Page 74 - Electrical Safety of Low Voltage Systems

The Earth 57

one, is added. The image electrode leaks the same current as the real
one.
This configuration, which is electrically equivalent to the actual
arrangement given in Fig. 4.12, allows a simpler determination of the
electrode’s earth potential and earth resistance, as the interface air–
soil is eliminated and the electrode medium is rendered homogeneous
again. We also assume that the depth Dis much greater than the radius
r 0 of the sphere.
The earth potential in correspondence of a generic point P from
the spherical electrode can be analyzed by superimposing the contri-
butions of actual and virtual spheres. Thus, by using Eq. (4.5) and the
2
lateral area of the sphere (i.e., 4 r ), we obtain
0
2 I 2 I
V P∞ = + (4.17)
4 r 1 4 r 2
The total earth potential V G , that is, the potential difference be-
tween any point on the actual electrode’s surface, the point S, and
infinity, can be calculated as follows:

2 I 2 I 2 I 2 I
V G = + = + (4.18)
4 r 0 4 r 3 4 r 0 4 2 2
r + 4D
0
Thus, dividing Eq. (4.15) by current I, the earth resistance R G of
the spherical electrode is

⎛ ⎞
1 1
2
R G = ⎝ + ⎠ (4.19)
4 r 0 2 2
r + 4D
0
The values of R G (and V G ) decrease as D increases, but the rate of
change with high values of D is very modest, as the values tend to
saturate (Fig. 4.14). Therefore, large, and therefore expensive, depths
are not necessary.
The earth potential in correspondence of a generic point Q, located
at distance x over the soil surface (Fig. 4.15), can be calculated by
superposing the effect of the actual and the virtual spheres:

2 I 2 I 2 I
Q
V = + = √ (4.20)
x∞ 2 2 2
4 r 1 4 r 1 x + D
An example of the variation of the surface potential as a function
of distance x from the electrode’s center buried at 0.1 m is shown in
Fig. 4.16.