Since publishing this article last May in the Atlanta-based magazine TechLINKS, it has drawn more responses and requests than any other article.  This business article is appropriate for anyone requiring an understanding of the quality, and reliability requirements for electronic components.

Calculating Failures In Time (FITs) is an important design step in predicting the likelihood of a failure, or the Mean Time Between Failures (MTBF).

Arrhenius Equation - Acceleration Models
A well known failure mode in semiconductors is metal migration; a chemical reaction, which we know can be accelerated by temperature. Temperature is an acceleration factor by itself, but also accelerates other failure mechanisms too, such as humidity, which can sometimes be modeled simultaneously, i.e. the Peck model. Eyring, Black, and Kenney provide acceleration models that combine temperature with other parameters such as voltage, or current.

The three variables in the Arrhenius model are the Use (T₁) and Test (T₂) temperature in degree Kelvin, and the Activation Energy (E_A) in eV. Selecting the appropriate Activation Energy for the failure mode being modeled is important, and values can be found in publications from suppliers, or publications like Telcordia GR-486 CORE, or MIL HDBK 217. It can be seen from the Arrhenius equation that the Acceleration coefficient (A_F), changes significantly with small changes in Activation energy (E_A), so that value must be chosen with care.

Censored Data
Data typically comes from two types of experimental models; time-censored (Type I), or failure-censored (Type II). Type I data is most common, an example being when x units are placed on test for y hours, and at the end of y hours the failures (if any) are reported, and no failed devices are replaced during the test. There are other Type I experiments, where the experiment records failures in fixed intervals of time, i.e. every 100 hours, and where failed devices are immediately replaced. These experiments are much more complex to design, and their expense precludes them from most general testing. An experiment to monitor Type II data is demonstrated when x units are placed on test until y failures are observed. A test may be truncated at the first failure. This technique is common when the device under test is very expensive, when the testing is destructive, or when few devices are available.

Distribution Models
There are many distribution models; normal (or Gaussian), Poisson, Weibull, exponential, or lognormal. It is important to choose a distribution which fits the data, and allows appropriate analysis of the data. If it was required to know the median lifetime, and also might be required to extrapolate that data over various temperatures (or other acceleration factors), the lognormal distribution and analysis would be a more efficient solution.

The exponential distribution model is used in this example, and many other applications in reliability. A unique value of the exponential distribution for reliability use is that it is the only distribution which has a constant failure rate λ, which is also the reciprocal of the MTBF. With the characteristic bathtub curve, we think of the first part as early failures, the middle as the intrinsic failures, or normal life expectancy, and the end as wear-out failures. The long, flat middle section, or normal life expectancy can be modeled as an exponential distribution, where we can then view those failures as random failures, occurring at a constant rate.

Chi-Square: Goodness-of-Fit
The Chi-square is a standard, non-parametric statistical test to evaluate the goodness-of-fit to an assumption. All we need to know is the number of failures, and then choose a confidence level to test the assumption. This point estimate of observed data becomes our Chi-Square test about the estimated data, or exponential distribution. A 90% confidence level would mean that data is likely to fall outside our confidence interval only 10% of the time. The Chi-Square test has upper and lower bounds of confidence level which can be evaluated. But, when we do not immediately replace the failed units, and/or there are zero failures, general usage confines evaluation to the upper confidence level (UCL). This single-tail, or UCL is used in our FITs calculation.

FITs Calculator
The number of steps involved and the mathematics required are generally slow to achieve with a calculator, but well suited for the capabilities of a spreadsheet. Using a Microsoft Excel spreadsheet, Corbitt Associates has created a FITs calculator that accepts the inputs required to solve the Arrhenius equation for thermal acceleration, and predict the failure rate for a given confidence level using a Chi-Square model. The FIT solutions are provided for the typical Upper Confidence Limits (UCL) of 60% and 90%, and for failures from zero to ten. The FITs calculator also has the ability to accept any UCL of your choosing, and provide the FIT data for that selection. Examples of the computed results for the user input is shown below in Table 1, and Table 2.

The example below uses the Arrhenius equation and Chi-Square distribution, respectively:

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Table 1

Table 2

Chi-Square Table