Page 205 - Electrical Safety of Low Voltage Systems

188 Chapter Eleven

If the protective device takes the time t f to interrupt the fault, in
order to calculate the total heat accumulation in the conductor during
this time, we integrate Eq. (11.1) and obtain Eq. (11.3):

t f 2 2 f d cS f d
2
i dt = cS = (11.3)
0 0 0 0 (1 + )
where 0 isthetemperatureoftheprotectiveconductorattheinception
of the fault and f is its final temperature when the fault is cleared.
The left-hand side of Eq. (11.3) is known as Joule integral and is
2
2
measured in A s. It is also referred to as “I square t,(I t),” “let-through
energy,” or “specific energy.”
Substituting y for (1 + ) and differentiating, we obtain
1
d = dy (11.4)

With this substitution, Eq. (11.3) yields

2 2
t f cS 1+ f dy cS
2 1 + f
i dt = = ln (11.5)
0 0 1+ 0 y 0 1 + 0
The adverse effects of fault-to-ground currents to the protective con-
ductor are prevented if the final temperature f reached by the PE
does not exceed the maximum values M its insulation can with-
stand (e.g., the maximum temperature for conductor insulation PVC
is 160 C). Therefore, Eq. (11.5) can be rewritten as an inequality. If we
◦
define
c
2 1 + M
k = ln (11.6)
0 1 + 0
where k depends on the material of the protective conductor, its type
of insulation, and the initial and final temperatures that are reached.
Through the PE no current normally circulates, therefore, its initial
temperature 0 corresponds to the standard ambient temperature (i.e.,
30 C). 3
◦
By substituting Eq. (11.6) into Eq. (11.5) and integrating, we obtain

t f 2 2 2

i dt ≤ k S (11.7)
0
By solving Eq. (11.7) for S, we obtain cross-section values that guar-
antee the protection of the PE against damages caused by thermal
effects.