Apr 10, 2000 - Elevated Temperature Acceleration of. Chemical Reaction Rates. ⢠The Arrhenius model Reaction Rate, R(temp), is. R(temp) = γ0 exp. (. âEa.
Accelerated Testing Background
Accelerated Testing Obtaining Reliability Information Quickly
• Today’s manufacturers face strong pressure to:
Improve productivity, product field reliability, and overall quality using new technology. Develop newer, higher technology products in record time.
William Q. Meeker Department of Statistics and Center for Nondestructive Evaluation Iowa State University Ames, IA 50011
• Implies increased need for up-front testing of materials, components and systems. • Accelerated tests provide timely information for product design and development. • Users must be aware of potential pitfalls
1
2
Overview What is Reliability? • Different kinds of accelerated tests • R(t) = 1 − F (t)
• Example 1—Evaluation of an insulating structure
• The probability that a system, vehicle, machine, device, and so on, will perform its intended function under encountered operating conditions, for a specified period of time. • Quality over time
• Example 2—New-technology microelectronic logic device • Accelerated Degradation Tests • Importance of physics of failure and physical/chemical models (and sensitivity analysis)
• A powerful marketing tool
• Example 3—Microelectronic RF amplifier device
• An engineering discipline requiring support from
• Connecting with the field
Physics and chemistry
• Example 4—Appliance field reliability
Statistics
• Areas for further research
3
4
Breakdown Times in Minutes of a Mylar-Polyurethane Insulating Structure (from Kalkanis and Rosso 1989)
Some Applications of Accelerated Tests
• Assess component or material reliability or durability.
10
4
10
3
10
2
10
1
10
0
• Make design decisions to improve reliability or lower cost
• System test to simulate field-use at accelerated conditions. • Predict product field performance. • Identify and fix potential failure modes at system/subsystem level (HALT and STRIFE tests). • Screening (100% or audit) testing of manufactured product (e.g. ESS and burn-in).
Minutes
• Verify predictions produced with physical models (e.g. FEM)
•• • • • •
• •• • •• •
• • • • • •
• •• •• • • •• • • • •
-1
10
100
150
200
250
300
350 400
kV/mm
5
6
Plot of Inverse Power Relationship-Lognormal Model Fitted to the Mylar-Polyurethane Data (also Showing 361.4kV Data Omitted from the ML Estimation)
Inverse Power Relationship-Lognormal Model
The inverse power relationship-lognormal model is
Pr[T ≤ t; volt] = Φnor
log(t) − µ σ
10
5
10
4
10
3
10
2
10
1
••• • • •
Minutes
where
• µ = β0 + β1x, and • x = log(Voltage Stress).
10
• σ assumed to be constant.
• ••• •• •
• ••• •• •
• •• •• •
90%
• •• • • • •
0
-1
10
50
100
200
50% 10%
500
kV/mm 7
8
Lognormal Probability Plot of the Inverse Power Relationship-Lognormal Model Fitted to the Mylar-Polyurethane Data
Methods of Acceleration Three fundamentally different methods of accelerating a reliability test:
.99 .98 .95
• Increase the use-rate of the product (e.g., test a toaster 400 times/day). Higher use rate reduces test time.
.9
Proportion Failing
.8 .7
• Use elevated temperature or humidity to increase rate of failure-causing chemical/physical process.
.6 .5 .4 .3
• Increase stress (e.g., voltage or pressure) to make degrading units fail more quickly.
.2 .1 .05
Use a physical/chemical (preferable) or empirical model relating degradation or lifetime at use conditions.
.02 219.0
.01 10
0
1
10
157.1
122.4
10
50 kV/mm
100.3
2
10
3
10
4
10
5
Minutes
9
Interval ALT Data for a New-Technology IC Device
10
New-Technology Integrated Circuit Device ALT Data
• Tests run at 150, 175, 200, 250, and 300◦C.
• Failures had been found only at the two higher temperatures. • After early failures at 250 and 300◦C, there was some concern that no failures would be observed at 175◦C before decision time. • Thus the 200◦C test was started later than the others.
11
Hours
• Developers interested in estimating activation energy of the suspected failure mode and the long-life reliability.
10
7
10
6
10
5
10
4
10
3
10
2
x x x
100
150
200
250
x x x
300
350
Degrees C
12
The Arrhenius-Lognormal Regression Model Elevated Temperature Acceleration of Chemical Reaction Rates The Arrhenius-lognormal regression model is
• The Arrhenius model Reaction Rate, R(temp), is
−Ea −Ea × 11605 = γ0 exp kB (temp ◦C + 273.15) temp K where temp K = temp ◦C+273.15 is temperature in Kelvin and
Pr[T ≤ t; temp] = Φnor
R(temp) = γ0 exp
kB = 1/11605 is Boltzmann’s constant in units of electron volts per K. The reaction activation energy, Ea, and γ0 are characteristics of the product or material being tested. • The reaction rate Acceleration Factor is
log(t) − µ σ
where
• µ = β0 + β1x, • x = 11605/(temp K) = 11605/(temp ◦C + 273.15)
R(temp) R(tempU ) 11605 11605 = exp Ea − tempU K temp K
AF(temp, tempU , Ea ) =
• and β1 = Ea is the activation energy
• When temp > tempU , AF (temp, tempU , Ea) > 1.
• σ is constant
13
Arrhenius Plot Showing ALT Data and the Arrhenius-Lognormal Model ML Estimation Results for the New-Technology IC Device.
14
Lognormal Probability Plot Showing the Arrhenius-Lognormal Model ML Estimation Results for the New-Technology IC Device
.95
10
7
10
6
10
5
10
4
10
3
.9
.7
Proportion Failing
Hours
.8 .6 .5 .4 .3 .2 .1 .05 .02
10
x x x
x x x
2
100
150
200
250
300
.01 .005 .002 50%
.0005
10% 1%
300 Deg C
.0001
350
10
2
250
200 3
10
175 10
4
150
100 10
5
10
6
10
7
Hours
Degrees C on Arrhenius scale 15
16
Lognormal Probability Plot Showing the Arrhenius-Lognormal Model ML Estimation Results for the New-Technology IC Device with Given Ea = .8 Pitfall 4: Masked Failure Mode .95 .9
• Accelerated test may focus on one known failure mode, masking another!
.8
Proportion Failing
.7 .6 .5 .4 .3
• Masked failure modes may be the first one to show up in the field.
.2 .1 .05 .02 .01 .005
• Masked failure modes could dominate in the field.
.002 .0005 300 Deg C
.0001 10
2
250
200 3
10
175 10
4
150
100 10
5
10
6
10
7
Hours
17
18
Unmasked Failure Mode with Lower Activation Energy
6
10
6
10
5
10
5
10
4
10
4
10
10%
10
10
3
3
Mode 1
2
10
2
10
Hours
Hours
Possible results for a typical temperature-accelerated failure mode on an IC device
Mode 2
10% 10%
10
10
1
40
60
80
100
120
1
40
140
60
80
100
120
140
Degrees C
Degrees C
19
20
Percent Increase in Resistance Over Time for Carbon-Film Resistors (Shiomi and Yanagisawa 1979)
Advantages of Using Degradation Data Instead of Time-to-Failure Data
• Degradation is natural response for some tests. 10.0
173 Degrees C
Percent Increase
5.0
• Can be more informative than time-to-failure data. (Reduction to failure-time data loses information)
133 Degrees C
• Useful reliability inferences even with 0 failures.
1.0
83 Degrees C 0.5
• More justification and credibility for extrapolation. (Modeling closer to physics-of-failure) 0
2000
4000
6000
8000
10000
Hours 21
22
Percent Increase in Operating Current for GaAs Lasers Tested at 80◦C
Limitations of Degradation Data
10 0
• Analyses more complicated; requires statistical methods not yet widely available. (Modern computing capabilities should help here)
5
• Substantial measurement error can diminish the information in degradation data.
Percent Increase in Operating Current
• Obtaining degradation data may have an effect on future product degradation (e.g., taking apart a motor to measure wear).
15
• Degradation data may be difficult or impossible to obtain (e.g., destructive measurements).
• Degradation level may not correlate well with failure.
0
1000
2000
3000
4000
Hours
23
24
Device-B Power Drop Accelerated Degradation Test Results at 150◦C, 195◦C, and 237◦C (Use conditions 80◦C)
Arrhenius Model Temperature Effect on Chemical Degradation
A1 k1 -A2 and the rate equations for this reaction are
0.0
Power drop in dB
-0.2
• • •• ••• • • • • •
••• • •• • •• • ••• • • •• • ••• •• •• • • • • • •• •• •
-0.4
• • • •• • • • •• • • •• • • • •
-0.6
-0.8
-1.0
• •
-1.2
-1.4
•• •• •• • •• • • • • • ••• • ••• • • •• •• • • • ••• • • • • • • • • • • ••• •• • • • • •• • • • • • • • • • • • • •• •• • •
••• ••• •• • • • •• • ••• • •• •• • • • 150 Degrees C • •• • •• • • • •• • •• ••• • •• •• •• •• • • • • • • • • • • • • •• • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • •• •• • • • • • • •• •• • • • • • • • • • •• • • •• •• • • • • • •• • • •• •• • • • •• •• • ••• • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • 195 Degrees C • • • • • • ••• • • • • • • • • • • • • • • • • •• • • 237 Degrees C • • • • ••
0
1000
2000
•• • • • • •• • • • •• ••• •
dA1 = −k1A1 dt Solving these gives
• • • • • • • • • •• • • • •
and
dA2 = k1A1, dt
k1 > 0.
(1)
A1(t) = A1(0) exp(−k1t) A2(t) = A2(0) + A1(0)[1 − exp(−k1t)] where A1(0) and A2(0) are initial conditions. The Arrhenius model describing the effect that temperature has on the rate of a simple first-order chemical reaction is
3000
k1 = γ0 exp
4000
−Ea kB × (temp + 273.15)
Hours 25
26
Lognormal-Arrhenius Model Fit to the Device-B Time-to-Failure Data with Degradation Model Estimates
.99
What Do Accelerated Test Results Tell Us About Field Reliability? Need information on:
.98 .95
Proportion Failing
.9 .8
• Effects of acceleration (e.g., cycling rate).
.7 .6 .5 .4 .3
• Distribution of use-rates in actual use.
.2 .1
• Distribution of environmental conditions (e.g., stress spectra distributions).
.05 .02 .01 .005 237 Degrees C
195
150 Degrees C
80 Degrees C
.001 10^1
10^2
10^3
10^4
These factors may be given or, in some situations, inferred from the available data.
10^5
Hours
27
28
Establish a Transfer Function Relating Laboratory Tests and Field Performance
Component-A Laboratory Test Cycles to Failure
• Carefully compare laboratory tests results and field failures.
Same failure mechanisms operating in laboratory tests? Same factors (environmental noises) exciting the failure mechanisms? Identify laboratory/field discrepancies to improve test procedures. Seek understanding of reasons for lack of agreement. • Find a model (transfer function) to relate laboratory test to field use. • Understanding the relationship between the laboratory test results and product field reliability will provide stronger basis for using future laboratory tests to predict field performance.
29
0
10000
20000
30000
40000
50000
Cycles
30
Appliance Use-Rate Distribution (discretized lognormal distribution)
Example Use-Rate Model
• Life of a component in cycles of use, has a distribution
0.15
FC (c) = P (C ≤ c) = Φ
log(c) − µ σ
0.05
0.10
• Actual use-rate has a distribution given by the proportion of users πi (i = 1, . . . , k) that use the appliance at constant rate Ri, where ki=1 πi = 1. • Then the failure probability as a function of time is
0.0
FT (t; θ ) = P (T ≤ t) =
k
πi Φ
i=1
log (t) − µi σ
where θ = (µ1, . . . , µk , σ) and µi = µ − log(Ri).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Relative Frequency of Appliance Uses per Week
31
32
Predicted Field Reliability of Component-A as a Weighted Average of Lognormal Distributions
Predicted Field Reliability of Component-A as a Weighted Average of Weibull Distributions
.999 .98 .9
.95 .9
.7 .5
.8 .7
.3 .2
.6 .5
Fraction Failing
Probability
.1 .05 .02 .01 .005 .003
.4 .3 .2 .1 .05 .02
.001
.01
.0005
.005 .0002
.002 .001
.0001 50
100
200
500
1000
2000
5000
50
100
200
Mon Apr 10 14:01:32 CDT 2000
Weeks of Service
500
1000
Weeks of Service
2000
5000
Fri Mar 23 21:27:13 CST 2001
33
Fitted Use-Rate Model for the Wear Failure Mode
34
Field Variability Lognormal f (r; ηR, σR ) Density for the Wear Failure Mode (unloaded cycles relative to field days of use)
Lab: subset AccWear Appliance B Wear Failure Mode ALT data Field: subset Field Appliance B Wear Failure Mode .95 Laboratory Field
.9
Lab: subset AccWear Appliance B Wear Failure Mode ALT data Field: subset Field Appliance B Wear Failure Mode
Fraction Failing
.8 .7 .6 .5 .4 .3 .2 .1 .05 .02 .01 .005 .002 .0005 .00005 .00001 2
5
10
20
50
100
Lab time: Test Cycle
200
500
Field time: Weeks
1000
2000
5000
Thu May 10 22:23:56 CDT 2001
0.01
0.05
0.20
1.00
5.00
20.00
Thu May 10 22:24:23 CDT 2001
Test Cycles per Week 35
36
Simulation of a Proposed Accelerated Life Test Plan
Planning Accelerated Tests
• Most basic ideas of traditional DOE still hold
Temp= 78,98,120 n= 155,60,84 centime= 183,183,183 parameters= -16.7330, 0.7265, 0.6000
10
5
10
4
10
3
10
2
10
1
Log time quantiles at 50 Degrees C Average( 0.1 quantile)= 8.014 SD( 0.1 quantile)= 0.4632 Average( 0.5 quantile)= 9.138 SD( 0.5 quantile)= 0.5116 Average(Ea)= 0.7266 SD(Ea)= 0.08594
• In censored accelerated life tests (failure time is response) allocate more test units to low acceleration factor level than high acceleration factor levels.
• Consider including some tests at the use conditions. • Use simulation to investigate properties of alternative ALT plans.
Days
• Limit, as much as possible the amount of extrapolation used.
10
10%
0
Results based on 500 simulations Lines shown for 50 simulations
40
60
80
100
120
140
160
Degrees C
37
38
Concluding Remarks
Areas for Further Research
• Physical/statistical models for failure acceleration
• Accelerated Testing can be valuable tool when used carefully
• Methods for sensitivity analysis when empirical models must be used
• There is no magic in Accelerated Testing
• Prediction of service life in complicated environments • Physical/statistical models the field environment • Bayesian methods for analysis and planning (especially adaptive test plans) • Accelerated degradation test planning • Degradation analysis and planning with coarse (e.g. ordered categorical and censored) data. • Physical comparison of lab and filed failures to validate testing methods
• Cross-disciplinary teams are needed to deal effectively with all issues
Product/reliability/design engineers to identify productuse profiles, environmental considerations, potential failure modes or weaknesses that need to be evaluated, etc. Experts in materials and the chemistry/physics of failure to help in the understanding of an suggest/develop appropriate models for acceleration of particular failure modes. Statisticians to help with stochastic modeling, plan tests, fit models, and to help quantify uncertainty in results. • Users of Accelerated Testing must beware of pitfalls
39
40
References References • D. Byrne, J. Quinlan, Robust function for attaining high reliability at low cost, 1993 Proceedings Annual Reliability and Maintainability Symposium, 1993, pp 183-191. • L. W. Condra, Reliability Improvement with Design of Experiments, 1993, New York: Marcel Dekker, Inc. • M. Hamada, Using statistically designed experiments to improve reliability and to achieve robust reliability, IEEE Transactions on Reliability R-44, 1995 June. • M. Hamada, Analysis of experiments for reliability improvement and robust reliability, in Recent Advances in Life-Testing and Reliability, 1995, N. Balakrishnan, editor, Boca Raton: CRC Press. • Meeker, W.Q. and Hamada, M. (1995), Statistical Tools for the Rapid Development & Evaluation of High-Reliability Products, IEEE Transactions on Reliability R-44, 187-198. 41
• Meeker, W.Q. and Escobar, L.A. (1998a), Statistical Methods for Reliability Data. John Wiley and Sons, Inc. • Meeker, W.Q. and Escobar, L.A. (1998b), Pitfalls of Accelerated Testing. , IEEE Transactions on Reliability R-47, 114-118. • W. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, 1990, New York: John Wiley & Sons, Inc. • G. Taguchi, System of Experimental Design, 1987; White Plains, NY: Unipub/Kraus International Publications. • T. S. Tseng, M. Hamada, C. H. Chiao, (1995), Using degradation data from a factorial experiment to improve fluorescent lamp reliability, Journal of Quality Technology, 363-369.
42
