Page 63 - Electrical Safety of Low Voltage Systems

46 Chapter Four

FIGURE 4.1 Current radially leaving a hemispherical electrode.

electrodes, which, due to their physical symmetry, uniformly leak cur-
rent in any direction of the earth. Such electrode is, indeed, never used
in practical applications, but its behavior can predict that of any other
differently shaped electrode. In fact, at sufficient distance from any
electrode and regardless of the shape, the current can be considered
to radially flow, just as it were impressed by a hemispherical electrode.
Let us apply a potential difference between two identical hemi-
spherical electrodes, of radius r 0 , one transmitting and the other re-
ceiving current, displaced by a sufficiently large distance, in theory
infinity. Because of its physical symmetry, the transmitting electrode
will radially leak current I to ground, and in the same fashion the
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remote electrode will receive it. The current leaving the electrode
uniformly distributes in any available direction (Fig. 4.1).
The current is limited by the resistance of the soil surrounding
the electrode, which can be modeled as the series of hemispherical
“shells,” each shell of increasingly large radius and infinitesimal thick-
ness dx (Fig. 4.2).
We can consider the elementary resistance dR of a generic shell of
radius r, and, then, once its expression is known, we can sum up the
contribution of all the elementary resistances as they succeed from the
electrode surface to the remote earth (in theory, the infinity).
Our assumptions of the uniformity of the soil allow us to apply
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the same formula as the resistance of a conductor : R = (l/S), where
is the resistivity of the material ( · m), l the length of the conductor,
and S the cross-sectional area the current flows through. In our case,
is the earth resistivity, also referred to as specific earth resistance.
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The resistivity is defined as the resistance of 1 m of earth.
By using this formula, we will obtain the resistance of the generic
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hemispherical shell of Fig. 4.2, as follows:
dx
dR = · (4.1)
2 r 2