 |
|||
|
Life Distributions (Page 2 of 4)
<Back to Page 1 - Life Distributions> <Proceed to Page 3 - Exponential and Lognormal Life Distributions> <Proceed to Page 4 - Weibull Life Distribution>
Life distributions are described mathematically by life distribution functions. Three of these functions are very important descriptors of life distributions, and should be understood by every reliability engineer. These are the cumulative failure distribution function F(t), the failure probability density function f(t), and the curve of failure rate l(t).
The cumulative failure distribution function F(t), or simply cumulative distribution function, gives the probability of a failure occurring before or at any time, t. This function is also known as the unreliability function. If a population of devices is operated from its initial use up to a certain time t, then the ratio of failures, c(t), to the total number of devices tested, n, is F(t). Thus, F(t) = c(t)/n. F(t) is therefore always less than 1, which is consistent with the fact that it's just a probability number after all.
The unreliability function F(t) has an equivalent opposite function - the reliability function R(t). R(t) = 1 - F(t), so it simply gives the ratio of units that are still good to the total number of devices after these devices have operated from initial use up to a time t.
The failure probability density function f(t), or simply probability density function, gives the relative frequency of failures at any given time, t. It is related to F(t) and R(t) by this equation: f(t) = dF(t)/dt = -dR(t)/dt.
The curve of failure rate l(t), also known as the failure rate function or the hazard function, gives the instantaneous failure rate at any given time t. It is related to f(t) and R(t) by this equation: l(t) = f(t)/R(t). Thus, l(t) = f(t)/[1-F(t)].
More details on how these functions describe the various life distributions may be found at Life Distribution Functions.
The Normal Life Distribution
A normal life distribution is one that consists of time-to-failure or life data that constitute a normal distribution. Thus, it is a symmetric bell-shaped curve whose mean, median, and mode are equal. The spread of the normal life distribution is determined by the standard deviation s of its life data. The failure rate of a normal life distribution monotonically increases with time, which is failure rate pattern typically exhibited by failures due to wear-out.
Normal distributions are often a result of the additive effects of random variables. Thus, normal life distributions are generally applicable to failures that are affected by additive factors, such as mechanical system failures that occur as a result of the accumulation of small and random mechanical damage. Such mechanical failures are often observed as the system wears out with use.
Figure 1. The f(t), F(t), and l(t) of a normal life distribution; source: D. S. Peck and O. D. Trapp, Accelerated Testing Handbook, Technology Associates.
We all know that the over-all failure rate of semiconductors do not increase monotonically with time. In fact, there aren't too many semiconductor failure mechanisms that fit the normal life distribution. Thus, the normal life distribution is generally not used by reliability engineers to model semiconductor survival in the field.
Note, however, that the bath tub curve representing the failure rate curve of semiconductor devices does include a wear-out phase in the end. This wear-out phase, although just the end portion of a semiconductor's life, may be modeled by a normal life distribution.
<Back to Page 1 - Life Distributions> <Proceed to Page 3 - Exponential and Lognormal Life Distributions> <Proceed to Page 4 - Weibull Life Distribution>
Copyright © 2004 SiliconFarEast.com. All Rights Reserved. |
