Applied Reliability Page 1 APPLIED RELIABILITY Techniques for Reliability Analysis with Applied Reliability Tools (ART) (an EXCEL Add-In) and JMP Software AM216 Class 5 Notes Santa Clara University Copyright David C. Trindade, Ph.D. STAT-TECH Spring 2010
Applied Reliability Page 2 AM216 Class 5 Notes Accelerated Testing (continued from Class 4 Notes) Accelerated Test Example (Analysis in JMP) Degradation Modeling Sample Sizes for Accelerated Testing System Models Series System Parallel System Analysis of Complex Systems Standby Redundancy Defective Subpopulations Graphical Analysis Mortals and Immortals Models Case Study Class Project Example Modeling the Field Reliability Evolution of Methods General Reliability Model AMD Example
Applied Reliability Page 3 System Models Series System Consider a system made up with n components in series. If the i th component has reliability R i (t), the system reliability is the product of the individual reliabilities, that is, R ( t) R t R t... R t s 1 2 which we denote with the capital pi symbol for multiplication R t s n i 1 R t i n The system CDF, in terms of the individual CDF s, is F t s n i 1 The system failure rate is the sum of the individual component failure rates. The system failure rate is higher than the highest individual failure rate. 1 1 F t i
Applied Reliability Page 4 System Models Parallel System Consider a system made up with n components in parallel. The system CDF is the product of the individual CDF s, that is, F t s n i 1 F t i The system reliability is R t s n i 1 1 1 R t i System failure rates are no longer additive (in fact, the system failure rate is smaller than the smallest individual failure rate), but must be calculated using basic definitions.
Applied Reliability Page 5 System Failure Rate Two Parallel Components A component has CDF F(t) and a failure rate h(t). Two components are used in parallel in a system. Determine the failure rate of the system. SOLUTION The CDF for the two components in parallel is F 2 (t) and the PDF, by differentiation, is 2F(t)f(t). The failure rate of the system is hs t 1 1 s t Fs t 2 F t f t F t F t h t F t 2F t 1 1 f 2F t f t 2F t 1 The result shows that the system failure rate is a factor 2F/(1+F) times the component failure rate. The smaller the component CDF, the bigger the improvement. Redundancy makes a larger difference in early life, and much less difference later on.
Applied Reliability Page 6 Class Project System Models A) A component has reliability R(t) = 0.99. Twenty-five components in series form a system. Calculate the system reliability. B) A component has reliability R(t) = 0.95 Three components in parallel form a system. Calculate the system reliability.
Applied Reliability Page 7 Reliability Block Diagrams For components in series: A B For components in parallel: A B
Applied Reliability Page 8 Example of Series-Parallel System: Big Rig G H I J Trailer C D E F Cab A B I J G H E F C D B A Reliability Block Diagram (RBD)
Applied Reliability Page 9 Class Project Complex Systems A system consists of seven units: A, B, C, D, E, G, H. For the system to function unit A and either unit B or C and either D and E together or G and H together must be working. Draw the reliability block diagram for this setup. Write the equation for the CDF of the system in terms of the individual component reliabilities, that is, the R i, where i = A, B, C,..., G, H. Hint: Consider the three subsystems:a alone; B with C; and D,E,G,H.
Applied Reliability Page 10 Standby Versus Active Redundancy In contrast to active parallel redundancy, there is standby redundancy in which the second component is idle until needed. Assuming perfect switching and no degradation of the idle component, standby redundancy results in higher reliability and less maintenance costs than active parallel redundancy. An illustration, assuming exponentially distributed failure times, is shown below. System Failure Rates (2 Components) 0.012 0.01 0.008 0.006 0.004 Single Parallel Standby 0.002 0 0 50 100 150 200 250 300 350 400
Applied Reliability Page 11 Series, Parallel Reliability in ART In ART, select System Reliability... Enter necessary information. Click OK.
Applied Reliability Page 12 Reliability Experiment Consider... We test 100 units for 1,000 hours. There are 30 failures by 500 hours, but no more by the end of test. Question : Are we dealing with two populations or just censored data? Question : If we continue the test, will we see only a few more failures, or will the other 70 fail with the same life distribution?
Applied Reliability Page 13 Defect Models Mortals versus Immortals The usual assumption in reliability analysis is that all units can fail for a specific mechanism. If a defective subpopulation exists, only a fraction of the units containing the defect may be susceptible to failure. These are called mortals. Units without the fatal flaw do not fail. These are called immortals. The model for the total population of mortals and immortals becomes : CDF = (fraction mortals) x CDF(mortals) Reliability analysis focuses on the life distribution of the defective subpopulation and the mortal fraction.
Applied Reliability Page 14 Example of a Defective Subpopulation A Processing Problem Suppose we have 25 wafers in a lot, but only two wafers are contaminated with mobile ions due to a processing error. If components are assembled from the 25 wafers, assuming equal yield per wafer, only 2/25= 8% of the components can have the fatal defect that makes failure possible. The components from the non-contaminated wafers will not fail for this mechanism since they are defect free; that is, we have a defective subpopulation.
Applied Reliability Page 15 Spotting a Defective Subpopulation Graphical Analysis Assume that a specified failure mode follows a lognormal distribution. Plot the data on lognormal graph paper. If instead of following a straight line, the points seem to curve away from the cumulative percent axis, it s a signal that a defective subpopulation may be present. If test is run long enough, expect plot to bend over asymptotic to cumulative percent line that represents proportion of defectives in the sample.
Applied Reliability Page 16 Defective Subpopulations Graphical Analysis Plot based on total sample (mortals and immortals). Plot based only on mortal subpopulation.
Applied Reliability Page 17 Defect Model Mortals and Immortals The observed CDF F obs (t) is F obs (t) = p F m (t) where F m (t) is the CDF of the mortals and p is the fraction of mortals (units with the fatal defect) in the total sample size. For example, if there are 25 % mortals in the population, and the mortal CDF at time t is 40%, then we would expect to observe about 0.25x0.40 = 0.10 or 10% failures in the total random sample at time t.
Applied Reliability Page 18 Major Computer Manufacturer Reliability Data Gate Oxide Fails Time (hours) 24 48 168 500 1000 Rejects 201 23 1 1 1 Sample Size 58,000 57,392 10,000 2,000 1,999 Censored 407 47,369 7,999 0 1,998 Analysis by Company Using Lognormal Distribution T 50 : 1.149E32 hours Sigma: 26.175
Applied Reliability Page 19 What Do These Numbers Mean? Analysis by Company Using Lognormal Distribution T 50 : 1.149E32 hours Sigma : 26.175 Plus and minus 3 sigma range of time to failure distribution extends from 33 seconds to 1.66E62 years! It takes seconds to get to 0.1% cumulative failures, but over 412,000 hours (that is, 47 years) to get to 1.00%! Assuming everything can fail is misleading and unnecessary.
Applied Reliability Page 20 Modeling with Defective Subpopulations The same data, assuming 99% of the failures have occurred by 48 hours, can be modeled by a fraction defective subpopulation of 227/58,000 = 0.39% and a lognormal distribution of failure times for the mortals T 50 =10.6 hours and sigma = 0.68. Practically 100% of failures occur by 168 hours. Any failures thereafter are probably not related to the defective subpopulation. For example, handling induced failures are a possibility.
Applied Reliability Page 21 Defective Subpopulation Models If we don t consider mortals vs. immortals, we will incorrectly assume that all units can fail. Projections of field reliability will be biased unless we identify the limited defective units.
Applied Reliability Page 22 Statistical Reliability Analysis and Modeling: A Case Study Analysis of Reliability Data with Failures from a Defective Subpopulation
Applied Reliability Page 23 Reliability Study Background One lot of a device type with initial burn-in results at 168 hours, 125 o C : Over 50% fallout due to bake recoverable failures Since other lots, with similar manufacturing, might have escaped to a few customers, we needed to assess the field impact. We were able to impound this lot, containing about 300 devices not burned-in.
Applied Reliability Page 24 Reliability Study Design Two static stresses: 179 Units : 125 o C ambient 90 Units : 150 o C ambient 30 Units: Control Frequent readouts at 2, 4, 8, 16, 32, 48, 68, 92, 116 hours
Applied Reliability Page 25 Purpose of Study Reliability Modeling Determine if fraction defective (mortals) model applies Determine failure distribution (lognormal, parameters) Determine if true acceleration is present Determine activation energy for acceleration factors Determine recovery kinetics with and without bake - Is 24 hours at 150 o C necessary? - Do devices recover at room temperature?
Applied Reliability Page 26 Modeling Procedure Statistical Analysis Plan Analyze cumulative percent failures plot versus time, both linear and probability plots. Estimate fraction mortals for stress cells. Test for significant difference. Plot fallout of mortals (reduced sample size) on lognormal probability graph. Check for linearity and equality of slopes. Run maximum likelihood analysis. Test for equality of shape factors (sigmas). Estimate single sigma. Estimate median life T 50 for both cells. Check model fit against original data.
Cum ula tive P e rc e nt Applied Reliability Page 27 Reliability Study Bake Recoverable Failures L in e a r P lo t o f C u m u la tiv e F a ilur e s V e rsu s T im e 80% 70% 60% 50% 40% 30% 20% 1 5 0 o C 1 2 5 o C 10% 0% 0 20 40 60 80 100 120 S tre s s Tim e (P ow e r on H our s ) S am ple S iz e s : 1 5 0 oc =9 0 ; 1 2 5o C = 1 7 9
S ta ndar d Norm a l V a ria te : Z Applied Reliability Page 28 Reliability Study Bake Recoverable Failures P r o b a bi li ty Pl o ts (N o A d ju s tm e n t fo r M o r ta ls) 1 0.5 0 0 1 2 3 4 5-0.5-1 -1.5 150oC 125oC -2-2.5 Ln (Tim e t o Fa ilure ) S a m pl e S iz e s : 1 5 0 o C = 9 0 ; 1 2 5 o C = 1 7 9
S ta n d a rd No rm a l V a ria te : Z Applied Reliability Page 29 Reliability Study Bake Recoverable Failures P robability P lot (Adjusted for Mortals) 2.5 2 1.5 1 0.5 0-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 150oC 125oC -1-1.5-2 L n (T im e to F a ilu re ) M o rta l S a m p le S iz e s: 150o C = 64; 125o C = 113
Applied Reliability Page 30 APL PROGRAM FOR MLE GENLNEST ENTER NUMBER OF CELLS: 2 CHOOSE CONF. LIMIT FOR BOUND IN PERCENT: 90 ENTER ANY EXACT TIMES OF FAILURE FOR CELL 1 ENTER START AND ENDPOINT OF ALL READOUT INTERVALS (INCLUDE ZERO S) SPREAD 2 4 8 16 32 48 68 92 116 ENTER CORRESPONDING NUMBERS OF FAILS PER INTERVAL (INCLUDE ZERO S) 34 6 21 2 0 0 0 1 0 ENTER TIMES ALL FAILED UNITS WERE REMOVED FROM TEST (INCLUDING END OF TEST) 116 ENTER CORRESPONDING NUMBERS REMOVED 0 ENTER ANY EXACT TIMES OF FAILURE FOR CELL 2 ENTER START AND ENDPOINT OF ALL READOUT INTERVALS (INCLUDE ZERO S) SPREAD 2 4 8 16 32 48 68 92 116 ENTER CORRESPONDING NUMBERS OF FAILS PER INTERVAL (INCLUDE ZERO S) 5 0 36 8 42 7 3 4 3 ENTER TIMES ALL FAILED UNITS WERE REMOVED FROM TEST (INCLUDING END OF TEST) 16 116 ENTER CORRESPONDING NUMBERS REMOVED 2 3 MAXIMUM LIKELIHOOD ESTIMATES VARIANCE VARIANCE COVARIANCE CELL T50 SIGMA MU SIGMA MU MU SIGMA 1 1.90 1.208.444.0322.0373e-1.643e-2 2 15.08 1.060 2.714.0059.0104e-3.266e-5 ESTIMATE BOUNDS (90 PERCENT CONFIDENCE) NUM. NUM. CELL ON TEST FAIL T50 LOW T50 UP SIGMA LOW SIGMA UP 1 64 64 1.38 2.63.909 1.508 2 113 108 12.74 17.86.933 1.187 WANT EQUAL T50 S OR SIGMAS OR BOTH IN SOME CELLS (Y/N)? Y CELLS: 1 2 TYPE 1 FOR EQUAL SIGMA S, 2 FOR EQUAL MU S, 3 FOR BOTH THE SAME: 1 THE ASSUMPTION OF QUAL SIGMA S CAN NOT BE REJECTED AT THE 95 PERCENT LEVEL. UNDER THIS ASSUMPTION, RESULTS LIKE OBSERVED OCCUR ABOUT 41.9 PERCENT OF THE TIME. (THE SMALLER THIS PERCENT, THE LESS LIKELY THE ASSUMPTION.) MAXIMUM LIKELIHOOD ESTIMATES VARIANCE VARIANCE COVARIANCE CELL T50 SIGMA MU SIGMA MU MU SIGMA 1 2.02 1.090.704.0051.0247e-2.538e-3 2 15.08 1.090 1.713.0051.0110e-2.250e-5 ESTIMATE BOUNDS (90 PERCENT CONFIDENCE) NUM. NUM. CELL ON TEST FAIL T50 LOW T50 UP SIGMA LOW SIGMA UP 1 64 64 1.56 2.63.972 1.207 2 113 108 12.68 17.54.972 1.207 WANT EQUAL T50 S OR SIGMAS OR BOTH IN SOME CELLS (Y/N)? N
C umum ative Pe r ce nt Fa ilur e s Applied Reliability Page 31 Reliability Study Bake Recoverable Failures Model Fit to Actual 80% 70% 60% 50% 40% 30% 20% 10% 1 50oC 1 25oC M LE F it: 1 50oC M LE F it: 1 25oC 0% 0 20 40 60 80 1 00 1 20 1 40 Tim e (P ow er o n Ho ur s)
Applied Reliability Page 32 Projection to Field Conditions Acceleration Statistics Estimate acceleration factor between two stress cells : AF = 15.08 / 2.02 = 7.465 Estimate activation energy, based on Tj s, 35 o C above ambient: E A = 1.375 ev Estimate field T 50 based on Tj at 55 o C ambient : field T 50 = 18,288 hours Using field T 50, sigma = 1.090, lognormal distribution: -project fallout and failure rates for various mortal fractions -use customer field data to determine which mortal fraction applies
C um u lat iv e P e r ce nt Applied Reliability Page 33 Projection to Field Use Bake Recoverable Fails Pr o je ct e d F ie ld F a llo u t w it h Va rio u s M o rt a l P e r c en t a g es 2 0% 1 8% 1 6% 1 4% 5% 1 0% 2 0% 1 2% 1 0% 8% 3 0% 4 0% 5 0% 6 6% 6% 4% 2% 0% 0 2 4 6 8 10 T ime i n F ie ld ( K H o ur s )
Applied Reliability Page 34 A Note of Caution Analysis When Mortals Are Present Since the analysis which took into account the presence of a defective subpopulation, parameter estimates were accurate. The two customers, notified of the affected lots, used analysis for decisions on how to treat remaining product in field. If assessment is not done correctly and there is a low incidence of mortals, the T 50 s and sigma s for a lognormal distribution may become very large and inaccurate.
Applied Reliability Page 35 A Side Benefit Screening a Wearout Mechanism Note that it may be possible to screen a wearout failure mechanism if only a subpopulation of the units are mortal for that mechanism and sufficient acceleration is obtainable. See Trindade paper Can Burn-in Screen Wearout Mechanism? Reliability Models of Defective Subpopulations - A Case Study in 29 th Annual Proceedings of Reliability Physics Symposium (1991)
Applied Reliability Page 36 Class Project Defect Models 50 components are put on stress. Readouts are at 10, 25, 50, 100, 200, 500, and 1,000 hours. The failure counts at the respective readouts are 2, 2, 4, 5, 4, 3, and 0. 1. Estimate the CDF for all units using the table below with n = 50. Cum # Time Fails 10 2 25 4 50 8 100 13 200 17 500 20 1000 20 CDF Est All Units (%) 2. Plot the data on Weibull probability paper on the next page. Does the data appear distributed according to a Weibull distribution or does a defect model seem possible?
Applied Reliability Page 37 Weibull Probability Paper
Applied Reliability Page 38 Note: Percent Failure scale on Weibull Probability paper is faint. Values are 99.9, 98.0, 90.0, 70.0, 50.0, 30.0, 20.0, 10.0, 5.0, 2.0, 1.0, 0.5, 0.2, 0.1, etc.
Applied Reliability Page 39 Class Project Defect Model Estimates Weibull Parameter Estimates for Mortal Population: Characteristic Life (c) Shape Parameter (m) F( t) 1 e t / c m How could we confirm that the Weibull model for the mortal population fits the data? We estimate the CDF at three times and compare to observations. Mortal CDF Time (Weibull Model) Mortal Fraction 25 0.221 0.4 100 0.632 0.4 1000 1.000 0.4 Model CDF for All Units Empirical CDF All Units
Applied Reliability Page 40 Defective Subpopulations in ART Enter failure information (readout times, cumulative failures) into columns. Under ART, select Defective Subpopulations Enter required information. Click OK.
Applied Reliability Page 41 System Models A General Model for the Field Reliability of Integrated Circuits An Evolution in the Projection of Field Failure Rates
Applied Reliability Page 42 Failure Rate Calculations Primitive Method Assumptions Constant failure rate Single overall activation energy Ambient temperatures No separation of failure modes
Applied Reliability Page 43 Primitive Method Problems with Calculations Example 100 units are stressed for 1,000 hours at 125 o C. Assume no self heating. One unit fails at 10 hours for mechanism with E A of 1.0 ev. Second unit fails at 500 hours for failure mechanism with E A of 0.5 ev. Primitive Method Calculation Overall average activation energy : 0.75 ev Acceleration Factor (125 o C to 55 o C): AF = 106 IFR (constant) at 55 o C : [1E9x2/(10+500+98x1000)]/AF = 192 FITS
Applied Reliability Page 44 Primitive Method Comparative Calculation Individual Analysis by Failure Mechanism Mechanism 1: E A = 1.0 ev, AF = 501 IFR (constant) at 55 o C: [1E9/(10+500+98x1000)]/AF = 20 FITS Mechanism 2: E A = 0.5 ev, AF = 22, IFR (constant) at 55 o C: [1E9/(10+500+98x1000)]/AF = 461 FITS Total IFR = 481 FITS
Applied Reliability Page 45 Failure Rate Calculations Later Improved Method Early failures (infant mortality) reported separately Long-term life modeled with activation energy specific to failure mechanisms Constant failure rate for long term life Temperature acceleration calculated with junction temperatures
Applied Reliability Page 46 Later Method Problems Defective subpopulations not adequately modeled Competing failure modes not adequately modeled with constant failure rate Zero rejects and unidentified mechanisms often not treated Bathtub curve approximated in flat region only because of constant failure rate
Applied Reliability Page 47 An Alternative Model Three categories of possible failures: Test Escapes Defective Subpopulations Competing Failure Mechanisms The three D s: Dead Defective Deficient
Applied Reliability Page 48 Non-Functional Test Escapes Dead on arrival (DOA) Quality issue Inadequate testing at manufacturer or damaged after testing prior to customer receipt Rejects discovered at customer; called mistakenly reliability failures Assume zero in model
Applied Reliability Page 49 Defective Subpopulations There are proportions of the total population at risk of failure. Defective units are called mortals. The ones without the defect are called immortals. Defective subpopulations are generally associated with processing problems. There are physical reasons why defective subpopulations should exist. Always question the assumption (common in the traditional approach) that any observed failure type will eventually affect all other devices.
Applied Reliability Page 50 Competing Risks There are failure mechanisms that can affect all units. We call these mechanisms competing risks because several different types may exist and any one can cause the unit to fail. These mechanisms are typically associated with design, processing, or material problems. We model the failures using Weibull or Lognormal distributions
Applied Reliability Page 51 General Reliability Model F F F 1 F T e d N where F N = 1 - R 1 R 2... R N Activation energies are specific to failure mechanisms. Zero rejects and unidentified mechanisms are included. Generates complete bathtub curve!
Applied Reliability Page 52 General Reliability Model In Use at AMD AMD Reliability Brochure 1994 Data
Applied Reliability Page 53 AMD Reliability Brochure 1994 Data
Applied Reliability Page 54 Appendix
Applied Reliability Page 55 Class Project System Models A) A component has reliability R(t) = 0.99. Twenty-five components in series form a system. Calculate the system reliability. R s (t) = (0.99) 25 = 0.778 or 77.8% B) A component has reliability R(t) = 0.95 Three components in parallel form a system. Calculate the system reliability. R s (t) = 1- (1-0.95) 3 = 0.9999 or 99.99%
Applied Reliability Page 56 Class Project Complex Systems A system consists of seven units: A, B, C, D, E, G, H. For the system to function unit A and either unit B or C and either D and E together or G and H together must be working. Draw the reliability block diagram for this setup. B D E A C G H Write the equation for the CDF of the system in terms of the individual component reliabilities, that is, the R i, where i = A, B, C,..., G, H. Hint: Consider the three subsystems:a alone; B with C; and D,E,G,H. 1) R A 2) R BC =1- (1- R B )(1- R C ) 3) R DEGH = 1- (1- R DE )(1- R GH ) = 1- (1- R D R E )(1- R G R H ) The system CDF is F S = 1 - R S = 1 - R A R BC R DEGH
Applied Reliability Page 57 Class Project Defect Models 1. Estimate the proportion defective p and the number of mortals in the sample. Fill in the mortal CDF column in the table below. Time Cum # Fails CDF Est All Units (%) 10 2 2/50 = 4% 25 4 4/50 = 8% 50 8 8/50 = 16% 100 13 13/50 = 26% 200 17 17/50 = 34% 500 20 20/50 = 40% 1000 20 20/50 = 40% CDF Est Mortals (%) 2. Plot the data for the mortal subpopulation on the same sheet of paper. Does the fit look reasonable? 4. Estimate the characteristic life c = T 63, the 63rd percentile. 5. Estimate the shape parameter m by drawing a line perpendicular to the best fit by eye line through the estimation point on the Weibull paper and reading the beta estimation scale.
Applied Reliability Page 58 Class Project Defect Model Example n = 50 Time Cum # Fails CDF Est All Units (%) CDF Est Mortals (%) 10 2 2/50 = 4% 2/20 = 10% 25 4 4/50 = 8% 4/20 = 20% 50 8 8/50 = 16% 8/20 = 40% 100 13 13/50 = 26% 13/20 = 65% 200 17 17/50 = 34% 17/20 = 85% 500 20 20/50 = 40% 20/20 = 100% 1000 20 20/50 = 40% 20/20 = 100% Estimated mortal fraction, p : 0.40 or 40% CDF estimate for mortals is based on sample size of defective subpopulation.
Applied Reliability Page 59 Weibull Probability Plot
Applied Reliability Page 60 Class Project Defect Model Example Model Check Weibull Parameter Estimates for Mortal Population : Characteristic Life (c) 100 Shape Parameter (m) 1.0 F( t) 1 e t / c m Mortal CDF (Weibull Model) Model CDF for All Units Empirical CDF All Units Mortal Time Fraction 25 0.221 0.4 0.088 0.08 100 0.632 0.4 0.253 0.26 1000 1.000 0.4 0.400 0.40
CDF Applied Reliability Page 61 Class Project Defect Model p x Weibull CDF Plot Defect Model Example 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 100 200 300 400 500 600 700 800 900 1000 Times (Hrs)