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Figure 1.10 Electrode ground-resistance as an equivalent one-port.

      The ground symbol (from IEC 60417 “Graphical symbols for use on equipment,” symbol 5017) does not represent the soil, but a point at sufficient distance from the electrode where the surface potential is negligible.

      From the graph of the ground potential of Figure 1.9, it can be observed that the radius r0 of the hemisphere identifies the point from where the hyperbolic distribution starts. For a given hemisphere, different values of the product ρi determine different hyperbolae, whose distance for the horizontal axis depends on the soil resistivity and the fault-current.

      The rate-of-change of the potential with the distance r from the hemisphere (i.e., the potential gradient) is defined in Eq. 1.11.

(1.11)

      which shows that the maximum variation of the ground potential occurs in proximity of the hemispherical electrode (i.e. r ≈ r_o).

      1.6.1 Area of Influence of a Ground-electrode

      The electric field is a long-range field and is zero only at infinite distance from its source, and so is the ground potential. In engineering practice, however, the design of ground electrodes is based on the area of influence, which defines the zone beyond which the ground potential can be considered negligible. If we evaluate the ground potential at the distance r = 5r0 , we obtain:

       (1.12)

      At a distance 5r₀ from the center of the hemisphere, the ground potential reduces to 20% of the ground potential rise, and this result has a general validity, independently of the shape of the electrode. It can conventionally be assumed that the hemispherical volume of the earth of radius 5r₀ is the area of influence of the electrode. For differently-shaped electrodes (e.g., rods, rings, grids, etc.), their maximum dimensions can be used in lieu of the radius; for instance, for grounding grids, the largest diagonal can be employed to identify the area of influence.

      Two unconnected ground-electrodes are defined as independent from each other if they are outside of their respective areas of influence.

      1.7 Hemispherical Electrodes in Parallel

      Two identical hemispherical electrodes of radius r₀ are connected in parallel into a uniform soil of resistivity ρ, and each leaks the current i/2. The electrodes are buried at a distance d.

      The hemispheres attain the same ground-potential rise, which can be calculated by superimposing the potentials of each electrode, supposed isolated from the other (Eq. 1.13).

(1.13)

      Thus, the total ground resistance is given by Eq. 1.14.

(1.14)

      If d > 5ro, the second factor in Eq. 1.14 is ≈1, and R _G is mathematically expressed by the formula of the parallel of the ground resistances of each electrode.

      If the electrodes are closer than d > 5ro, they are not independent from each other, and their total resistance is greater than their mere parallel (Figure 1.11).

Figure 1.11 Ground-electrodes in parallel.

      1.8 Hemispherical Electrodes in Series

Let us consider two identical hemispheres embedded into the soil at the distance 5ro center-to-center, so that they can be considered independent from each other. The first electrode leaks the current i, whereas, the second electrode receives the same current ( Figure 1.12).

      Figure 1.12 Hemispherical electrodes connected in series.

      The resulting surface potential V(r) at a generic point r along a line joining the two hemispheres is given by the superposition of the potentials imposed by each electrode (Eq. 1.15).

(1.15)

      If r = d/2, which is the center point between the two hemispheres, the value of the surface potential is V(d/2) = 0.

      The point of the soil at which the ground potential is zero is the location that must be identified for the correct measurement of the ground resistance RB of an electrode, as further elaborated.

      1.9 Person’s Body Resistance-to-ground and Touch Voltages

      In the case of the pathway hands-to-feet, the current will flow into the soil through the feet, and its amount will depend on the series between the body resistance RB and the person’s resistance-to-ground RB. The person’s resistance-to-ground limits the circulation of the body current, therefore is beneficial for the electrical safety of individuals.

      For an approximate calculation of RBG, the adult human foot can be modeled as a circular

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