Page 48 - Electrical Safety of Low Voltage Systems

Mathematical Principles of Electrical Safety 31

FIGURE 3.2 Safety offered by a circuit breaker.

We can then use the negative exponential distribution to quantify the
safety of the generic ith protective measure as follows:

N − F(t) − i t
S i (t) = = e (3.2)
N
where i represents the failure rate, defined as the mean number of
failures per unit-time, for example, years, of the ith protective mea-
2
sure (e.g., the failure rate for a circuit breaker is 0.0052 failure per
year). We will, herein, assume a constant failure rate, that is, the fail-
ure associated with the steady-state period of the life of the protective
component. The failures, therefore, will be considered as due to ran-
dom causes and not due to infant mortality or deterioration caused
by the age of the PM.
Safety as offered by a circuit breaker to a piece of equipment is
shown in Fig. 3.2.
If n protective measures are simultaneously deployed, we need to
superpose their protective effects in order to calculate the total safety
achieved by the item against electric shock. If all the n measures must
simultaneously operate to ensure safety, the protection system is de-
fined as “serial.” If, on the contrary, all PMs must fail in order for
safety to fail, the system is defined as parallel or redundant.
Safety for serial and parallel systems can be, respectively, evalu-
ated through Eqs. (3.3) and (3.4):

S S (t) = n S i (t) = S 1 (t) ... S n−1 (t)S n (t) (3.3)
1
S P (t) = 1 − n [1 − S i (t)] = 1 − [1 − S 1 (t)] ... [1 − S n (t)] (3.4)
1