exploring reliability with jmp: accelerated testing€¦ · accelerated testing is a powerful...

Copyright © 2009, SAS Institute Inc. All rights reserved.Copyright © 2009, SAS Institute Inc. All rights reserved.

Exploring Reliability with JMP: Accelerated TestingDavid C. Trindade, Ph.D.March 2010

Copyright © 2009, SAS Institute Inc. All rights reserved.

Topics

Reliability Concepts and Terminology

Life Distribution Fitting

Competing Cause Analysis

Accelerated Life Testing

Recurrence Analysis

Degradation Studies


The Benefits of Accelerated TestingReveal failure mechanisms

Reduce sample size requirements

Reduce stress times

Save money

Provide a way to model expected field behavior

Define limits for stress testing

Speed up product introduction with confidence


Accelerated TestingWhat is Accelerated Testing ?

Stressing at higher than normal conditions to make failures occur earlier.

True acceleration occurs when levels of increasing stress cause “things to happen faster”.The failure mechanisms are exactly the same as seen under normal stress, only the time scale has been changed. It’s like watching a movie in fast forward mode.True acceleration is, therefore, just a transformation of the time scale.


Physical Acceleration ModelsLinear Acceleration

If we know the life distribution of units operating at a high stress, and we apply an appropriate time scale transformation to a lower stress condition, we can derive the life distribution at that lower stress.

When every time of failure at high stress is multiplied by the same constant value to obtain the projected results at a low stress, we have linear acceleration.

In other words,

failure time low stress = AF x failure time high stress, where AF is called an “acceleration factor.”


Acceleration Factor AFThe acceleration factor is the ratio of the time to a given

CDF value under low stress conditions to the time to that same CDF value under high stress conditions, that is:

A Ftt

S

S F F

l o w

h i g hS l o w S h i g h

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

=

For true, linear acceleration, the acceleration factor, once determined, is the same for any percentile value. Thus,

A Ft

t

t

t

t

tS l o w

S h i g h

S l o w

S h i g h

S l o w

S h i g h

= = =5 0

5 0

1 0

1 0

1

1

For example, it takes 1,000 hours to reach 10% failures under high stress. If the AF is 25 to field use, it should take 25,000 hours to achieve 10% failures in the field.


Lifetime Distributions Time TransformationsExponential

Exponential distribution is the only case where the hazard rate varies inversely with the acceleration factor. Not true in general.

Weibull

Lognormal

For both the Weibull and the lognormal distributions, equality of shape parameters is a requirement for linear acceleration. This stipulation translates into parallel slopes for different stress levels on probability plots.

U S U SAFα α β β= × =

50 50 U S U ST AF T σ σ= × =

( ) or ( ) SU S U

h tMTTF AF MTTF h tAF

= × =

shape parameters


Checklist for Accelerated ModelingPart 1

1) Run at least two stress cells per accelerating factor under different accelerated conditions.

2) Confirm the failure distribution, for example, lognormal or Weibull, for each set of conditions. Probability plots are useful here.

3) For each cell, estimate the failure distribution parameters, and verify that shape parameters across cells are statistically equal.


Checklist for Accelerated ModelingPart 2

4) Choose the acceleration model for the mechanism involved, check graphically, and estimate the parameters of the model.

5) For the specified field conditions, apply the appropriate acceleration multiplier to transform the scale parameter (e.g., T50 or characteristic life) under stress to the scale parameter in the field.

6) Use the failure distribution model and the common shape parameter along with the field scale parameter to estimate the field failure distribution.


Accelerated Testing Involves Two Models

1. A parametric model (e.g., lognormal, Weibull) for the life distributions obtained by finding the best fitting parametric model for the data.

2. An acceleration model for the extrapolation of the parameters of the stress distributions to use conditions obtained by physical considerations and empirical confirmation.


Temperature Acceleration:Arrhenius Model

The Arrhenius model states that the log of the time to a given CDF value is proportional to an activation energy (EA or ΔH) and inversely proportional to the temperature in degrees Kelvin (which is 273.16 + degrees Celsius). Express the relationship for T50 as

where A is a constant, and the Boltzmann constant k = 8.617E-5 eV/oK. The acceleration factor between temperatures T1 and T2becomes :

50 0/

50

ln( ) /

e A

AE kT

T E kT

T A

β= +

=

( )( )

( ) ( ) ( )1 21

2

/ 1/T 1/T50 1

50 2

at Tat T

AE kTAF e

T⎡ − ⎤⎣ ⎦= =


Activation Energy EAThe activation energy is specific to a given failure mechanism

and must be determined empirically or obtained from reliability literature. The higher the number, the greater the acceleration by temperature. Some typical values are shown below for integrated circuit failure mechanisms.

Mechanism EA

Gate Oxide 0.2 - 0.3 eVElectromigration 0.5 - 1.2 eV

Corrosion 0.3 - 0.6 eVMobile Ion Movement 1.0 - 1.4 eV


The Eyring Model for Acceleration by Temperature and Voltage

The Eyring model is a general solution to the problem of additional stresses. The Eyring model for temperature and a second stress S1 is:

( ) 1//5 0

A B C T SE k TT A T e eα ⎡ + ⎤⎣ ⎦=Additional stress terms may be added.

For example, a temperature and voltage acceleration model can be written from the Eyring model as

/5 0

AE k TT A e V β−=


Acceleration Model Parameters

We’ll use the electromigration model formula/ T

5 0AE knT A J e−=

Take logs of both sides to get

5 01ln ln lnTAT A n J E

k⎛ ⎞= − + ⎜ ⎟⎝ ⎠

For fixed current density J, plot of ln T50 vs.

should produce a straight line plot with slope EA.

For fixed temperature T, plot of ln T50 vs. ln J should give a straight line plot with slope -n.

1kT

Empirical Determination by Regression


Reliability Studies

Generally run minimum of two cells, preferably three, for each accelerating factor (temperature, voltage, etc.)

For two factors, run a factorial design (full or fractional) with minimum of two, preferably three levels, for each factor. Referred to as a matrixstudy.


Planning Matrix Reliability StudiesA few rules of thumb to keep in mind :

1) Allocate the units to the individual stress cells to obtain approximately an equal number of failures per cell. The higher stress cells will produce higher cumulative percent failures than lower stress cells. So we use more units in the lower stress cells and fewer in the higher stress cells.

2) Analyze each cell data via probability plots to check fit to the assumed failure distribution. The shape parameters for each cell should be roughly the same. Verify using MLE methods.

3) Use regression analysis for estimating the acceleration model parameters.


Accelerated Test Example

Company ABC has developed a new integrated circuit and wishes to estimate the reliability of the product after one year (8,766 hours) under use conditions at 55° C.

An accelerated stress is run for 2000 hours on a total of 400 units. Four temperature cells are run, with temperatures 85°, 100°, 125°, and 150° C. Allocated sample sizes are 160, 120, 80, and 40, respectively.

Readouts are conducted at 0, 24, 48, 96, 168, 336, 500, 750, and 1000 hours. The results are shown in the next table.


Accelerated Test Results


JMP Data Table


JMP Fit Life by X ModelSelecting Analyze > Reliability and Survival >

Fit Life by X, we then fill in the dialog box.


Fit Life by X Results: Scatterplot

Not too informative: shows each temperature cell had the same readouts starting at 24, 48, 96, …1000 hours.


Fit Life by X Results: Nonparametric Overlay

Adjusting scale limits, display shows nonparametric fraction failures at the end of each readout interval on a lognormal probability plot scale. Lognormal looks promising for the failure distribution.


Fit Life by X Diagnostics

Lognormal fit looks good. Constrained equal slopes looks good also. Regression acceleration model appears good also.


Fit Life by X Distribution Profiler

At field use temperature of 55° C, the expected fraction failing after one year (8766 hours) is about 18%.


Fit Life by X: Density Curves and Quantile Lines

Quantile lines at 0.1,0.2, and 0.5.


Fit Life by X: Quantile Profiler

Time to reach 5% fraction failure at use conditions.


Fit Life by X: Acceleration Factor ProfilerAcceleration factor between 125° and 85° C.

EA estimate.

Common σ estimate.

Equation to estimate lognormal mean at any temperature.


Lessons Learned

Accelerated testing is a powerful approach to conduct reliability studies in a reasonable time .

There are two models of interest in accelerated stress testing: the lifetime distribution model and the acceleration model. If the slopes are not roughly equal across different stress cells, suspect different failure causes are present.


Review of AR-1 Continued

• Accelerated Testing (continued)• Activation Energy• Eyring Models• Determining Model Parameters• Matrix Reliability Studies and Example• Planning Guidelines• Degradation Modeling• Sample Sizes for Accelerated Testing

• System Models• Series System• Parallel System• Analysis of Complex Systems• Standby Redundancy

• Defective Subpopulations

exploring reliability with jmp: accelerated testing€¦ · accelerated testing is a powerful...

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