
ReliabilityReliability, Safety, Security definitionsReliability: ability of an entity to perform under NORMAL conditions Safety: ability of an entity to perform under ABNORMAL conditions Security(ability of an entity to perform in the presence of MALEVOLENT environment) Reliability Prediction: A reliability prediction is simply the analysis of parts and components in an effort to predict and calculate the rate at which an item will fail. A reliability prediction is one of the most common forms of reliability analyses for calculating failure rate and MTBF. Failure definitionFailure is defined as the lack of ability of a component, equipment, sub system, or system to perform its intended function as designed. Failure may be the result of one or many faults. Applied to the supercapacitors,the failures may be :
There are 2 basic types of failures, Early Failures and Wear Out Failures. These are reflected in the curve known as the Bathtub curve. Manufacturers test the components at the end of the production process to eliminate "Early Failures" cases. The presentation below concerns the statistical behavior of the components considering the "Wear Out" failures. Supercapacitor failuresEarly Failures
Failures in operation
Survivor function F(t)This is the number of elements of the statistical sample which have not failed at time t. In other words, it is the number of elements which have not broken down and are still working Probability density function f(t)This is the number of failures dF occuring at time t during a unit time period dt in the sample considered. Failure rate λ(t)As noted the failure is an event which corresponds to an object breakdown. It is equal to the inverse of the MTBF (Mean Time Between Failure). The failure rate increases with the aging time. λ(t) isn't the number of failures observed in a unit of time; but the ratio of the failures and the number of survivors (or number of breakdowns observed for 1000 elements for example). Different survivor models
Exponential modelThe simplest statistic model is the particular case of the exponential model. Probability density function Survivor function Failure rate Mean Time Between Failures (MTBF) Weibull modelThe Weibull model is widely used in electrotechnics. Probability density function Survivor function Failure rate Mean Time Between Failures (MTBF) If p is set to 1, it comes out that the exponential model is a particular case of the Weibull model Lognormal modelThe logarithmic mean time The standard deviation Probability density function Survivor function UnitsThe failure rate may be expressed with different type of units: Weibull survivor curve
The survivor function corresponds to the proportion of capacitors in the batch which didn t failed after the time t. It may also be considered as the remaining capacitance of a capacitor after a time t. Weibull survivor curveThe cumulated failure function (=1  survivor function) may be represented in a special "Weibull scale" in order to exhibit a straight line. Log(Log(1/(1F(t)))) Probability density function f(t)This is the number of failures dF occuring at time t during a unit time period dt in the sample considered. Failure rate λ(t)As noted the failure is an event which corresponds to an object breakdown. It is equal to the inverse of the MTBF (Mean Time Between Failure). The failure rate increases with the aging time. λ(t) isn t the number of failures observed in a unit of time; but the ratio of the failures and the number of survivors (or number of breakdowns observed for 1000 elements for example). The failure rate is given in FIT (Failure In Time ) which is the number of failures occurring during 10^{9} hours of working of an object. λ(t) and &lambda _{o} must not be confused. This latter is a constant (from the time, but is depending on the temperature and on the voltage) which corresponds to the inverse of the time necessary for 63 % of the sample to fail. Weibull failure rateThe failure rate value given in the manufacturer specifications, as it is increasing with time, corresponds to the value for the end of life. Failure modesThe ultracapacitor failure modes are:
