Page 70 - Electrical Safety of Low Voltage Systems

The Earth 53

system, we obtain
V GTOT = R GTOT I = V A (r 0 ) + V B (d − r 0 )
I I I 1 1
= + = +
2 2 r 0 2 2 (d − r 0 ) 4 r 0 (d − r 0 )
I d − r 0 + r 0 I d
= =
4 r 0 (d − r 0 ) 4 r 0 d − r 0
1
= I (4.9)
4 r 0 1 − (r 0 /d)
Thus, R GTOT equals:

1
R GTOT = (4.10)
4 r 0 1 − (r 0 /d)
The multiplicand in Eq. (4.10) corresponds to the parallel of the
earth resistances of two identical hemispherical electrodes, obtained
by dividing by 2 the result of Eq. (4.2). The equivalent ground resis-
tance of the two electrodes of Fig. 4.8, then, is not merely the parallel
between their corresponding earth resistances, as shown by the pres-
ence of the multiplier in parenthesis in Eq. (4.10). Such multiplier may
be ≥1 and be considered 1 only if the mutual distance d r 0 , in which
case the two electrodes will result connected in a “true” parallel.
As the multiplier increases the earth resistance of the parallel,
it can be thought as an additional resistance R a in series with the
aforementioned parallel. In formulas:
1
R GTOT = = + R a (4.11)
4 r 0 1 − (r 0 /d) 4 r 0
1 1
R a = − = − 1
4 r 0 1 − (r 0 /d) 4 r 0 4 r 0 1 − (r 0 /d)
(r 0 /d) 1
= = (4.12)
4 r 0 1 − (r 0 /d) 4 (d − r 0 )
The earth resistance R GTOT , calculated in Eq. (4.10), is shown in
the circuit of Fig. 4.10, where R GA (or R GB ) is the earth resistance of
the electrode A (or B) when the other is not present.
In some cases, real estate constraints may impose a very close
placement of earth electrodes. This arrangement limits the grounding
system “interface” capability with the earth and lowers the effective-
ness of its performance.
In sum, two or more ground electrodes can be considered inde-
pendent when, due to their own geometries and relative positions,