Weibull analysis of component failure data from accelerated testing

Weibull analysis of component failure data from accelerated testing

Reliabilio, Engineering 19 (1987) 237-243 Weibull Analysis of Component Failure Data from Accelerated Testing Martin S h a w Product Quality and Reli...

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Reliabilio, Engineering 19 (1987) 237-243

Weibull Analysis of Component Failure Data from Accelerated Testing Martin S h a w Product Quality and Reliability, IBM (UK) Ltd, MP06D, PO Box 30, Greenock, Scotland, UK (Received: 5 June 1987)

A BS TRA C T The Weibull distribution ts contntonh' used to describe the Atilure rate parameters q[electronic Conlponents and has the advantage q f presenting an 'easy-to-understand'picture O[COnlponentjclih~re nlodes with respect to tinle. This allows conclusions to he ch'awn on whether an increasing, decreasinjz or constanl JZtilure rate is present which is undouhledh' a key.[itctor in nlakin~ any reliability statements. This paper describes in detail how a transistor prohh, m on a colour l,ideo display unit ( V D U ) was screened out and the linle-to-/ail data plotted on Weihul/ paper to allow conclusions to he ~h'awn. Two d(tTbrent lratlSivlor.v were erallmted in the same application to determine.[)'on1 their respeclice Weihull plots, the reliahilitv characteristics ql'each type. As the product under di.scu.vsion was in its early production stage, the Weihu/I p/ots were no! used to predict field [+Hlout /)'ore the test data, bu! to lmderstand the /~li/urc t}l(~CiIl[in{'ittl, Ttle paper also describes the way in wtlic'll weak comt~om, nt /a/lure distributions can he isolated in l~'ihull plots By us'in~, a./m'nt o / l h e Bayesian .s'talislica/ approac/l. Lsohtth N the weak tfislrihution in tills' manner a/h~w.s a more exact calculation o/thc required hum-in ~hn'alion which nla r q/'ten hare a ntarked ~ff~'c't on hllpro~ing pro~htcl re/iahililr.

TEST D E S C R I P T I O N In order to accelerate the transistor failures the V D U s were stressed at constant high temperature with power-on. The high temperature environmerit was created by covering the monitor with a plastic bag to simulate 237 Reliahilitr Engineering 0143-8174/87/$0350 ,~ Elsevier Applied Science Publishers Lid, England, 1987. Printed in Great Britain

Martin Shaw

238

testing in a 40~C environment. Temperature measurements taken during testing gave an average internal m/c ambient temperature of 45-C with a temperature of 42'~'C around the outside of the m/c while inside the polyethylene bag, these temperatures being in line with the m/c specification of operation in a 40°C ambient environment. Time to failure {TTF) for both transistor types was recorded during the exercise enabling Weibull plots to be drawn although the test duration for type 2 transistors was shorter than that for type 1, hence the Weibull plol is incomplete. As will be explained in the latter part of the paper this does not affect the test conclusions.

WE1BULL ANALYSIS In the following text analysis of the failure pattern t\~r these transistors has shown evidence of a weak/strong or bimodal failure distribution. The easiest method of portraying the presence of bimodality is to consider the probability density function as illustrated in Fig. I. Translating the T T F data onto a Weibull plot gives Fig. 2. The Weibull plot is an illustration of the cumulative distribution curve, the first 'knee" in the curve indicating that the weak population of components has been screened out. The value p is the proportion of weak component failures screened out during the high-temperature burn-in and the intersect time of 0-632p provides the mean time to failure of the weak component distribution or 'population'. This value is termed the characteristic lifetime and is denoted by r/. The same procedure for defining p and fi for the main or 'strong' distribution also applies to the values taken fiom the second knee in

MAIN DIST~TBUTION

WEAK DISTRIBUTION

I

7 Fi~. I.

I

140

mean T . T , F { h r s )

T h e p r o b a b i l i l ? d e n s i l v f u n c t i o n o1 type I n a n s i s t o r f a i l u r c ~

239

Weihull analysis o1 component.fililure data

the curve. The shape parameter fl, relevant in both distributions, is simply the slope of the asymptotic line drawn next to the curve in question and is calculated using a special scale normally found on Weibull paper. This parameter depicts whether a constant, an increasing or a decreasing failure rate is present. Values of fl = 1 mean there is a constant failure hazard rate, greater than one an increasing failure rate, and less than one a decreasing failure rate. ANALYSIS OF FAILURE DATA Figure 2 illustrates the Weibull plot of the actual component time to failure for both types. Type 1 shows very similar characteristics to type 2 although type 2 transistors were not tested for as long as type 1 and fewer were tested, hence the plot is not as complete. The data derived from the above plots of the weak distribution are as follows /91 = 25%

132 = 18%

ql = 7 h

t/2 = 8h

= 3

1 2= 3

To derive conclusions on the main distribution for type 2 the assumption

TYPE 'I

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hl

IE

l.IJ O.

1

I

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Fig. 2.

TIME (HRS) Weibull plot of m,'c tlmc to Failure.

i 00

240

Martin Shaw

must be m a d e that it will have the same slope parameter as type I. Applying this assumption provides the data below: t/1 = 1 4 0 h

r/2 = 3 0 0 h

= 1

1

The conclusions to be drawn from the weak distribution plots are that a seven-hour burn-in of type 1 transistors would be sufficient to remove the weak components and an eight-hour burn-in of type 2 transistors to do the same. The next portion of the type 1 Weibull plot (part 2) is simply a flat portion lasting from approximately 35 to 60 h during which there are no failing units until failures from the main distribution are induced. Analysing the type 1 weak distribution further, there is an obvious upturn in failure rate before the flat portion of the curve (2), hence a burn-in of 7 h may not be sufficient to remove a high proportion of the weak components. This analysis however poses the question 'does the failure at the start of part 2 in the curve belong to the weak distribution'?' Plots on Weibull paper often make it difficult to determine the weak distribution parameters, hence a technique is required that solves this problem. The technique used is based on the Bayes' theorem which is an expression for the handling of conditional probabilities to determine the probability of a particular failure time belonging to a particular distribution which has a probability density function, fl t. The Weibull distribution is simply a cumulative failure distribution where: F(t) = weak component cumulative distribution function = 1 - exp(-(t/~lO/~)

and F(t) = strong c o m p o n e n t cumulative distribution function = 1 - exp(-(t/rl2)/~2)

To obtain the probability density functions for each distribution a differentiation of the cumulative distribution function is required to give the following; (fl It1) = [:ta exp --(t/tlOth(t/rll)/h r/1

--1

(f21t2) =-/]-z exp --(t/rl2)a'-(t/rl2) t~'- 1 r/2

Weihull anal 3'sis q[ component/ailure ¢klta

241

To determine the probability of failure i belonging to the weak distribution requires the use of the following expression P=

probability of failure in weak distribution probability of failure in weak or strong distribution

This equation of course requires the necessary parameters to describe both the weak and strong distributions which enables (fl I tl) and (f2 I/2) to be calculated, hence: p =

(fl I tl)

(fl [tl) -[- (f2 ] 12) Using the test data with the weak and strong distribution parameters (r/2,/~2,P2) it is now possible using the above equation to carry out a Bayes analysis of the burn-in data of the type 1 transistors (Table 1). TABLE I

Analysis of Burn-in Data, Type I Transistors

Cumulatit~e no. J~tilures

Time to [~il (tt)

Probability ~?/Jitilure belonging to the weak distribution

6 22 30 34 35 36 51 58 64

5 7 l0 15 20 30 38 78 90

0.97 0.96 0.88 0.02 0.00 0.00 0.00 0.00 0.00

P = 25%

The analysis shows all failures that occur in less than 15 h belong to the weak distribution and those after that time to the strong component distribution. Plotting only the failures which occurred before 15 h gives the Weibull plot in Fig. 3. The three points define a straight line with a shape parameter, fl = 2"7 and a characteristic lifetime q = 8-5 h, both of which equate closely to the initial estimates. Also from this plot it can be seen that 90% of the weak components can be screened out by a burn-in that lasts 12h. The Bayes' analysis performed has isolated the weak failure distribution to allow calculation of the exact burn-in time required, which is a definite advantage when planning a product burn-in whether it be to

242

Martin Shaw 0 °

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hi

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6

8

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TIME (HRS) Fig. 3.

Weibull plot of weak distribution.

TABLE 2 Median Ranks lbv the Weak Distribution Time to /~dl (ht

Median rank ('!4,1

5 7 10

20.6 500 79.4

remove all weak c o m p o n e n t failure types or to concentrate on a particular distribution. The other advantage of using the Bayes theorem is to enable optimisation of the burn-in duration, i.e. to calculate the exact burn-in period to remove only the weak distribution with a given level of confidence rather than the approach used earlier in the text which gave an estimated burn-in time of only 7 h. CONCLUSIONS .

Short term burn-in is very successful in removing weak component distributions.

Weihull analysis (4['componentfailure data

2.

3.

243

Separation of weak and strong distributions using the Bayes' method makes it less difficult to evaluate parameters of a particular distribution when several failure modes are competing and the 'knee' is not clearly defined. With enough data to plot the product failure pattern, it is a simple task to determine the optimum product burn-in duration by applying the Bayesian method.

REFERENCE 1. Jensen, F. and Petersen, N. E. An Engineering Approach to the Design and AnaO'sis g/Burn-#7 Procedures, John Wiley & Sons Ltd, New York, 1982, pp. 38-40.