is strongly affected by the parameters of both layers especially for frequencies when the wavelength is close to the length of the conductor. For longer conductors, these effects are observed in the lower frequency range – from few kHz up to 1MHz. At very high frequencies over 1MHz, the effects of the two-layer soil model is approaching that of a homogeneous soil model.

• The two-layer soil parameters have strong influence on the current distribution along passive parallel conductor in the lower frequency range, especially when the neighbor layer is much more resistive.

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[9] V. Arnautovski-Toseva, L. Grcev, K. El Khamlichi Drissi, “Calculation of the harmonic impedance of a single horizontal grounding wire in two-layer soil by the electromagnetic model”, IEEE International Symposium on Electromagnetic Compatibility (EMCEurope04), 23-25 Sept. 2004, Eindhoven, The Netherlands, pp. 887-892.