the actual surface of the electrode. Points belonging to the same equipotential surface have equal potential both on the surface and deep in the soil. Current lines are perpendicular to such surfaces.
The ground-potential rise on the surface of the hemisphere, that is, the potential at the distance r0 from its center, is
We define the resistance RG of the hemisphere-electrode to earth (from now on the ground-resistance) as the ratio of the ground-potential rise VG to the leakage current i (Eq. 1.10).
The ground-resistance of a ground-electrode can be seen as an equivalent one-port (Figure 1.10): one terminal of the one-port is the metal connection to the electrode (generally named grounding electrode conductor in codes and standards), whereas the other terminal represents a point at zero potential (i.e., a point at sufficient distance from the electrode where the potential is negligible).
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.
which shows that the maximum variation of the ground potential occurs in proximity of the hemispherical electrode (i.e. r ≈ ro).
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:
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).
Thus, the total ground resistance is given by Eq. 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.
