Reliability and Failure Rate – A Case of “The Weakest Link”

October 13, 2016 // By EDN Europe
Jeff Smoot, VP of Application Engineering, CUI
In considering reliability in an earlier blog we focused on the term MTBF (mean time between failures) to understand what it attempts to measure.

Hopefully what readers took away from this discussion was an appreciation that the life expectancy of a component is not directly tied to its MTBF value, but rather the MTBF parameter provides us with a probability for how long that component might continue to operate and can really only be used to assess the comparative reliability of products.

What we only touched on in that blog was that the failure rate of any product built from multiple components, such as a power supply, will be determined by the failure rates of each of those individual components. Also, while the failure rate used for reliability calculations comes from the constant portion of the “bathtub” curve of failure rate over time, ignoring the early-life and wear-out failure phases, that period can still be a long time over which to measure a statistically significant number of component samples. Consequently, as engineers, we owe it to ourselves to understand how various standardized component databases may be used to predict component failure.

Previously we saw that reliability R(t)is the probability that a component, operating under specified conditions, will work correctly for a specified period of time t and can be calculated with the following formula:

R(t) = e -λt   where λ is the component’s failure rate (expressed as a rate per million hours)

For a system made up of many components we can substitute λA for λ in the reliability formula, where:

λA = λ 1n1 + λ 2n2 + λ 3n3 + … + λ ini   and   MTBF = 1/ λ

If we have 3 components each with an MTBF of 50,000 hours, then individually 20 will be expected to fail every million hours (i.e. given by λ = 1,000,000/MTBF). Together, λ A = 20 + 20 + 20 = 60 and hence the overall MTBF is 16,667 hours, i.e. one third of the single component MTBF. Intuitively we can appreciate this since with 3 equally reliable components the probability of failure is increased 3 times and conversely failure is likely in a third of the time.

On the other hand if we take 3 components with differing MTBF figures, one of which is significantly lower than the other two, then we get a notably different result. MTBFs of 50,000, 40,000 and 5,000 give failure rates of 20, 25 and 200 respectively. Summing these give an overall failure rate of 245 that converts back to a system MTBF of 4,082. This resulting MTBF is not that much worse than the component with the lowest MTBF, but is significantly worse than the other two components, by an order of magnitude.

This demonstrates that at the system level, reliability is very much determined by the weakest link in the chain and that designers should pay attention to this and take care during component selection. It also brings us to needing to understand how the failure rates for components are actually calculated.

Determining the reliability of today’s many and varied system designs is clearly