Page 413 - Offshore Electrical Engineering Manual

400 CHAPTER 1 Reliability

With parallel components, calculation becomes more complicated. In a system X
where two parallel components are redundant, i.e., both are designed to be function-
ing but only one is required for system operation (Fig. 8.2.10), the unavailability of
the system is
A x = A A · A B

If there are three components in parallel and two out of the three are required for
successful system operation, then

A x = 1 − A A · A B − A B · A C − A C · A A + 2A A · A B · A C
With complicated systems, as in Fig. 8.1.11, a truth table can be drawn.
For the truth table in Fig. 8.1.11:

A x = A A · A B + A A · A B · A C + A A · A B · A C · A D + A B · A C ·
A D + A A · A C + A B · A D + A A · A D
The Boolean equation may be written out directly from inspection of the diagram
as follows:
{( ) ( )}
A x = A A + A B · A C + A D + A A · A B
For more complex situations consult Reference 1.
As each component availability is usually a decimal fraction which approaches a
value of unity with improving reliability and reducing repair time, the value is often a
figure such as 0.999999945, where only the last two figures are significant. Therefore,
whether calculators or computers are used, care must be exercised to ensure either that
these figures are not lost because the particular hardware cannot handle the required
number of decimal places or that the significant figures are not lost by rounding.
Spreadsheet programs such as Lotus 1-2-3 or Excel are very useful in producing a
tabular output of failure rates, repair times, unavailabilities, etc., and some examples
of evaluations are given in the following sections.
If the system unavailability is multiplied by the number of hours in a year (8736),
the result is the annual system downtime in hours.

CONFIDENCE LIMITS
Failure rates are statistical values based on samples of known population. However,
neither the total population, the mean value of failure rate for all components of a
particular type, nor the way the values vary over the range from the worst to the least
is known.
The electrical engineer needs to know how closely the sample mean (x) agrees
with the total population mean value of failure rate (μ).
Failures in similar components will tend to reach a peak after a certain time and
then tail off, again producing a bell-shaped characteristic called a normal distribution
(see Fig. 8.1.7).