IEEE Transactions on Electrical Insulation Vol. EI-20 No.1y February 1985 79

LIFE PREDICTION FOR CABLE INSULATION MATERIAL BASED ON

WEIBULL ACCELERATED TESTING WITHOUT FAILURES

Ralph E. Thomas

Batte lie-Columbus LaboratoriesColumbus, Ohio

ABSTRACT

Under strong assumptions associated with Weibulldistributions for normal and accelerated test conditions,a statistical hypothesis-testing procedure is developedthat permits the life of cable insulation materialunder normal stress conditions to be inferred fromaccelerated life-test data in which few, if any, break-downs occur. The procedure also yields an approximaterelation between the number of tests required fortesting under normal stress conditions and the numberof tests required under accelerated test conditions.This relation, in turn, provides a graphical methodfor judging the feasibility of proposed acceleratedlife tests. Subject to the validity of the assumptionsthese procedures may have application to a broad rangeof materials, components, and systems.

INTRODUCTION assumptions required to support such predictions areexceptionally strong, and must be carefully examined

Constant-stress accelerated life testing of cable in detail for each actual application.insulation materials is difficult. If the overstresstest conditions are too severe, then the insulation tes procre was develo dionmanaccelertedmaterial nay fail electrically for the wrong reasons. tetpormo al nuainmtra. Th tsForexample,thematerialnayfailelcally fa nthe vinisof program experienced the possibility that the only valid

For example, the material mayfaiinthevicinityof breakdown "data" might consist of "runouts". Althougha termination where electrical fields are known to be . dexceptionally high. If the overstress test conditions cable insulationmaterial, the resultling procedure mayare not severe enough, then the insulation material cabe insulation material theresulti roeuenmay not fail even over relatively long test durations. havenaplian to aflSuch "runouts" are the principal concern of this paper. components, and systems.

In standard statistical settings the absence of APPROACHtimes-to-failure is essentially equivalent to an ab-sence of useful "data". Run-outs frequently do not Accelerated life testing of insulation material isprovide a suitable quantitative basis for data analy- performed in order to predict the life of the materialsis. Nevertheless, it seems clear that, under suffi- under normal operating conditions. To do this, speci-ciently strong assumptions, it must be possible to mens are tested at relatively high stress levelsmake inferences about the probability of obtaining mensiare testediat relativelythighrstress levels,

longinslatonife t opratig sreses henpossibly involving voltage, temperature, and frequency,,long insulation life at normal operating stresses when for relatively short durations, say fractions of a

sufficiently large numbers of cable specimens are year. The accelerated test data are then consideredtested at sufficiently high constant stresses for as a base for extrapolation to obtain the estimationsufficiently long times with the occurrence of few, of the life that could be expected when such materialsif any, electrical breakdowns. This paper identifies are operated at relatively low (normal) levels ofone set of assumptions associated with Weibull distri- stress for relatively long time periods, say decadesbutions under normal and accelerated test conditions or evren hundreds of years [1,2,3].that permit inferences of predicted life under normalconditions to be made based on accelerated test datain which virtually no failures occur. Of course, the

001 8B-93567/85/0200£-0079$0l .0OO 1985 IEEE

80 IEEE Transactions on Electrical Insulation Vol. EI-20 No.1, February 1985

We consider the probabilities that specified lengths environment for a time period of D years under over-of cable insulation material will have long lifetimes stress test conditions characterized by a known andunder normal operating conditions. The lifetimes of constant acceleration factor f, where fl.0 for accel-the cable lengths are considered to be described by erated testing and f=l.O for normal stress testing.Weibull distributions [4]. The Weibull distribution By definition, an acceleration factor equal to f meansfor the times-to-breakdown under accelerated test con- that one unit of test time under an associated over-ditions is obtained by an explicit transformation of stress condition is equivalent to f units of oPeratingthe parameters of the Weibull distribution for the time under normal stress. In what follows D years oftimes-to-breakdown under normal stress. accelerated testing are taken to yield performance of

the insulation material that would be expected afterfD years of operation under normal stress conditions.

HYPOTHESIS TESTING FRAMEWORKThe test hypotheses may be stated as follows:

In a statistical setting we formulate two testhypotheses. The "long" life hypothesis is labeled HI: the probability that a randomly selected lengthH1. Under this hypothesis the probability of having of cable will last l years under normal stress condi-a specified long lifetime for the cable insulation tions exceeds P1, where P1 is relatively large, saymaterial is denotedbyP1, where P1 is assumed to be 0.90relatively large, say 90%. An alternative hypothesis,called the "short" life hypothesis, is labeled H0. H0: the probability that a randomly selected length ofUnder this hypothesis, the probability of a long life- cable will last L years under normal stress conditionstime is taken to be relatively small, say 50%. The is less than PO, where PO is relatively small, sayanalysis of the data is intended to identify which 0.50.hypothesis is to be rejected and, consequently, which With T denoting a random time to failure, and t de-is to be accepted. The values assumed for P0 and Pl noting a specified operating time, the probabilityand the resulting zone of indifference [5] betweenthem should be based on the consequences associated that th insulatn metime t iS givren by:with making an incorrect conclusion. An incorrectacceptance of the long life hypothesis, when in fact PfT>t|HI = exp(-(ft/-c1A > PIthe short life hypothesis is true, is called the Type pI error. An error of this type might result in the when Hl is true and byinstallation of short life cable and could eventually wprove to be very expensive. An incorrect acceptance P{T>t[H1} = exp(-(ft/-u1)) < P, (2)of the short life hypothesis, when in fact the longlife hypothesis is true, is called the Type II error. when Ho is true, for Weibull distributions of failureAn error of this type might result in costs associated times having a common shape parameter and character-with a continuing unnecessary search for a suitable istic lifetimes of Ti under Hi, with 0, 1.insulation material. Roughly, the data analysis con-sists of determining which of the two hypotheses is We note that multiplication of the operating time tmore consistent with the data. If sufficiently few by the acceleration factor f indicates that t hoursbreakdowns occur under the accelerated test conditions, of testing at overstress conditions are taken to bethen the long life hypothesis will be accepted; other- equivalent to ft hours of operation at normal stresswise, the short life hypothesis will be accepted. At conditions. With this interpretation the Weibullspecified probabilities for making the Type I and characteristic time is taken to be invariant to changesType II errors, the procedure described below is in- in stress. The quantity (ft/c) may also be written astended to determine how many cable specimens must be t/(r/f. In this form it is seen that time is takentested and how many breakdowns will be allowed and to flow at the usual calendar rate at overstress condi-still permit the acceptance of the long life hypothesis. tions, but the Weibull parameter T for such a condi-The allowed number of breakdowns is called the accept- tion is reduced by the factor l/f. These interpreta-ance number and is denoted by c. The resulting de- tions are equivalent and are taken to represent acision rule then has the following form: if the number simple transformation that may account for the effectsof breakdowns is less than or equal to c, accept the of operation at overstress conditions; more comfplexlong life hypothesis; otherwise accept the short life transformations involving power laws have been studiedhypothesis. by Nelson [7]. With regard to the shape parameter,

it is further assumed that F is known and that F isFor some accelerated life tests the procedure may invariant over different levels of stress. In practi-

make it desirable to test more specimens at lower cal applications F frequently lies in the intervalstress levels, with a consequent reduced concern between 1.0 and 3.0. The usefulness of the procedureabout changes in failure mechanisms at high stress proposed below is limited to those nstances where thelevels, and accept in return stringent requirements assumed relations between the Weibuln parameters andon the number of failures that can be permitted with- stress are met.out rejecting the hypothesis of long life at normaloperating stress. Under certain additional assump- With input values for Po, P1, and F either assumedtions, the procedures described below constitute a or inferred from previous data, the above expressionsgeneralization of the sampling procedures associated may be solved for normal stress conditions by puttingwith normal life testing described by Goode and Kao f-1 and t=L to obtain the characteristic lifetimes[6]. and mean times to failure when equality holds on the

right sides of Eqs. (1) and (2):

DEVELOPMENT OF THE PROCEDURE li= /llpi / ,

We assume that a random sample of n lengths of andcable are to be tested in a controlled laboratory Ui. = -ir.(1+(1/5), i = 0, 1. (3)

Thomas: Life Prediction Based on WeibLill Accelerated Testing 81

Suppose n lengths of cable are tested for an assumed This relation can be obtained by requiring the ex-time D under stress conditions associated with an pected number of failures to be equal for tests ofacceleration factor f-1. The decision rule requires duration D under the accelerated and normal stressthe rejection of H1 whenever the number of break- conditions. To show this let Q and q denote thedowns exceeds c, and requires the rejection of Ho probabilities of failure under normal and acceleratedwhenever the number of breakdown is less than, or stress conditions, respectively. The requirement thatequal to c. The probability of incorrectly rejecting NQ-nq yieldsH1, when in fact H1 is true, is the probability of aType II error and is given by NQ = n(l-exp(-(fD/T)F)). (7)

cN N-j For small values of (fD/T) a linear approximation to

= - ( )A1 f(1-A1) , the exponential yieldsj=O i

(4) NQ - n(1-(1-(fD/T) = nff(D/T) (8)

A1 = exp(-fD/T1) ) Applying the exponential approximation in reverseshows that

Similarly, the probability of incorrectly rejecting NQ f 1 (9)Ho, when in fact Ho is true, is the probability of a fly f-&ID/T))) nf(1-exp(D/T) )Type I error and is given by with the result that NQ nf Q and Eq. (6) follows.

The approximation is valid provided fD<T; that is,X =~ C-)AoNjNl-Ao)i the equivalent normal stress test time must be small

.= a relative to the characteristic life. This conditionJ v (5) often holds in accelerated life testing.

AO = exp(-(fD/TO) ) * The sample size relation given in Eq. (6) showsthat the number of cable lengths required for acceler-ated life testing can be reduced relative to that re-

These equations for a* and FA can be used to determine quired for testing at normal stress. For example,the smallest number of cable lengths n for which the for the exponential case, the shape parameterb=1,probabilities of incorrect conclusions, computed using and the normal stress sample size is reduced by athe right sides of Eqs. (4) and (5), are less than the factor of 1/f at the higher stress corresponding topreviously specified acceptable values of cX and F4 the acceleration factor f. A relatively large accel-that occur on the left sides. The computations may be eration factor, say f=10, and a relatively large shapecarried in the following sequence. Using To, T1, F, factor, say F=3, reduce the normal stress sample sizef, and D, we first compute Ao and Al. Next, take c=0 by the factor 1/1000. That such reductions are use-and find the smallest sample size n for which the com- ful is suggested by the fact that in Table 4e of [6]puted probability of a Type I error a is less than a* many of the tabulated normal stress sample sizes lieand compute the corresponding probability of a Type II between one thousand and ten million. For this Table,error F. If F is less than F*, take the resulting n the probability of a Type I error is set at 10%, theand c as the solution; otherwise, increase the accept- shape parameter is equal to 2.0, and the acceptanceance number c by unity and repeat the sequence of cal- number c varies between 0 and 15 failures.culations. The calculations will terminate at thesmallest n and smallest acceptance number c for whicha < a* and F < F', if a solution exists, A CRITERION FOR JUDGING THE FEASIBILITY

By generating additional solutions to Eqs. (4) and OF ACCELERATED LIFE TESTS(5) statistical power curves for the hypothesis test Eq. (6) shows that a plot of n versus f on log-login the indifference zone can be constructed. For scales yields a straight line with a slope equal to -F,testing at normal stress the required sample sizes for and an "intercept' at the normal stress sample size Ne= 0.0 can be obtained using the tables in [6]. corresponding to f=1. Fig. 1 shows the general struc-

These tables may also be used to infer the required ture of such a plot. The line shown in Fig. 1 is asso-sample sizes under overstress conditions by replacing ciated with values of L, Dm, Po, P1, F, where Dthe tabulated quantity (t/WJ by (ft/l). represents the maximum accelerated test duration that

is considered feasible. The figure also shows twopoints labeled (fmax nmax) The fmax coordinate de-

A SAMPLE SIZE RELATION notes the acceleration factor associated with the max-imum overstress condition that does not change theThe required number of cable lengths depends on the failure modes known to be associated with normal stress

acceleration factor f. Eqs. (4) and (5) can be solved conditions. This constraint must be derived from afor the required sample size N under normal stress con- knowledge of the physical rate processes associatedditions, corresponding to t=L and f=1, and then solved with the degradations of the insulation material overagain for the required sample size n for overstress time. To reflect economic constraints the quantityconditions, corresponding to t=D and f>1. An approxi- nmax denotes the maximum number of cable lengths thatmate relation between the normal sample size N and the can be subjected to an accelerated life test. Asoverstress sample size n is found to be shown in the Figulre, if the plotted point (nm fmax:)

F6 lies below the line given by Eq. (6), then the accel-

n- Nf * (6 erated life test is labeled not feasible. Conversely,if (fm<=,n?v~) falls above the line given by Eq . (6),then a range of acceleration factors and correspondingsample sizes is defined by the interval labeled AB on

82 IEEE Transactions on Electrical Insulation Vol. EI-20 No.1, February 1985

the line. Any point in the interval AB thus defines tions, the risk of an incorrect conclusion is 20%.a feasible accelerated life test with corresponding Corresponding calculations made for normal stressvalues of (f,n%_ffmx,nmax)- testing show that AO=0.99827, AI=0.99974, and the min-

imum sample size is found to be 930. Thus, at a riskIt should be noted that the normal stress sample of 20% the corresponding test carried out a normal

size N is a function of L, D, Po, P1, F, and c. stress requires 95:. specimens to be tested for a yearChanges in one or more of these parameters can be ex- without failure in order to conclude that the proba-pected to change the position of the line, and thereby bility exceeds 90% that the items will last 20 years.alter the feasible-infeasible regions. In contrast, Finally, it is noted that N/fF=930/102 and this re-the point (fmax,nmax) is fixed by the constraints, and sult agrees well with n=10 as indicated by Eq. (6).will generally fall in one region or the other. Over- For this example the equation of the line in the feasi-all the plot reflects the constraints imposed by eco- bility plot of Fig. 1 is given by log n=log(930)-2nomics (costs of test facilities and test time), phy- log(fD. The general feasibility of accelerated test-sics (limitations on magnitudes of acceleration factors ing involving various additional combinations of sam-and associated stress levels that will yield normal ple sizes and acceleration factors can be easilystress failure modes), and statistics (numbers of assessed using this line.specimens required to control risks of reaching incor-rect conclusions).

ACKNOWLEDGMENTS

The author is pleased to acknowledge the encourage-ment of Emanuel L. Brancato, formerly of the U.S.

6 Naval Research Laboratory, and Gordon B. Gaines and10 * Michael M. Epstein, both at the Battelle-Columbus

c N Laboratories. Important revisions suggested byPaul I. Feder of Battelle-Columbus are also gratefully

log N-~ log facknowledged. This work was supported by the U.S.

aJ \ log n = log N-F log f Department of Energy Contract No. DEAC01-77-ET29239._- | 8at Battelle-Columbus Laboratories.

Accelerated testsV) \ are not feasible

4- 3- ~~~~~~~~~~~~~~REFERENCESo 1O3 - - a99 nmax) Accelerated tests [1] J. McCallum, R. E. Thomas, and J. H. Waite,EI are easibe Accelerated Testing of Space Batteries, NASA SP-

z . * 1 *u-.rf n ) 323, Scientific and Technical Information Office,av nfmax)

0 IA\ | maxs maNational Aeronautics and Space Administration,S_ § \ .Washington, D.C., 212 pages, 1973.

tX [2] R. E. Thomas, G. B. Gaines, "Procedure for Devel-O Bl \ oping Experimental Designs for Accelerated Life

100 Tests for Service-Life Prediciton", Thirteenth

O 1 2 IEEE Photovoltaic Specialists Conference,100 l 10 Washington, D.C., June 5-8, (1978).

Acceleration Factor, f[3] R. E. Thomas, G. B. Gaines, M. M. Epstein,

"Methodology for Estimating Remaining Life of

Fig. 1: Feasibility pZot for accelerated Zife tests Components Using Multifactor Accelerated Lifebased on the WeibuZZ distribution Tests", U.S. Nuclear Regulatory Commission

Workshop on Nuclear Power Plant Aging, Bethesda,MD, August 4-5, (1982).

A NUMERICAL EXAMPLE[4] N. R. Mann, R. E. Schafer, N. D. Singourwalla,

As a numerical exampgle of this approach, let H0o de- Methods for Statistical Analysis of Reliabilitynote the hypothesis that less than 50% of the cable and Life Data, John Wiley Sons, (1974).lengths will last 20 years under normal stress; let H1denote the hypothesis that more than 90% of the cable [5] S. S. Wilks, Mathematical Statistics, page 398,lengths will last 20 years under normal stress. John Wiley F Sons, Inc., New York, 1962.Suppose a one-year accelerated life test can be runat an overstress condition having an acceleration [6] H. P. Goode, J. H. K. Kao, "Sampling Plans Basedfactor equal to 10. Further, suppose that the times- on the Weibull Distribution", Proceedings Seventhto-failure are Weibull distributed with an invariant National Symposium on Reliability and Qualityshape parameter equal to 2.0. With D=1.O year, F-"2.0, Control, pp. 24-40, (1961).Po=0.50, and P1=O.90, Eqs. (1)-(3) give To=24.0 yearsand T1=61.6 years. Eqs. (4) and (5) show that [7] W. Nelson, W. Q. Meeker, "Theory of OptimumAO=0.8406 and Al=0.9740. With c=O it follows that a* Accelerated Censored Life Tests for Weibull and=(O.8406)N and F*=1- (O.9740)N. If each risk is re- Extreme Value Distributions", Technometrics, 20,quired to be less than 20%, the smallest acceptable N (1978).is determined by &X and is found to be 10 . Thlus, if10 specimens are subjected to an accelerated life testhaving an acceleration factor of 10 for a year without Manuscript was received 27 Septemrber 1984.failure, then it is concluded that the probability ex-ceeds 90% that the material will1 last 20 years undernormal stress. Subject to the validity of the assump-