App Notes

Scope: Establish a method for calculating the standard reliability values Failure Rate (?), Failures in Time (FIT) and Mean Time to Failure (MTTF) using the Arrhenius High Temperature Operating Life (HTOL) model.  The focus of this paper is to present the applicable equations, terms and definitions along with an example of an Excel driven reliability calculator used to perform these calculations.
Terms & Definitions: Reliability is defined as the probability that a component or system will continue to perform its intended function under stated operating conditions over a specified period of time.  The reliability level is derived by monitoring the functional stability of a number of representative subjects operating under elevated stress conditions resulting in a statistical prediction of reliability.   Two approaches to establishing a reliability level is to evaluate either the probability of survival or the probability of failure.  Either method is equally effective, but the most common method is to calculate the probability of failure or Rate of Failure (?).  The values most commonly used when calculating the level of reliability are FIT (Failures in Time) and MTTF (Mean Time to Failure) or MTBF (Mean Time between Failures) depending on type of component or system being evaluated. 
Equations & Calculations

• Failure Rate (?) in this model is calculated by dividing the total number of failures or rejects by the cumulative time of operation.  In the HTOL model, the cumulative time of operation is referred to as Equivalent Device Hours (EDH): EDH = D × H × A_f

Where:
D = Number of Devices Tested 
H = Test Hours per Device
A_f = Acceleration Factor derived from the Arrhenius equation
Acceleration Factor (Af) is the test time multiplier derived from the Arrhenius equation.  This equation calculates the time acceleration value that results from operating a device at an elevated temperature.  The test type used to achieve this is generally referred to as High Temperature Operating Life (HTOL) or Burn-in.  More specific terms for these tests depend on the type of technology under test.  Tests such as High Temperature Reverse Bias (HTRB), DC Burn-in and Alternating Current Operating Life (ACOL) are technology specific tests within the HTOL category.  The Acceleration Factor (Af) is shown by:

Activation energy (Ea) is an empirical value that is the minimum energy required to initiate a specific type of failure mode that can occur within a technology type. Oxide defects, bulk silicon defects, mask defects, electro-migration and contamination are some examples of such failure modes, each with an unique associated Ea.  In lieu of using empirical data for each individual failure mode, it is generally accepted that a standard single value of Ea = 0.7 eV provides a reasonably accurate average Ea value for diode type semiconductors.
To derive a more statistically accurate calculation for Failure Rate (?), the number of rejects (r) is replaced with the probability function using Chi-squared (X2).
FIT (Failures in Time) is a standard industry value defined as the Failure Rate (?) per billion hours.
MTTF (Mean Time to Failure or ?) is another standard industry value which provides the average time to failure of Non-repairable Items such as light bulbs and diodes or unserviceable systems such as satellites or other unmanned space craft.  For items with long life expectancies, it is often a more useful to report MTTF in years rather than hours.
MTBF (Mean Time between Failures) is used to describe Repairable Items such as compressors and aircraft.  MTBF uses MTTF as one factor and Mean Time to Repair (MTTR) as the other to capture the complete break-down and repair cycle.  The primary purpose of MTBF is to identify appropriate preventive maintenance schedules to avoid, perhaps indefinitely, catastrophic failures due to predictable piece part wear-out.  As a rule of thumb, component reliability centers around MTTF since most components cannot be repaired.  MTBF is shown by:
Reliability Calculation Tools
Over the years, there have been several methods used to derive the value for X2 (Chi-squared); everything from the use of probability tables to the application of approximation equations.  More detail on this subject is discussed in “Calculating Chi-squared (X2) for Reliability Equations” by this author.  The following is an excerpt.   In MS Excel 2003 and later versions, the CHIINV function calculates X2 from the input data probability and deg_freedom rendering accurate values throughout the entire use range up to 4-decimal places.  This method is currently used as a standard in the statistics community for calculating values for the X2 distribution. In Excel, the formula is expressed as: =CHIINV(probability,deg_freedom)
In determining ? (nu), degrees of freedom or deg_freedom, we use the equation described previously where r = number of failures or rejects:
Since reliability calculations require the left portion of area under the X2 distribution curve and CHIINV happens to calculate the right side, we must use the equation below to determine the correct value for probability: