# Quality Progress - November 2015 - (Page 54)

STATISTICS ROUNDTABLE BY NECIP DOGANAKSOY, GERALD J. HAHN AND WILLIAM Q. MEEKER Reliability Upsizing How to adjust for size effect in product life data analysis IN MANY RELIABILITY evaluations, the occur due to the degradation of the vert the estimates for electrodes shown in units placed on life test are reduced-size organic material, causing reduced voltage Table 2 to full-length bars. versions of the actual product. This is of- strength. ten the result of practical limitations such Because the expected lifetime at the Correcting for size effect as available test facilities. In the example use condition of 120 volts/mm was ex- General method: The simple and widely we will consider, electrodes were used in pected to appreciably exceed that which used method described here for correcting an accelerated test to obtain time-to-fail- could be observed in life tests, a voltage- for size is based on reliability modeling for ure information on dielectric insulation accelerated stress test was conducted on components in series. In series systems, of generator armature bars that were four electrodes with new insulating material. the failure of any one component results times as long as the test electrodes. Fifteen randomly selected electrodes were in the failure of the entire system. Suppose tested at each of the five accelerated volt- that a component (or segment) of length ages from 170 to 220 volts/mm. L0 has reliability R0 (t). Given R0 (t), the The term "size" is used in a generic sense. In addition to length, it can, for example, also represent area or volume. After nine months, half of these 75 following approach can be used to obtain In such situations, you must "upsize" the electrodes had failed, mostly at the higher reliability R(t) for a system (or full) prod- results of the statistical data analysis on voltages. The electrode lifetime data were uct of length L > L0. the reduced-size test units to the full-size analyzed-assuming both a Weibull and product. a lognormal distribution, with constant of as being comprised of L / L0 nominally The full product, which can be thought shape parameters, for time to failure at identical segments, each of length L0, fails to account for the effect of product size each voltage (see the sidebar, "Back- when the first segment fails. Therefore, on reliability evaluations. These methods ground on Weibull and Lognormal Distri- assuming statistically independent times can be implemented using popular soft- butions," p. 56). An inverse power model to failure for the individual segments ware packages such as JMP, Minitab and (linear relationship between log life and (covered later), full product reliability is Weibull ++. log voltage) was fitted to the data using R(t) = [R0(t)]L/L0. Similarly, the q quantile This column will cover a simple method the method of maximum likelihood. This of the time-to-failure distribution of a full Example led to the estimated model parameters product of length L corresponds to the An example from a previous Statistics shown in Table 1. 1 - (1 - q)L0/L quantile of the time-to-failure Roundtable column1 addressing a new Table 2 shows the point estimates and distribution of a segment of length L0. insulation for generator armature bars the associated confidence intervals for the The preceding approach can be used to will be used to illustrate the methods. The 0.01 and 0.05 distribution quantiles and for obtain not only point estimates, but also insulation consists of a mica-based system the reliability estimates at five years and confidence intervals and bounds. bonded with an organic binder. Failures 10 years at the (120 volts/mm) use condition, using the methods Parameter estimates of two time-to-failure distributions at 120 volts/mm / TABLE 1 Example: In our example, L / L0 = 4. Estimating bar reliability: Assuming described in the earlier independence of failures, bar reliability column. R(t) at time t can be calculated from elec- Due to test equipment trode reliability R0(t) as R(t) = [R0(t)]4. limitations, the electrodes Thus, assuming a Weibull distribution for used in the accelerated test were 50 inches long. electrode time to failure, 10-year bar reliability is estimated to be Rˆ(10) = [0.9972]4 Weibull distribution Lognormal distribution The actual product units = 0.9890, using Rˆ0 = 0.9972 from Table Scale parameter ηˆ = 133.78 years Scale parameter exp( μˆ ) = 125.71 years (referred to as "bars") 2. Also, the lower bound of the associ- Shape parameter βˆ = 2.27 are 200 inches long. It is, ated 95% confidence interval for R(10) is Shape parameter σˆ = 0.6299 therefore, desired to con- obtained as [0.8647]4 = 0.5591. The results, 54 QP * www.qualityprogress.com http://www.qualityprogress.com

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