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Four-electrode method is one of the well-known methods in measuring ground resistivity. But, most faults currents and lightning currents have high frequencies components. It is proposed to develop this method to study ground frequency characteristics. A step like current was injected into ground to measure the ground impedance. The ground impedance is assumed to be frequency dependent parallel resistance/capacitance. Two equations were proved to estimate ground resistivity and permittivity from four-electrode method. An analytical model was proposed to model studied cases. The four electrodes are divided to equal spheres and complex image method had been used to satisfy the boundary conditions and penetration depth effects. The calculated results show good agreement with the measured results.

When designing a grounding system for a specific performance objective, it is necessary to accurately measure the ground resistivity of the site where the ground is to be installed. Grounding system design is an engineering process that removes the guesswork and “art” out of grounding. It allows grounding to be done “right, the first time”. The result is a cost savings by avoiding change orders and ground “enhancements” [

The ground impedance frequency characteristic plays an important role in understanding and designing grounding systems. To investigate this issue, samples of ground are tested in laboratories [2,3]. The characteristics of these samples will be changed due to ground excavation, temperature and humidity. There are other methods used in prediction of ground parameters and it depends on electromagnetic wave transmitted and reflected from ground or grounding system analysis [4,5].

Grounding resistivity measuring methods depend on injecting a current through the ground via the probe electrodes. The current flowing through the ground (a resistive material) develops a voltage/potential difference. There are different methods [6,7] such as, four electrode method, deep electrode method and two electrode method to measure and obtain ground resistivity. The most accurate method in practice of measuring the average resistivity of large volumes of undisturbed earth is the four-electrode method. The electrode configurations commonly used for ground resistivity measurements are the Wenner and Schlumberger, illustrated in Figures 1(a) and (b), respectively. Approximating the current electrodes by hemispheres, the apparent soil resistivity ρ_{app }can be computed using the following Equations [

Wenner Method: (1)

Schlumberger Method: (2)

When the adjacent current and potential electrodes are close together, the measured ground resistivity is indicative of surface ground characteristics. When the electrodes are far apart, the measured ground resistivity is indicative of average deep ground characteristics throughout a much larger area.

This paper studies the frequency dependence of ground impedance by injecting a step like current in outer electrode of four electrodes method. By using successive image method, four electrodes are modeled in ground with permittivity and conductivity parameter of ground. The ground impedance by the proposed method is studied for different ground parameters and different frequencies.

The ground resistivity and permittivity is obtained from measured voltage and current waveforms due to wave propagation in the ground to study the effect of frequency.

The current rise time 20 ns is injected for different distances between electrodes. Four electrodes are buried in the ground, as shown in

A current, as shown in

The ground impedance is obtained from the measured voltages and currents (C1, C2, P1, P2). The four electrode method at low frequency is used to obtain the grounding resistance by dividing the potential difference

between two inner electrodes by the injected current at the outer electrodes as follows:

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In the same manner the ground impedance at different frequencies Z_{G}(f) is obtained. The current at injected points to ground (C1, C2) is distorted due to the induced voltage between ground and connection wires. The voltage waveforms at any frequency (f) at inner electrodes are reformed to be as a result of current I by multiplying them V_{p}_{1} and V_{p}_{2} by I/I_{C}_{1} and I/I_{C}_{2} as follows:

The ground impedance is assumed to be consists of parallel resistance R_{G}(f) and capacitance C_{G}(f). From the obtained ground impedance Z_{G}(f) the ground resistance and capacitance are obtained. The ground resistivity is obtained for non-equal four electrode method by the following equation:

As the ground permittivity can be obtained.

Assume a wave travels into a conducting medium [

where: d = penetration depth m.

It gives the variation of E_{y} or J_{y} in both magnitude and phase as a fuction of x. The electric field E_{y} or current density J_{y} deceases to 1/e (36.8%) of its initial value, while the wave penetrates to a distance d called penetration depth [

where: f = frequency Hz, µ = ground permeability, s = ground conductivity.

_{y} or currrent density J_{y} as a function of penetration depth, based on the magnitude of Equation (7). Integrating the absolute value of Equation (7) from x = 0 to ∞ results in E_{0}/d or J_{0}/d. Areas under step functional and exponential curve are equal when step function width is equal to the penetration depth [

It is assumed that all of injected current pass in the area of 1/e depth and the ground resistivity below the penetration depth is proposed to be infinity [

The analytical method used to calculate the surface potential profile of the four electrodes and grounding resistance/capacitance assumed each electrode driven into

the ground as a sphere. As electrode length is very short, each electrode is considered as equipotential surface. The relationship between the voltage and current can be written as:

where I_{j} is the current of the j^{th} electrode (j = 1; 2; 3; 4), V_{i} is the voltage of the j^{th} electrode, Z_{mn} is the mutual impedance element (i.e., mutual impedance between electrode number m and electrode number n), Z_{nn} is the self-impedance of the n^{th} sphere.

The elements of the impedance matrix are calculated as equal to:

where r_{mn} is the distance between m^{th} electrode and n^{th} electrode, r_{mnp}_{1,2,3,4} are the distances between the m^{th} electrode and the image of the n^{th} sphere and equal to:

_{r} = 10 and 50 and (a = 1 m, r = 1000 Ω×m) with frequency. The apparent ground impedace decreases more sharbly as the ground relative permittivity increases. This is due to the decrease of the capacitive part of apparent ground impedance with the increase of apparent relative permittivity.

_{r} = 10) with frequency. The apparent ground impedances decreases as the ground resistivity decreases. This is due to the decrease of the decrease of gound resistance.