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Package ReliabilityTKK 2009 L 3 P A & B1

James E. MorrisDept of Electrical & Computer Engineering

P l d S U i i

TKK 2009 Lecture 3 Parts A & B1

Portland State University

Package Reliability

A. Reliability TheoryR li bili (b h b ) PDF l i f il f iReliability (bathtub curve), PDF, cumulative failure functionHazard rate, mean time to failureExponential distributionWeibull distributionSystem reliabilityAccelerated testing

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gPlotting failure distributions

B. Package Failure Mechanisms

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Potential consequences of poor reliability

Customer Supplier/VendorLoss of Product Warranty claims

Loss of product capability Production downtime

Production downtime Test and repair cost

Spare parts and maintenance Diminished confidence and image

Loss opportunities Loss of future business

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Reliability & FailureReliability: The reliability of a product is the probability that the product will perform satisfactorily for a given time at a desired confidence level under specified operating and environmental conditions.Failure is defined as the loss of the ability of the product to perform a required operation in a specific environment.Three types of failure according to when a failure

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Three types of failure according to when a failure occurred during a product’s operating life:

• Infant mortality• “Useful” life• Wear-out failure

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The Bathtub CurveIf a plot of the rate failure of a product versus its operating life is constructed from data taken a large sample of identical products placed in operation at t=0

High level stress

Medium stress level

Low stress level

stressRate of failure

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Low stress level

Mission timeInfant mortality

(stress screening)Useful life

(random failures)Wear out

Reliability TheorySample size = Number failed at time =

ont )(tn fNumber failed at time =

Number still operating satisfactorily =

Reliability = and “Unreliability” = Probability device "ok”

t)(tns

osf ntntn =+ )()(

)(tn f

)(tR )(tQ

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Probability device ok)(tR

)(1 /)(1

/)()(

tRntn

ntntQ

os

of

−=−=

=

)(1

/)(1/)(

tQ

ntnntn

of

os

−=

−==

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Failure Probability Density Function (PDF)Failure probability function )(tf

)()/1()( tndtdntf fo=

dttdR

dttdQ )()( −==

Cumulative failure function

Similarly and

)()(

)()(0

tFn

tndftQ

o

ft

===∴ ∫ ττ

dt dtdt

∫∞

==t

dfn

tntR s ττ )()()(0

1)(0

=∫∞

ττ df

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F(t) here corresponds to the three regions of the bathtub curve

Hazard Rate (“force of mortality”)The Hazard Rate is defined as the number of failures per unit time per number of operational parts left.

tRdtdQtdnt f )]([1)(1)(1)( −

===λ)(tf

=

Note λ(t) = [dnf(t)/dt]/ns ≈ f(t) = [dnf(t)/dt]/n0 if ns ≈ n0i.e. if few failures yet

Cumulative Hazard Rate: )( ln)()( tRttHt

−== ∫λ

dttRdttRdttnt

s )()()()( ===λ )(tR

since R(0)=1Mean Time to Failure (MTTF):

if R(t)→0 for t « ∞.

0∫

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∫∞

=0

)(. dttftMTTF ∫∞

⇒0

)( dttR

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Function Definition General 2-parameter Weibull (γ=0)

Exponential (η=1, β=1)

Rayleigh (β=2, k=2/η2)

Probability density function

f(t)

kt.exp-½kt2

Reliability relationships, for nf(t) failed and ns(t) surviving at time t of an initial sample of n0.

Survivor function (reliability) R(t)

exp-½kt2

Cumulative failure function (unreliability)

Q(t)=1-R(t)=F(t)

1- 1-1- exp-½kt2

Hazard rate λ(t) kt

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Cumulative hazard rate H(t)

λ0t kt2

Mean time to failure MTTF

1/

Example: Exponential DistributionMost commonly used with wide application in electronic systems. It is a single parameter distribution.

)(exp)( )( tuttf oo

λλ −=

oλ)(tuwhere is the Heavyside step function andis called the chance hazard rate.

)(.exp)( tutf oλ=

∫∞

=t

dftR ττ )()( ]∞∞

−−=−=∫ t0 )exp( ).exp( τλττλλt

oo d )(exp 0tλ−=

and

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t t

)(1)( tRtQ −= 1)(0

=∫∞

ττ df

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Exponential Distribution

0.80

1.00

1.20Q (t)Q (t)

λ0

f (t)

Area Q (t)

Total Area = 10.00

0.20

0.40

0.60

0 1 2 3 4 5 6

R (t) time t

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0 1 2 3 4 5 6t

Area R (t)ttime

Hazard Rate ("force of mortality")

)()()()(

1

)()(

1)( )0(

tRdt

tdRdt

tdQtR

ndt

tdntn

t f

s

−=⇒

⎯⎯ →⎯= ÷λ

( )0

0

0

0

expexp λλ

λ

λ

=−

−⇒ −

−

t

t

)( dtdttR expfor exponential (Failure frequency at time t)

Mean Time to Failure MTTF∫∞

=0

)(. dttftMTTF0

)(⇒∞

∫ dttR

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00

0

0

1exp1 ] λλ

λ=⎟

⎠⎞

⎜⎝⎛ −=

∞

tfor exponential distribution

i.e. interpret Reliability ⎟⎠⎞

⎜⎝⎛ −

=MTTF

ttR exp)(

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Weibull Distribution

β

ηγβ

ηγ

ηβ

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

tttfw exp1

)(

)(tfw is Weibull probability distribution function, where"3-parameter distribution"

ηβγ)(t

exp)(−−

=tR

0=γ "2-parameter Weibull"

101 −λβ e ponential

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,0 , 1 === oληγβ exponential

Weibull Distributionβ > 0 is "shape factor"

0< β< 1 early failureβ =1 constant rateββ >1 wear out

η>0 is "scale factor"

γ is location parameter

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γ pγ < 0 pre-existing failure (storage, transport)γ = 0 failure begins t = 0γ > 0 failure free period t = 0

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Weibull Distribution1.4)(tfw

0.2

0.4

0.6

0.8

1

1.2

β=1

β=2

β=3γ=0

η=1

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0

0.2

0 0.5 1 1.5 2

mission time

Effect of shape parameter

Weibull Distribution

)(tf2,0 == βγ

0 80.9

1

)(tfw 1=η

2=η

3η =0 10.20.30.40.50.60.70.8

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t

Effect of scale parameter

00.1

0 0.5 1 1.5 2 2.5mission time

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0=γ 2=β 1=η

Weibull Distribution1)(tfw

0<γ0>γ

γ

0.2

0.4

0.6

0.8

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-0.2

00 0.5 1 1.5 2 2.5 3

mission time

Weibull Distribution: Hazard Rate(Bathtub Curve)

4

5

β=3

1

2

3

haza

rd r

ate

β=2

β=3/2

β=1

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00 0.5 1 1.5 2 2.5

time

β=1/2

Effects of shape parameter β

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Series ReliabilityFailure of a system if any components fails. It is function of the weakest link in the system.

n

n

iss RRRRRR ,..... 321==∏

ssss

n

iss

ni

iss

MTTFλ

λλ

11

3211

=

=∑

∏=

P ll l R li bilit

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Parallel ReliabilityIf there are redundant components

note: for n=2n )1(11

1∏=

−−=−=n

iipsps RQR

ttt

ttt

ps eeeeee)(

)(2121

2121

2121 )(λλλλ

λλλλ λλλλλ +−−−

+−−−

−++−+

=

Effect of series complexity on system reliability

No. ofC

36.60%90.48%99.01%99.90%100

90.44%99.004%99.90%99.99%10

99.00%99.90%99.99%99.999%

System reliability for individual component reliability ofComponentsin series

4/21/2009200.004%36.77%99.48%99.01%1000

0.66%60.64%99.12%99.50%500

8.1%77.87%99531%99.50%250

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Accelerated TestingFailure Mechanism (diffusion, corrosion, etc)

kTE /

So, accelerated at higher temperature

kTErr /0

0exp−=

/rα 1 failure totime

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So, Times to failure at , at1t 1T 2t 2T

)11(exp21

0

2

1

TTkE

tt

−= Acceleration factor1/ 21 >tt 21 TT <

Plotting :Weibull Distributionβαexp1)( )/(−= −tQ t ))](1/(1log[log tQ−

αββα β

lnln)](1ln[ln{)/()}(1ln{

exp1)(

−=−−−=−

=

ttQttQ

tQ

or use Cumulative Hazard ratet tlog

slope β

Intercept gives α

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)](ln[)()(0

tRdtttHt

−== ∫λ [ ])(1ln tQ−−=

plot )(log tH vs tlog

tlog

∴

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Plotting: Exponential Distribution

)(1/1 tQ−

ttQ

tH

tQ t

λ

λ

=−

=∴

−= −

)(11ln)(

exp1)( 1000

100

10

1

)(Q

tslope =λ 1−MTTF

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0 100 200 300 4001 t

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B. Failure Mechanisms and Modeling1. Mechanical and Thermomechanical Degradation mechanisms

a. Fatigue (Fracture Mechanics) b. Creepc. Stress corrosion cracking

2.Thermo- and Electrotransport Fail Mechanisms. e o d ec o spo ec s sa. Electromigrationb. Thermomigration

3. Electrical and Thermal Degradation Mechanismsa. Dielectric degradation & breakdownb. Contact resistance degradation due to oxidation

4. MIL-HDBK-217 and Physics of Failure5. Chemical and electrochemical failure mechanisms

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a. Corrosionb. Wet and Dry Migration

6. Plastic Package Failuresa. Popcorningb. Dry Packing

7. Distributions of Failures

Plastic Package Failure Mechanisms

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1. Thermomechanical Deformation in solder joints

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Maximum stress at the edge due to a large DNP(distance from neutral point)

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Nucleation and propagation of fatigue crack in solder joints

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Fatigue life for MCM solder joints on the various substrate materials vs. material's CTE

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Mechanical and Thermo-mechanical Degradation Mechanisms

Mechanical Elastic and plastic deformation

stress

strainε

deformation

σ

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Ideal elastic material

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Stress control Strain control

Cyclic dependent hardeningTypical of soft fully annealed

metal samples

ε σ

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Cyclic dependent softening Typical of initially

hard cold worked metal

samples

σε

tcoefficienstrength fatigue'

)2('2

=

=Δ

= NE

f

bff

ea

σ

σεσ

Cyclic Mechanical Testing: Coffin-Manson EquationElasticElastic

exponentstrength failure

failure toreversals load of no. 2

fail tocycles

)"12( reversal load oneat intercept stress"

=

=

=

==

b

N

N

N

f

f

f

f

Pl i

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Plastic

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Palmer-Miner Cumulative Damage LawIf number of cycle at stress which would cause failure cycles, etc

iniN

iay ,

Failure predicted at point where1

1=∑

=

k

i i

i

Nn

iN

ii

i

σnσkwhere

liff iat cycles of no.

level sterss th i levels stressdifferent of no.

===

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ii σN at life fatigue =

1

whenFailure

3

3

2

2

1

1 =++Nn

Nn

Nn

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