.
experimental conditions. It is calculated by dividing the total
number of device 鈥?hours by the number of failures. It is
failure. If each part has a 0.1% chance of failure before 1
by that time. The MTTF will be the same in both cases. 1
both produce an MTTF of 10 device 鈥?hours.
fail per unit of time. The conditional probability is the
given that it survived at the start of the interval.
device 鈥?hours.
is simply failure rate scaled from failures per device 鈥?/div>
hour to failures per billion device 鈥?hours.
ON TO THE DETAILS
In the definition section MTTF is defined as the average
time, in device 鈥?hours, per failure observed under specific
experimental conditions such as a life test. Here at Burr-
Brown we use a slightly modified formula for MTTF. We
calculate 2 times the total device 鈥?hours, T
dh
, divided by the
upper 60% confidence limit of a chi-square distribution with
2 times the observed number of failures + 2 degrees of
freedom, X
2
(2f + 2). Our formula is
MTTF
=
漏
Since both time and failures are doubled, these definitions
are roughly equivalent. Some explanation is in order.
If multiple life tests are run on the same type of device, it is
unlikely that all tests will have the same number of failures
for the same number of device 鈥?hours. Rather there will be
a distribution of failures. The minimum value must be 0 for
no failures. The maximum value could correspond to 100%
failures, but we can presume that we are running enough
parts that this will not happen. Rather the distribution will
taper off as the number of failures increases. Somewhere in
between there will be a concentration of failures.
The chi-square calculation provides us with a tool for adjust-
ing the actual number of failures from a limited life test to
make it more accurately reflect what we might expect from
the population as a whole. For example, applying a confi-
dence level of 60% to a chi-square distribution with 8
degrees of freedom will return a value into the denominator
of the MTTF calculation which is greater than or equal to
60% of the values in a chi-square distribution with a mean
of 8.
One intuitive interpretation of the chi-square calculation is
that the calculated value represents, roughly, a number of
failures which will be greater than 60% of the failures we
might get during multiple life tests. The upper 60% level is
selected because it represents an approximately average
estimate for MTTF and because it is widely accepted among
semiconductor manufacturers and users. This method of
estimating MTTF does not prevent further reliability calcu-
lations from being made at more conservative levels.
One more point remains to be explained regarding this
calculation. Why do we use 2 (# failures) +2? The technical
explanation for this is given later in this paper. Briefly, the
factor of 2 is necessary to achieve theoretical validity of the
X
2
distribution. Given the factor of 2, it can be seen that we
are merely adding 1 failure to the actual number of failures.
The added failure appears in the calculation as if a failure
occurred at the end of the test. This assures that the test
terminates with a failure, also a theoretical requirement, as
well as allows calculation of MTTF even if no failures were
observed.
The MTTF value by itself really only serves for comparison
purposes. Many more factors need to be considered before
predictive statements regarding the longevity of our compo-
nents can be made. The statistical concepts of reliability and
failrate allow us to make such predictions. I will present
here, with justification yet to come, the statistical formulas
which quantify these concepts.
2T
dh
蠂
(2f
+
2)
2
1993 Burr-Brown Corporation
AB-059
Printed in U.S.A. December, 1993
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