(QALT): Data Analysis and Test Design

Transcription

1 (QALT): Data Analysis and Test Design Huairui(Harry) Guo, Ph.D, CRE, CQE Thanasis Gerokostopoulos, CRE, CQE ASTR 2013, Oct. 9-11, San Diego, CA

2 Qualitative vs. Quantitative ALT (Accelerated Life Tests) Understand Quantitative ALT Ø Data Modeling Ø Examples Design Quantitative ALT Ø Theory Ø Example Outlines

3 Section I: Qualitative vs. Quantitative ALT

4 Qualitative vs. Quantitative ALT Many different names are used for different accelerated life tests. In general, we can classify them into 2 categories.

5 Qualitative Accelerated Tests An accelerated test that yields failure information (or failure modes) only is a qualitative test (commonly called shake & bake tests, HALT, MEOST, elephant tests, etc.). In such tests, stresses applied to the product are well beyond normal shipping, storage and in use levels for the purpose of finding and correcting design weaknesses. (Statistical properties of life distributions will not be measured.)

6 Qualitative Accelerated Tests Benefits Ø Increase reliability by revealing probable failure modes. Unanswered Questions Ø What is the reliability of the product? Ø Are the failure modes the same as the ones that will occur during the life of the product under normal use?

7 Quantitative accelerated life testing, unlike qualitative testing, is designed to provide reliability information on the product, component or system, through failure data obtained from the accelerated test.

8 Why Quantitative ALT? To be able to quantify the life characteristics, and/ or failure rate behavior, of an item in a shorter time. Ø If you are warranting a product for 5 years and it has a 12-month development cycle, you can t afford the time to do life tests under usual conditions.

9 Section II: Understand Quantitative ALT

10 Stress & Strength: Why Components Fails

11 Increasing Stress in an ALT to Generate Failures

12 Predict Life at Normal Usage Stress Level

13 Life Stress Relationship (LSR) Strictly Monotonic Function Ø It is a function that is increasing on its entire domain or decreasing on its entire domain. One-to-One Function Ø The function should be a oneto-one function (i.e., for each y value in the range of f, there exists exactly one value of x in its domain).

14 Overview of the Analysis Steps Accelerated life models usually consist of: Ø A life distribution (e.g., Weibull). Ø Life-stress relationship(s).

15 Weibull Lognormal Exponential Select the Failure Time Distribution A statistical distribution is used to describe the randomness of the failure times at a given stress level β η β η η β = t e t t f 1 ) ( 2 1 ln( ) 2 1 () 2 t f t e t µ σ σ π = = m t e m t f 1 ) (

16 Choose one Characteristic to Represent the Life In QALT, one of the most important assumptions is that the failure mode due to the test stress is the same at different stress levels. Failure mode is represented by the shape parameter such as β in Weibull distribution and σ in lognormal distribution. They are the same at different stress levels. The scale parameter is used as the life characteristic to represent the life at a given stress. They are different at different stress levels. Ø Weibull: η(s) Ø Lognormal: µ(s) Ø Exponential: m(s)

17 Select the Life-Stress Relation (LSR) functions will describe the life characteristic (L)/or its transformation as a function of stress (V). L ( V ) = f ( V ) Simple curves provide the best candidates. Ø General exponential curve: L( V ) = A BV e Ø General power curve: L( V ) = A V B

18 Using the Relationship in a Distribution In QALT, one must first choose the underlying life distribution. Once the distribution has been chosen, the distribution s scale parameter is usually modeled using the life-stress Reliability vs Stress Surface relationship. For the Weibull distribution then: f η( V ( t; β, η( V ) = L( V )) )

19 Common Life-Stress Relationships Found widely in literature and used in QALT application (for constant stresses): Ø Arrhenius Ø Eyring Ø Inverse Power Law Ø

20 Arrhenius Relationship Commonly used for analyzing data for which temperature is the accelerated stress. R( T ) = A e K T Ø where R is the speed of reaction, A is a nonthermal constant, EA is the activation energy (ev), K is Boltzman s constant ( E-5 ev/k) and T is the absolute temperature (kelvins). The activation energy is the energy that a molecule must have to participate in the reaction. In other words, the activation energy is a measure of the effect that temperature has on the reaction. E A

21 A More General Arrhenius Relationship Since the activation energy is a measure of the effect that temperature has on the reaction, when used in QALT, it would be the effect of temperature on life. This must then be determined from the data. Since K is a constant, a parameter B can be defined and solved for based on data, thus generalizing the model. e E A K 1 T = e where B B T = E K A

22 A More General Arrhenius Relationship(cont d) For a stress V, the life characteristic of interest as a function of the stress, L(V), is given by an exponential relationship which assumes that life is proportional to the inverse reaction rate of the process: L( V ) C e The term acceleration factor is used to describe the ratio of the life characteristic at the use and accelerated test conditions, or: A F = = Ø Thus, if the life characteristic (i.e., MTTF) is half of what it is at use conditions, then the acceleration factor is 2. B V L( Use) L( Accelerated )

23 Common Formulations with Arrhenius Relationship Exponential- Arrhenius f(t;v)= 1/C e B/V e 1/C e B/V t Weibull- Arrhenius f(t;v)= β/c e B/V ( t/c e B/V ) β 1 e ( t/c e B/V ) β Lognormal- Arrhenius f(t;v)= 1/t σ 2π e 1/2 ( ln(t) ln(c) B/V / σ ) 2

24 Estimating the Parameters (MLE) D i ( t; V ) for i = 1 n f ( t; V ) = β C e B V C t e B V β 1 e t C e B V β Using the model and data, estimate the parameters. Maximum likelihood estimation (MLE) methods are utilized. β,c, B

25 Once the parameters are obtained, the model is fully defined for the current data set. As an example: Result of Interest β = 2.4; C = ; B = f ( t; V ) = e V t 9 e V e t e V 2.4

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27 Other Life-Stress Relationships Inverse Power Law: L( V ) Eyring Relationship: L( V ) = = K 1 V 1 V All the models have the same analysis procedure as the Arrhenius relationship e n B A V

28 Rule of Thumb Regarding LSRs Use exponential life-stress relationships (LSRs) for thermal stimuli. Temperature Humidity Use power LSRs for non-thermal stimuli. Voltage Mechanical Fatigue Other

29 Consider the Bulbs at 140V and 130V TTF (hr) at 140V Example 1 TTF (hr) at 130V

30 Power Relationship f 130 (t)= β/ η 130 ( t/ η 130 ) β 1 e ( t/ η 130 ) β Life Characteristic f 140 (t)= β/ η 140 ( t/ η 140 ) β 1 e ( t/ η 140 ) β Voltage (V)

31 f V (t)= 3.9/ 1/9.6E 31 V 13 ( t/ 1/9.6E 31 V 13 ) 2.9 e ( t/ 1/9.6E 31 V 13 ) 3.9

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33 Temperature-Non-Thermal(T-NT) Temperature-Humidity(T-H) General Log-Linear(GLL) Relationships for Two or More Stresses V B n n V B e U C KU Ae V U L = = 1 ), ( + = U B V e A V U L φ ), ( ( ) + = = = + i i i X X Y L e V U L i i i α α α α 0 ln ), ( 0 Note: Arrhenius, Inverse Power Law, T-NT, T-H and many other models can be represented by GLL.

34 Note on Data Requirements For one stress model ( X 1 is the stress) L( X = α + α X Ø We need at least two stress levels in order to solve for the two parameters α 0 and α 1. For multiple stress model 1 ) 0 Ø The stress combinations must be at least the number of stresses

35 Will These Test Plans Give Enough Data for Modeling? Case 1: Stresses are temperature and humidity, 2 test conditions. Temperature Humidity Case 2: Stresses are temperature and humidity, 3 test conditions Temperature Humidity

36 Design Evaluation Both designs are bad since the effects of temperature and humidity are confounded (or aliased). We do not have enough data (stress levels) to separate the effects of temperature and humidity, so the coefficients of the two stresses in the model cannot be estimated.

37 Section III: Design a Quantitative ALT

38 Design a QALT for Reliability Prediction The purpose of a Quantitative Test is to predict the reliability of a product. Enough failures should be obtained from the test in order to have an accurate prediction model. To design a test plan, we usually need to determine Ø How many test units Ø How long to test Ø What are the stress levels Ø What are the combinations of stresses Ø How to allocate test units to the experiment.

39 Commonly Used Test Plan For a single stress, the most commonly used test plans are: Ø The 2 Level Statistically Optimum Plan. The plan will recommend two stress levels. One will be the maximum allowable stress and the second will be computed so that the variance of the B(X) life is minimized. Ø The 3 Level Best Standard Plan. The plan will recommend three equally spaced stress levels with equal allocations. One stress will be the maximum allowable stress and the other two stresses will be computed so that the variance of the B(X) life is minimized.

40 Commonly Used Test Plan (cont d) Ø The 3 Level Best Compromise Plan. The plan will recommend three equally spaced stress levels using the same approach as the 3 Level Best Standard Plan. The difference is that the proportion of the units to be allocated to the middle stress level is defined by the user. Ø The 3 Level Best Equal Expected Number Failing Plan. The plan will recommend three equally spaced stress levels using the same approach as the 3 Level Best Standard Plan. The difference is that the proportion of units allocated to each stress level is calculated such that the number of units expected to fail at each level is equal.

41 Theory: Reduce the Model Uncertainty In QALT In QALT, it usually is required to minimize the uncertainty of the estimated model parameters. Ø Variance-covariance matrix of parameter estimation Var 1 2 Λ = 2 θi where θ i Λ are the model parameters, and = F is the log-likelihood function. Minimizing the variance-covariance matrix is the same as to maximize the determinant of Fisher information matrix objective : max F st. constraints on stresses, constraints on sample,...

42 Example: One Stress Test Plan A reliability engineer wants to design an ALT for an electronic component. Use temperature is 300K while design limit is 380K. The engineer has: Ø 2 months or 1,440 hours available for testing and 2 available chambers. From historical data: Ø The beta parameter of the Weibull distribution is 3. Ø At 1,440 hours, the probability of failure at use temperature is At the design limit, it is The engineer wants to determine: Ø The appropriate temperature that should be set at each chamber. Ø The number of units that should be allocated at each chamber.

43 Example: One Stress Test Plan (cont d) The inputs for a 2 Level Statistically Optimum Test Plan are:

44 Results for the Test Plan The following figure shows the output of the test plan. The results show that: 68.2% of the units should be allocated at 355.8K and 31.8% at 380K. This test plan will give a minimal variance for the estimated B10 life.

45 Conclusions Discussed Qualitative vs. Quantitative Accelerated Life Tests. The theory and an example of modeling the failure data in Quantitative ALT. The theory and an example of design an efficient Quantitative ALT

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