Page 72 - Electrical Safety of Low Voltage Systems

The Earth 55

The previous system yields the following solutions:
V GTOT (R 22 − R 12 )
I 1 = (4.14)
R 11 R 22 − R 12 R 21
V GTOT (R 11 − R 21 )
I 2 = (4.15)
R 11 R 22 − R 12 R 21

Thus, R GTOT from the two electrodes can be so calculated [Eq. (4.16)]:

V GTOT R 11 R 22 − R 12 R 21
R GTOT = = (4.16)
I GTOT R 11 + R 22 − (R 12 + R 21 )
8
If the two electrodes are independent, the mutual resistances are
negligible and the total resistance R GTOT coincides with the parallel of
the electrodes’ ground resistances.
It is important to note that the optimization of the grounding sys-
tem requires as little interactions as possible between electrodes. This
allows the minimization of the total earth resistance, which benefits
the safety of the installation. In practice, we consider as in parallel
electrodes separated by at least five times their (equivalent) radii, as
the earth potential has greatly decayed at that distance [Eq. (4.7)].
It may not be always economically convenient to connect more
electrodes to a given grounding system, when this addition reduces
their reciprocal distances. The value of the grounding resistance, in
fact, may “saturate,” and no longer linearly decreases with the number
of electrodes and therefore with the additional cost.

4.5 Spherical Electrodes
Another electrode never used in practical applications, but with a very
interesting behavior, is the spherical electrode. A spherical electrode
embedded at infinite depth within homogeneous soil would produce
spherical equipotential surfaces solely developing within the earth.
This does not happen if the spherical electrode, of radius r 0 and leak-
ing current I, is buried at a finite depth D. In this case, in fact, the
equipotential surfaces will develop through two different media of
different resistivity: soil (resistivity 2 ) and air (resistivity 1 equal to
infinity) (Fig. 4.12).
The resulting nonhomogenous medium can be studied by using
the theory of images, which yields the equivalent configuration of
Fig. 4.13.
The two semi-infinite media (i.e., air and soil) are replaced by
a single medium coinciding with the soil. An additional “virtual”
electrode, a symmetrical image with respect to the ground of the real