L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 1 MER301: Engineering Reliability LECTURE 14: Chapter 7: Design of Engineering

L Berkley DavisCopyright 2009

MER301: Engineering ReliabilityLecture 14

1

MER301: Engineering Reliability

LECTURE 14:

Chapter 7: Design of Engineering Experiments



2

Summary of Topics

Design of Engineering Experiments DOE and Engineering Design Coded Variables Optimization

Factorial Experiments Main Effects Interactions Statistical Analysis



3

Design of Experiments and Engineering Design

Applications of Designed Experiments Evaluation and comparison of design

configurations Establish Production Process Parameters Evaluation of mechanical properties of

materials/comparison of different materials Selection of ranges of values of independent

variables in a design (Robust Design) Determination of Vital x’s (Significant Few versus

the Trivial Many…..)



4

Design of Engineering Experiments Objectives of engineering

experiments include acquiring data that can be used to generate an analytical model for Y in terms of the dependent variables Xi

The model may be linear or non-linear in the Xi’s and it defines a Response Surface of Y as a function of the Xi’s

The model can be used to generate statistical parameters(means, std dev) for use in product design, as in DFSS



5

Factorial Experiments

Factorial Experiments are used to establish Main Effects and Interactions

Levels of each factor are chosen to bound the expected range of each Xi

,....),,...,( 2121 xxxxxfnY k

UBiiLBi xxx ,,



6

DOE Glossary Model- quantitative relationship ; also called

Transfer Function DOE- systematic variation of Xi’s to acquire data to generate

Transfer Function Factorial Experiment-all possible combinations of Xi’s are

tested Main Effects – change in Y due to change in Xi . Interactions –

joint effects of two or more Xi’s Replicates and Center Points

Response Surface- surface of Y generated by the Transfer Function Optimum Response- local max/min of Y

Partial Factorial Experiments- can run fewer points if can neglect higher order interactions

)( ixfnY

)( ixfnY



7

Coded Variables Coded Variables..

Each x typically has some (dimensional) range in which it is expected to vary….

In Designed Experiments the lower value of each x is often assigned a value of –1 and the upper value of x a value of +1

Coded Variables have two advantages First, discrete variables( eg, “yes/no”, Operator A/ Operator B) can

be included in the experiment Second, the magnitude of the regression coefficients is a direct

measure of the importance of each x variable Coded Variables have the disadvantage that an equation that

can be directly used for engineering design is not specifically produced

UBiiLBi xxx ,,

11 ix


DOE Process Map

8



9

Example 14.1-Text Example 7-1

Two level factorial design applied to a process for integrated circuit manufacturing Y= epitaxial growth layer thickness A= deposition time( ),levels are long(+1)

or short(-1) B= arsenic flow rate( ), levels are 59%

(+1) or 55%(-1) Experiment run with 4 replicates at each

combination of A and B

1x

2x



10

Union CollegeMechanical Engineering


8

Example 14.1(Text Example 7-1)

Two level factorial design applied to a process for integrated circuit manufacturing Y = epitaxial growth layer thickness A= deposition time( ),levels are

long(+1) or short(-1) B= arsenic flow rate( ), levels are

59%(+1) or 55%(-1) Experiment run with 4 replicates at each


1x

2x


Epitaxy is a kind of interface between a thin film and a substrate. The term epitaxy(Greek; epi "above" and taxis "in ordered manner") describes an ordered crystalline growth on a monocrystalline substrate

Epitaxial films may be grown from gaseous or liquid precursors. Because the substrate acts as a seed crystal, the deposited film takes on a lattice structure and orientation identical to those of the substrate. This is different from other thin-film deposition methods which deposit polycrystalline or amorphous films, even on single-crystal substrates. I f a film is deposited on a substrate of the same composition, the process is called homoepitaxy; otherwise it is called heteroepitaxy.

Homoepitaxy is a kind of epitaxy performed with only one material. In homoepitaxy, a crystalline film is grown on a substrate or film of the same material. This technology is applied to growing a more purified film than the substrate and fabricating layers with different doping levels.

Heteroepitaxy is a kind of epitaxy performed with materials that are different from each other. In heteroepitaxy, a crystalline film grows on a crystalline substrate or film of another material. This technology is often applied to growing crystalline films of materials of which single crystals cannot be obtained and to fabricating integrated crystalline layers of different materials. Examples inc lude gallium nitride (GaN) on sapphire or aluminium gallium indium phosphide (AlGaInP) on gallium arsenide (GaAs).

Heterotopotaxy is a process similar to heteroepitaxy except for the fact that thin film growth is not limited to two dimensional growth. Here the substrate is similar only in structure to the thin film material.

Epitaxy is used in silicon-based manufacturing processes for BJ Ts and modern CMOS, but it is particularly important for compound semiconductors such as gallium arsenide. Manufacturing issues inc lude control of the amount and uniformity of the deposition's resistivity and thickness, the c leanliness and purity of the surface and the chamber atmosphere, the prevention of the typically much more highly doped substrate wafer's diffusion of dopant to the new layers, imperfections of the growth process, and protecting the surfaces during the manufacture and handling

Doping An epitaxial layer can be doped during deposition by adding impurities to the source gas, such as arsine, phosphine

or diborane. The concentration of impurity in the gas phase determines its concentration in the deposited film. As in CVD, impurities change the deposition rate.

Additionally, the high temperatures at which CVD is performed may allow dopants to diffuse into the growing layer from other layers in the wafer ("autodoping"). Conversely, dopants in the source gas may diffuse into the substrate.



11

Example 14.1( con’t) What are the questions we

need to answer? What is the quantitative

effect of changes in A on the value of Y, ie the response of Y?

What is the response of Y to changes in B?

What is the interaction effect on Y when both A and B are changing, if any?

Are there values of A and B such that Y is at an optimum level?

Thickness time Arsenic14.037 30 0.5514.165 30 0.5513.972 30 0.5513.907 30 0.5514.821 60 0.5514.757 60 0.5514.843 60 0.5514.878 60 0.5513.88 30 0.5913.86 30 0.59

14.032 30 0.5913.914 30 0.5914.888 60 0.5914.921 60 0.5914.415 60 0.5914.932 60 0.59



12

Example 14.1(con’t)Two level factorial design applied to integrated

circuit manufacturing Y= epitaxial growth layer thickness A= deposition time( X1),levels are

long(+1) or short(-1) B= arsenic flow rate( X2), levels are

59%(+1) or 55%(-1)Experiment run with 4 replicates at each


How do we answer these Questions? What is the form of Experimental Design ? What is the quantitative effect

of changes in A on the value of Y?

What is the effect of changes in B on Y?





13

Factorial Design factorial design is used

when each factor has two levels Establish both main

effects/interactions Assumes linearity of

response Smallest number of runs

to test all combinations of x’s

Factor(Xi) levels often described as “+ or –” Called geometric or

coded notation

k2

k2

k2

7-8

Factors A B



14

Response of Y to A and B: Interaction/No Interaction In Fig 7-1, response of Y

to change in A is independent of B level; there is No Interaction between A and B

In Fig 7-2, the response of Y to change in A is shown with a different slope to illustrate an interaction between A

and B



15

Interaction/ No Interaction Interactions change the

shape of the response surface significantly

Experimental design must identify interactions and allow their impact to be quantified

7-3

7-4



16

Main Effects

Main Effect term captures the change in the response variable due to change in level of a specific factor



17

Main Effects

ab

a

b

)1( A

B



17

AB Interactions

Interaction terms show the effects of changes in one variable at different levels of the other variables For Eq 7-3, this would give the effects of A at

different levels of B(or vice versa)



17

Main Effects

ab

a

b

)1( A

B



18

Example 14.1: Data Set Two level factorial design applied to integrated circuit

manufacturing Y= epitaxial growth layer thickness A= deposition time(x1),levels are long(+1) or short(-1) B= arsenic flow rate(x2), levels are 59%(+1) or 55%(-1)

Experiment run with 4 replicates at each combination of A and B



17

Main Effects

ab

a

b

)1( A

B



19

Example 14.1: Effect Values



17

Main Effects

ab

a

b

)1( A

B



20

Coded Least Squares Model

The Regression Equation is of the Form

The Coded Least Squares Model is of the Form

X1 and X2 are coded variables and range from –1 to +1

is the average of all observations and the coefficients are

21322110ˆ xxxxY

21210 )2/()2/()2/(ˆ xxABxBxAYY

YY 002/,2/,2/ 321 ABandBA



21

Relationship between Regression Coefficients and the DOE Effects

The Regression Equation is

The Coded Least Squares Model is

so that

21322110ˆ xxxxY

21210 )2/()2/()2/(ˆ xxABxBxAYY

YxY 10ˆˆ

n

xy

x

xyy

xx

xxyy

S

Sk

i

n

jijij

i

n

jij

i

n

jijij

i

n

jij

i

n

jijij

xx

xy

k

k

k

k

k

2

)(

)(

)()(ˆ

2

1 1

2

1 1

2

2

1 1

2

1 1

2

2

1 11

22

......

2

1ˆ1

)1(,,1,,11

A

n

yyyyk

nbanab



17

Main Effects

ab

a

b

)1( A

B 1

0

ijx

x



22

Relationship between Regression Coefficients and the DOE Effects



For k=2

21322110ˆ xxxxY

21210 )2/()2/()2/(ˆ xxABxBxAYY

4

)(

4

ˆ)()

4

1()ˆ(

22

......

2

1ˆ

22)1(

22221

)1(,,1,,11

AV

nnnnn

nV

A

n

yyyy

baab

nbanab

1

0

ijx

x

0.....4/)()( 210 xbecausenVV



23

Example 14.1(con’t)Regression Analysis

Thickness time=A Arsenic=B interaction=AB coded time coded arsenic interaction14.037 30 0.55 16.5 -1 -1 114.165 30 0.55 16.5 -1 -1 113.972 30 0.55 16.5 -1 -1 113.907 30 0.55 16.5 -1 -1 114.821 60 0.55 33 1 -1 -114.757 60 0.55 33 1 -1 -114.843 60 0.55 33 1 -1 -114.878 60 0.55 33 1 -1 -113.88 30 0.59 17.7 -1 1 -113.86 30 0.59 17.7 -1 1 -1

14.032 30 0.59 17.7 -1 1 -113.914 30 0.59 17.7 -1 1 -114.888 60 0.59 35.4 1 1 114.921 60 0.59 35.4 1 1 114.415 60 0.59 35.4 1 1 114.932 60 0.59 35.4 1 1 1



24

Example 14.1 Excel Regression Analysis

Regression StatisticsMultiple R 0.958467416R Square 0.918659787

Adjusted R Square 0.898324733Standard Error 0.144187523Observations 16

ANOVAdf SS MS F Significance F

Regression 3 2.81764325 0.939214 45.17617 8.19196E-07Residual 12 0.2494805 0.02079

Total 15 3.06712375

Coefficients Standard Error t Stat P-value Lower 95%Intercept 14.388875 0.036046881 399.1712 4.11E-26 14.3103356

time 0.418 0.036046881 11.59601 7.08E-08 0.339460595% arsenic -0.033625 0.036046881 -0.93281 0.369306 -0.112164405interaction 0.01575 0.036046881 0.436931 0.66992 -0.062789405



25

Example 14.1:Regression Equation

Effects are calculated as A=0.836, B=-0.067, and AB=0.032 Large effect of deposition rate A Small effect of arsenic level B and interaction AB

Sample Variance( , pooled data set)=0.02079 Sample Mean( ) = 14.388900 YY

2121

21210

21210

01575.00336.0418.03889.14ˆ

)2/032.0()2/067.0()2/836.0(ˆ

)2/()2/()2/(ˆ

xxxxY

xxxxYY

xxABxBxAYY

2̂



26

Example 14.1(con’t)Two level factorial design applied to integrated

circuit manufacturing Y= epitaxial growth layer thickness A= deposition time( X1),levels are

long(+1) or short(-1) B= arsenic flow rate( X2), levels are

59%(+1) or 55%(-1)Experiment run with 4 replicates at each


How do we answer these Questions? What is the form of Experimental Design ? What is the quantitative effect

of changes in A on the value of Y?

What is the effect of changes in B on Y?





27

One Factor at a Time Optimization

The One Factor at a Time method of conducting experiments is intuitively appealing to many engineers

7-5 7-6



28

One Factor at a Time Optimization One Factor at a Time will

frequently fail to identify effects of interactions Learning to use DOE

factorial experiments often difficult for new engineers to accept

DOE’s however are the most efficient and reliable method of experimentation

7-7



29

Optimization Screening tests establish

factors (vital ’s) that affect Y

Range of ’s is a critical choice in the experimental design

Optimization will require multiple experiments

iXix

ix

7-7a



30

Example 14.1:Regression EquationOptimization

Take partial derivatives wrt X1 and X2 Set equal to zero and solve for X1 and X2 X1~2 and X2~-26

2121

21210

21210

01575.00336.0418.03889.14ˆ

)2/032.0()2/067.0()2/836.0(ˆ

)2/()2/()2/(ˆ

xxxxY

xxxxYY

xxABxBxAYY

No Optimum of Y in Range of X’s-1<=X,=1



31

Example 14-1:Statistical Analysis of the Regression Model-

There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample Data Sum of Squares based on Mean and Interaction Effects ANOVA/Significance Analysis of the Regression Equation

All of these methods are based on analysis of the single data set generated in the DOE.



32


There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample

Data Sum of Squares based on Mean and Interaction Effects ANOVA/Significance Analysis of the Regression Equation




33

Standard Error of the Effects calculated from Sample Data

The magnitude/importance of each effect can be judged by comparing each effect to its Estimated Standard Error.

The first step in the analysis is to calculate the means and variances at each of the i factorial run conditions using data from the n replicates. For the variances

The second step is to calculate an overall(pooled) variance estimate for the factorial run conditionsk2

k

ii

n

jijE yySS

2

1

2.

1

)ˆ(

)1(22

n

SSEk

abbaeitheryy ii ,,),1...(..



34

Calculation of Variance

The variances for the i factorial runs are

The overall(pooled) variance estimate for the 2k=4 factorial run conditions is

0625.0..,..0059.0,..0026.0,..0121.0 2222)1( abba and

0208.0)0625.00059.00026.00121.0(4

1

2ˆ

2

1

22

k

ik

i



35

Standard Error of the Effects calculated from Sample Data (con’t)…..

Given the overall variance, the effect variance is calculated as follows The Effect Estimate is a difference between two means,

each of which is calculated from half of the N measurements. Thus the Effect Variance is

where

The Standard Error of each Effect is then

)2/(ˆ)()()()( 22 knABseBseAseeffectse

2

2222

2/

ˆ

2/

ˆ2

2/

ˆ

2/

ˆ)()()()(

NNNNABVBVAVeffectV

knN 2




37

Relationship between RegressionCoefficients and the DOE Effects



For k=2

21322110ˆ xxxxY

21210 )2/()2/()2/(ˆ xxABxBxAYY

YxY 10ˆˆ

4

)(

4

ˆ)()

4

1()ˆ(

22

......

2

1ˆ

22)1(

22221

)1(,,1,,11

AV

nnnnn

nV

A

n

yyyy

baab

nbanab

1

0

ijx

x

U n i o n C o l l e g eM e c h a n i c a l E n g i n e e r i n g

M E R 3 0 1 : E n g i n e e r i n g R e l i a b i l i t yL e c t u r e 1 4

3 5

S t a n d a rd E r ro r o f t h e E ff e c t s c a lc u la t e d f r o m S a m p le D a t a (c o n ’t )… ..

G iv e n t h e o v e r a l l v a r i a n c e , t h e e ff e c t v a r i a n c e i s c a l c u la t e d a s fo l l o w s T h e E ff e c t E s t im a t e i s a d iff e re n c e b e t w e e n t w o

m e a n s , e a c h o f w h ic h i s c a l c u la t e d f ro m h a l f o f t h e N m e a s u re m e n t s . T h u s t h e E ff e c t V a r ia n c e i s

w h e re

T h e S t a n d a rd E r r o r o f e a c h E ff e c t i s t h e n

)2/(ˆ)()()()( 22 knABseBseAseeffectse

2

2222

2/

ˆ

2/

ˆ2

2/

ˆ

2/

ˆ)()()()(

NNNNABVBVAVeffectV

knN 2

etcAsese ),(2

1)ˆ( 1



37


Because is the same for all of the effects and is used to calculate the Standard Error of any specific Effect( ie, A, B, AB,…) the value calculated will be the same for each one…

For the Epitaxial Process

The value of the Effect is twice that of the coefficient in the Regression Equation. Similarly, the Standard Error for the Coefficient is half that of the Effect

)2/(ˆ)()()( 22 knEffectABseEffectBseEffectAse

072.0)24/(0208.0)2/(ˆ)( 2222 knEffectse

036.02/)()( EffectsetCoefficiense

knN 2



38


A Hypothesis Test is carried out on each of the Main Effects and the Interaction Effects. This is a t-test.

The A Effect is significant and the B and AB Effects are not

7-5



39

Example 14.1 Excel Regression Analysis

Regression StatisticsMultiple R 0.958467416R Square 0.918659787

Adjusted R Square 0.898324733Standard Error 0.144187523Observations 16

ANOVAdf SS MS F Significance F

Regression 3 2.81764325 0.939214 45.17617 8.19196E-07Residual 12 0.2494805 0.02079

Total 15 3.06712375

Coefficients Standard Error t Stat P-value Lower 95%Intercept 14.388875 0.036046881 399.1712 4.11E-26 14.3103356

time 0.418 0.036046881 11.59601 7.08E-08 0.339460595% arsenic -0.033625 0.036046881 -0.93281 0.369306 -0.112164405interaction 0.01575 0.036046881 0.436931 0.66992 -0.062789405



40




41


There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample Data Sum of Squares based on Mean and Interaction

Effects ANOVA/Significance Analysis of the Regression Equation




42

Sum of Squares:Main Factors and Interaction

The Sum of Squares for the Effects can be expressed as

Sum of Squares from the Main and Interaction Effects can be used to assess the relative importance of each term

The Total Sum of Squares is obtained from and the Mean Square Error from

222 .,.,. ABnSSandBnSSAnSS ABBA

)( ABBATE SSSSSSSSSS

2

1

22

1

4 ynySSn

iij

jT

k

2

2

2

ABn

Bn

An



43

Sum of Squares:Two Calculation Methods

The Total Sum of Squares,Effect Sum of Squares,and Mean Square Error are obtained from

0672.3)3882.14(44 2

1

22

1

2

1

22

1

n

iij

j

n

iij

jT yynySS

kk

2495.0)0040.00181.07956.2(0672.3

)(

E

ABBATE

SS

SSSSSSSSSS

0040.0032.04,0181.0)067.0(4,7956.20836.04 222 ABBA SSSSSS

0208.02

ˆ2

1

22

k

iki 2495.0

12

0208.0

)1(

ˆ 2

knSSE



44


There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample Data Sum of Squares based on Mean and Interaction Effects ANOVA/Significance Analysis of the Regression

Equation




45

Example 14.1 Regression Analysisand ANOVA (con’t)

7-6



46

ANOVA: for each term

The Significance of each term (A,B,AB) can be obtained from ANOVA

0040.0032.04,0181.0)067.0(4,7956.20836.04 222 ABBA SSSSSS

2495.012

0208.0

)1(

ˆ 2

knSSE

0208.0

2ˆ

2

1

22

k

iki

222 ˆ)1/(...ˆ)1/(....ˆ)1/( ABABBBAA SSfSSfSSf

A is SignificantB and AB are not Significant



47

Example 14.1 Regression Analysis and ANOVA(con’t)

U n i o n C o l l e g eM e c h a n i c a l E n g i n e e r i n g

M E R 3 0 1 : E n g i n e e r i n g R e l i a b i l i t yL e c t u r e 1 4

4 3

E x a m p le 1 4 .1 E x c e l R e g re s s i o n A n a ly s i s

Re gre s s i on S ta tis t ic sM u ltip le R 0.9 58 467 416R S qu are 0.9 18 659 787

A dju s te d R S q ua re 0.8 98 324 733S t and ard E rror 0.1 44 187 523O bs erva tio ns 1 6

A NO V Ad f S S M S F S ign if ic anc e F

R eg res s io n 3 2 .81 764 32 5 0 .93 92 14 45 .17 617 8.1 919 6E -07Re s id ual 1 2 0 .24 94 805 0.0 207 9

Tot al 1 5 3 .06 712 37 5

Co eff ic ie nts S ta nd ard E r ror t S ta t P -va lue Lo wer 95 %Int erc ept 1 4.3 888 75 0. 036 04 688 1 3 99 .17 12 4. 11E -26 14 .31 033 56

tim e 0. 418 0. 036 04 688 1 1 1.5 96 01 7. 08E -08 0.3 394 60 595% ars enic -0 .03 36 25 0. 036 04 688 1 -0.9 32 81 0. 369 306 -0.1 121 64 405inte rac tio n 0.0 157 5 0. 036 04 688 1 0 .43 69 31 0 .66 992 -0.0 627 89 405

7-6



48

Summary of Topics

Design of Engineering Experiments DOE and Engineering Design Coded Variables

Factorial Experiments Main Effects Interactions Optimization Statistical Analysis

Documents

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 1 MER301: Engineering Reliability LECTURE 14: Chapter 7: Design of Engineering