Ridge Regression Based Development of Acceleration Factors and Closed Form Life
Prediction Models for Leadfree Packaging
by
Dinesh Kumar Arunachalam
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
December 12, 2011
Keywords: Regression, Acceleration Factors, Life Prediction,
Electronic Packaging
Approved by
Pradeep Lall, Chair, Thomas Walter Professor of Mechanical Engineering
Jeffrey C Suhling, Quina Distinguished Professor of Mechanical Engineering
Peng Zeng, Associate Professor of Mathematics and Statistics
ii
Abstract
The thermomechanical mismatch caused by the difference in the coefficient of
thermal expansion between the electronic part and the printed circuit board results in
shear strains in the solder interconnects during thermal excursions. Widely used life
prediction models include the Manson [1966] and Coffin Model [1954, 1963] which
correlates plastic strain amplitude, Goldmann Model [1969] which correlates the
geometry and material parameters with cyclic life, ??p, with fatigue life, Norris
Landzberg`s Model [1969] which correlates the thermomechanical material and
geometry parameters with cyclic life. NorrisLandzberg acceleration factors for leadfree
solders have been developed based on ridge regression models (RR) and on PCR for
reliability prediction and part selection of areaarray packaging architectures under
thermomechanical loads. The principal component transformation has been used to rank
the new orthogonal principal components in the order of their importance. Scree plots,
Eigen values and proportion of total variance explained by each principal component are
then used to eliminate the least important principal components. Multiple linear
regressions have been performed with the original response variable and reduced set of
principal components. Ridge regression adds a small positive bias to the diagonal of the
covariance matrix to prevent high sensitivity to variables that are correlated. The
proposed procedure proves to be a better tool for prediction than multiplelinear
regression models. Models have been developed in conjunction with Stepwise Regression
iii
Methods for identification of the main effects. Package architectures studied include,
BGA packages mounted on coppercore and nocore printed circuit assemblies in harsh
environments. The models have been developed based on thermomechanical reliability
data acquired on coppercore and nocore assemblies in four different thermal cycling
conditions. Packages with Sn3Ag0.5Cu solder alloy interconnects have been examined.
The models have been developed based on perturbation of accelerated test thermo
mechanical failure data. Data has been gathered on nine different thermal cycle
conditions with SAC305 alloys. The thermal cycle conditions differ in temperature range,
dwell times, maximum temperature and minimum temperature to enable development of
constants needed for the life prediction and assessment of acceleration factors. Norris
Landzberg acceleration factors have been benchmarked against previously published
values. In addition, model predictions have been validated against validation datasets
which have not been used for model development. Convergence of statistical models with
experimental data has been demonstrated using a single factor design of experiment study
for individual factors including temperature cycle magnitude, relative coefficient of
thermal expansion, and diagonal length of the chip. Life prediction models have been
developed over the years trying to assess the influence of the different geometrical,
material and thermocycling parameters on the life of an electronic package. In this study
the influence of silver content on packages based on SAC alloys have been investigated.
Along with silver content, the solder ball configuration parameters such as ball pitch, ball
count, ball height, ball diameter and cycle conditions such as dwell time and delta T have
been considered .An assortment of packages such as CBGAs, PBGAs, flips chips based
iv
on a variety of SAC alloys with a set of different silver contents were considered for the
analysis.
v
Acknowledgments
I would like to express my sincere gratitude to my research advisor Dr. Pradeep
Lall for giving me the opportunity to work under his guidance and mentorship at NSF
Center for Advanced Vehicle and Extreme Environment Electronics (CAVE3) as a
Graduate Research Assistant at Auburn University. I would also like to thank my other
committee members Dr. Jeffrey Suhling and Dr. Peng Zeng for their support and
guidance while completing this thesis.
I would like to thank my parents Mr. Arunachalam M and Mrs. Shanthi A and my
sister Divya Senthil for having faith in me and providing endearing love, encouragement
and moral support. I would also like to thank all my friends and colleagues for their
priceless friendship and camaraderie all throughout my graduate studies.
vi
Table of Contents
Abstract ............................................................................................................................... ii
Acknowledgments............................................................................................................... v
List of Tables ................................................................................................................... viii
List of Figures .................................................................................................................... ix
Chapter 1 Introduction ........................................................................................................ 1
1.1 Overview .............................................................................................................. 1
1.2 Life Prediction ...................................................................................................... 1
1.3 Regression Analysis ............................................................................................. 3
1.4 Scope of Data ....................................................................................................... 6
Chapter 2 Literature Review ............................................................................................ 10
2.1 Statistical Prediction Models .............................................................................. 10
2.1.1 Ridge Regression ............................................................................................. 12
2.1.2 Principal Component Regression ..................................................................... 18
2.2 Physics of Failure Based Models ....................................................................... 22
2.3 Experimental Methods ....................................................................................... 24
Chapter 3 Extended Life Prediction Model ..................................................................... 26
3.1 Scope of Data ..................................................................................................... 26
3.2 Model Development........................................................................................... 27
3.3 Selection of Ridge Parameter ?K? ...................................................................... 29
vii
3.4 Model Validation ............................................................................................... 34
3.4.1 Dwell Time ...................................................................................................... 35
3.4.2 Percentage of Silver ......................................................................................... 35
3.4.3 Delta T ............................................................................................................. 36
3.4.4 Ball Diameter ................................................................................................... 37
3.5 Model Prediction ................................................................................................ 38
Chapter 4 Norris Landzberg Constants Developement ................................................... 40
4.1 Ridge Regression .............................................................................................. 42
4.2 Model Comparison............................................................................................ 46
4.3 Model Validation .............................................................................................. 46
4.3.1 Delta T ............................................................................................................. 48
4.3.2 Frequency Ratio ............................................................................................... 49
Chapter 5 Goldmann Constants Development ................................................................. 50
5.1 Principal Component Regression ...................................................................... 51
5.2 Ridge Regression .............................................................................................. 54
5.3 Model Comparison............................................................................................ 56
5.4 Model Validation .............................................................................................. 57
5.4.1 Model Validation (Parameters) ....................................................................... 58
5.4.2 Delta T ............................................................................................................ 60
5.5 Model Predictions ............................................................................................. 61
Chapter 6 Summary and Conclusions .............................................................................. 62
References ......................................................................................................................... 64
viii
List of Tables
Table 1: NLZ constants Comparison .................................................................................. 3
Table 2: Scope of Packaging Architectures ........................................................................ 8
Table 3 Pearson?s Correlation Matrix ............................................................................... 28
Table 4 MLR Results ....................................................................................................... 29
Table 5 RR Results ........................................................................................................... 33
Table 6 ANOVA ............................................................................................................... 34
Table 7 PCR Results (Transformed) ................................................................................ 41
Table 8 ANOVA for PCR NLZ ........................................................................................ 41
Table 9 PCR Results Transformed back ........................................................................... 42
Table 10 NLZ Results (Ridge) .......................................................................................... 45
Table 11 ANOVA NLZ (Ridge) ....................................................................................... 45
Table 12 Validation Dataset NLZ ..................................................................................... 47
Table 13 Transformed Z variable regression for Goldmann`s model of Cu Core ........... 52
Table 14 ANOVA Goldmann PCR................................................................................... 53
Table 15 Transforming Z back to Original Variables in the GLM for Copper Core........ 53
Table 16 Ridge Results (Goldmann Model) ..................................................................... 55
Table 17 ANOVA  Ridge (Goldmann)............................................................................ 56
Table 18 Model Validation Dataset (Goldmann).............................................................. 57
ix
List of Figures
Figure 1 Package Configurations ........................................................................................ 6
Figure 2 More representation of Scope of packages ........................................................... 7
Figure 3 Ridge Process ..................................................................................................... 17
Figure 4 Ridge Plot (based on VIF) .................................................................................. 31
Figure 5 Ridge Plot 2 (based on Coefficients) ................................................................ 32
Figure 6 Validation of Dwell Time ................................................................................... 35
Figure 7 Validation of Silver Content ............................................................................... 36
Figure 8 Validation of Delta T .......................................................................................... 37
Figure 9 Validation of Ball Diameter ............................................................................... 38
Figure 10 Model Prediction for Silver content model ...................................................... 39
Figure 11 Ridge Plot  VIF ............................................................................................... 43
Figure 12 Ridge Plot Regression Coefficients ................................................................ 44
Figure 13 Model Validation for NLZ (PCR Vs Ridge) .................................................... 47
Figure 14 Validation of DeltaT (NLZ) ............................................................................. 48
Figure 15 Validation of f Ratio (NLZ) ............................................................................. 49
Figure 16 Ridge Plot (coefficients) ................................................................................. 54
Figure 17 Ridge Plot (coefficients) ................................................................................. 55
Figure 18 Model Validation (PCR Vs Ridge)................................................................... 58
x
Figure 19 Effect of Die Length ......................................................................................... 59
Figure 20 Delta T Validation ............................................................................................ 60
Figure 21 Goldmann Prediction ........................................................................................ 61
1
Chapter 1
Introduction
1.1 Overview
The fast pace developments and improvements in the field of electronics demands
a faster method of determination of reliability and improvements in electronic packaging.
Institutional learning is a proven method of understanding the behavioral pattern of
electronics and hence knowing beforehand the mode of failure and the variables and
parameters responsible in contributing to failure. Understanding the behavioral pattern
helps one to be ready for what is going to happen when an electronic component is
deployed in a given operating condition. Improving reliability could be based on
changing or improving the parameter that is most contributing to the failure, given that
parameter can be learnt by understanding the behavior of the package to a given
condition with given geometrical and material configuration. Knowing the acceleration
factor for a thermomechanical process is critical in understanding the used and available
thermomechanical life of a package. Several Models to predict the Acceleration factors
of an electronic package have been proposed, proven and used over the years.
1.2 Life Prediction
The mismatch between the coefficient of thermal expansion between the chip and
the module due to the thermal cycling which the chip undergoes, results in shear strains
in the solder joint. Thus the mechanical strain along with the time and temperature factors
2
has to be taken into consideration while evaluating the fatigue behavior of solder
interconnections under accelerated conditions.
Manson and Coffin [1965, 1954] developed an equation that related plastic strain
??p, with number of cycles to failure N. Following Goldmann`s analysis and assuming
the interconnection to be a spherical segment, the plastic strain amplitude at any cross
section is given by the NorrisLandzberg`s Model [1969] for controlled collapse
interconnections.
Engelmaier [1990] developed a surface mount solder joint reliability prediction
model containing all the parameters influencing the shear fatigue life of a solder joint due
to shear displacement caused by thermal expansion mismatch between component and
substrate. Engelmaier developed separate equation for stiff solder joints and compliant
solder joints. The parameters of the model include effective solder joint area, solder joint
height, diagonal flexural stiffness, distance from neutral point and thermal coefficient
mismatch thermal cycling conditions, degree of completeness of stress relaxation and
slope of weibull distribution.
Knecht and Fox [1991] developed a strain based model using creep shear strain as
damage metric to determine the number of cycles to failure. The creep shear strain
included creep of component due to matrix creep alone ignoring the plastic work. The
equation was applicable to both 60Sn40Pb and 63Sn37Pb solder joints.
Vandevelde [1998] developed thermomechanical models for evaluating the
solder joint forces and stresses. Barker [2002] synthesized the Vandevelde models for
calculating the solder joint shear forces in ceramic and plastic ball grid array packages.
3
Clech [1996] developed a solder reliability solutions model for leadless and
leaded eutectic solder assemblies and extended it to area array and CSP packages. Clech
obtained the inelastic strain energy density from area of solder joint hysteresis loop and
developed a prediction equation correlating inelastic strain energy density with number of
cycles to failure.
Singh [2006] developed failure mechanics based models for solder joint life
prediction of ball array and flip chip packages. He calculated the maximum shear strain a
using a simplified DNP formula which was then used for initiating a hysteresis loop
iteration for both global and local thermal mismatch. Inelastic strain energy was then
calculated from the area of the hysteresis loop for both the thermal mismatch cases. The
number of cycles for failure was determined using Lall [2003] model.
Previously Hariharan [2006] demonstrated the power law dependencies of various
design parameters for flipchip, CBGA and CCGA packages using BoxTidwell
transformation. He compared the values obtained with those in the table.
Table 1: NLZ Constants Comparison
Variable
Model
Norris
Landzberg Goldmann
Delta T 2 2
Ball Height 2.7 2
Solder Volume 0.152 0.175
Solder ball radius 4 5.44
1.3 Regression Analysis
Regression is an effective life prediction tool that predicts the life of packages
based on historical behavior of the packages. Multiple linear regression attempts to model
4
the relationship between two or more explanatory variables and a response variable by
fitting a linear equation to observed data. Every value of the independent variable x is
associated with a value of the dependent variable y. The population regression line
for p explanatory variables x1, x2, ... , xp is defined to be y = 0 + 1x1 + 2x2 + ...
+ pxp. This line describes how the mean response y changes with the explanatory
variables. The observed values for y vary about their means y and are assumed to
have the same standard deviation . The fitted values b0, b1... bp estimate the
parameters 0, 1 ... p of the population regression line.
Since the observed values for y vary about their means y, the multiple
regression model includes a term for this variation. In words, the model is expressed as
DATA = FIT + RESIDUAL, where the "FIT" term represents the expression 0 +
1x1 + 2x2 + ... pxp. The "RESIDUAL" term represents the deviations of the observed
values y from their means y, which is normally distributed with mean 0 and variance
. The notation for the model deviations is . In common terms the matrix ?Y? is
called the response matrix and the matrix ?X? is called the predictor matrix.
The most predominant method of solving a regression problem uses the co
relation matrix X?X matrix and is given by the formula YXXX ')'( 1
^ ?
=? . This method is
good if and only if the determinant of the X?X matrix is nearly one. However the case
may not always be true as the determinant of the X?X matrix tends to move toward zero
if the columns of the X matrix are related to each other. In a lot of engineering
5
applications, some of the columns are derived functions of the other. In other words, the
factors that significantly contribute to the response can be derived functions of each
other. In such cases the regular solution to multiple linear regression, YXXX ')'( 1
^ ?
=?
would fail. There have been a lot of techniques developed over the years to circumvent
the resulting numerical snag. Since the determinant is approaching zero, the method fails
the coefficient tends to infinity losing the actual meaning and failing to explain the
actual significance of the variable.
The predominant methods are using Principal component Analysis and Ridge
Regression. The principal component method transforms the predictors into their
principal components and hence reduces the dimensions of the predictors and hence nulls
the effect of the corelation. The method is effective for curve fitting and for low
dimension data and is also useful for making life prediction models.
The other most commonly used technique is called Ridge Regression. The method
tries to circumvent the numerical snag by adding a small positive bias to the diagonal of
the matrix and hence avoiding the determinant to tend toward zero. The method visibly
reduces the mean square errors and is a lot flexible. The method is more of a prediction
tool than a curve fitting tool as we manually handle the flow of the process of regression.
Both methods have been used to re create the Norris Landzberg and Goldmann
Models for SAC alloys exposed to thermomechanical loading. The methods have been
validated against a validation data set to assess the prediction accuracy of the models and
their dependability.
Additionally the dependency of Life on the Silver content of the SAC alloy has
been investigated taking to account the other variables that are critical in deciding the life
6
of the electronic package. The other variables looked at include Ball Parameter,
Geometric parameters and the test conditions like Dwell time and Delta T. The results
have also been validated and checked for dependability.
1.4 Scope of Data
Accelerated test data is accumulated from the tests that conducted at the CAVE3
and from publications. The accumulated data is used to run the regression analysis. The
data accumulated is based on various package types and various material and geometric
configurations.
Figure 1 Package Configurations
7
Figure 2 More representation of Scope of packages
There is a wide scope of architecture made available through test conducted
previously at the CAVE3 and through publications; a small representation of the
available material is shown below.
8
Table 2: Scope of Packaging Architectures
Pa
ck
ag
e
typ
e
Ar
ray
typ
e
I/O
Pi
tch
(m
m)
I/O
Co
un
t
Ra
ng
e
So
lde
r
all
oy
Pa
ck
ag
e
siz
e
(m
m)
Di
e S
ize
Ra
ng
e
Pa
ck
ag
e
to
Di
e
siz
e r
ati
o
PBGA
Full
0.5 
1.00
49 ?
900
Pb
free
SAC
305
7.0 
31.0
4.00

24.0
1.00 
3.94
Perimeter
Mixed
FCPBGA Full
0.8
1.00
532 ?
1508
Pb
free
SAC
305
23.0 
40.0
Perimeter
MCMPBGA
Full
1.00 128 ? 324 22.0 Perimeter
HiTCE CBA
Full
1.27 360
Pb
free
SAC
305
25.0
CBGA
Full
1.27 483 29.0
CSP
Full
0.5 
0.8
132 ?
228
Pb
free
SAC
305
7.0 
12.0
Perimeter
Flip Chip
Full 0.25

0.45
48 ?
317
Pb
free
SAC
306
5.08 
6.35
Perimeter
Micro  Lead
Frame
0.40

0.65
44 ?
100
Pb
free
SAC
307
9.0 
12.0
Perimeter
QFP / LQFP
0.4 
0.5
100 ?
176
Pb
free
SAC
308
14.0 
20.0
Perimeter
9
The center has been publishing prediction models (Lall 04, 07, 08, 09) previously
based on regression techniques. Hence the behavioral properties of certain variables are
known for certain given conditions are known and hence it gives an approximate idea of
what to expect from the regression analysis in terms of positive or negative dependence.
The models were checked to see if they comply with the underlying physics of the
package.
10
Chapter 2
Literature Review
The rapid minimization of electronics and the fast paced change of technology
demand a faster method of determining the reliability of packages. Accelerated tests,
reliability models and life predictions are useful tools and hence making these tools more
accurate and reliable is critical. A reliability assessment numerical model that could take
into account the geometric and material properties of the widely operating conditions
could be of great help in obtaining the failure modes such as die cracking, solder joint
fatigue failure, delamination and total failure. Solder joint fatigue failure being a
dominant failure mode contributing 90% of all structural and electrical failures [Tummala
1997] demands greater focus for improving the mechanical reliability of the package. In
this section, traditional approaches for solder joint reliability prediction, including
physics of failure based models, statistical models, finite element models and
experimental techniques have been discussed. Elaboration of the two methods that are
mainly used, PCR and Ridge regression is also done.
2.1 Statistical Prediction Models
Statistical prediction models developed include cumulative failure distribution
functions for expressing the experimental failure data as a probability function of time to
failure for any failure distribution. Weibull distribution and Log normal distribution have
been most widely used failure distribution functions. Log normal distributions [Muncy
11
2004] have widely been used for modeling failure due to slow degradation such as
chemical reactions and other corrosions and weibull distributions have been used for
modeling failures due to weak link propagations such as solder joint failure.
Regression analysis and analysis of variance have been widely used by
researchers for correlating the reliability of a package with its geometic attributes,
material properties and operating conditions. Muncy [2004] conducted air to air thermal
cycling and liquid to liquid to liquid thermal shock tests on a flip chip package for 1200
test boards with four different die sizes, eight board configurations, two underfill
materials and two substrate metallization. The predictor variables considered for model
building include substrate metallization, substrate mask opening area versus the UBM
area of the flip chip bump, die size, perimeter or full area array flip chip interconnect
pattern, underfill material properties, location of the die on the test board, frequency of
cycling, number of interconnects, and percent area voiding. Multiple linear regression
modeling and regression with life data modeling methodologies were used for obtaining
the parameters of regression.
The variables of interest on a regression analysis for an electronic package are
often closely related to each despite the fact that they significantly contribute to the life of
the package by themselves. Multico linearity implies near linear dependence among
predictors. Multi co linearity can seriously disturb the least squares fit and in some cases
render the regression model almost useless. In some cases, the regression coefficients
can have the wrong sign or many of the predictors are not statistically significant when
the overall Ftest is highly significant. We make such intuitions from physical
significances and prior learning. Thus, when a model includes more than one predictor, it
12
is important to asses whether strong corelations exist among the predictors. Several
techniques have been proposed for detecting multico linearity. When the method of least
squares is applied to collinear data, very poor estimates of regression coefficients can be
obtained. The variance of the least square estimates of regression coefficient may
considerably inflate in the presence of near linear dependencies among predictors. This
implies that the least squares estimate of regression coefficient are very stable, that is,
their magnitude and signs may change considerably given a different sample.
The problem with the least squares estimation method is the requirement that ?g3552 be
an unbiased estimate of ?. Though ordinary least squares gives unbiased estimates and
indeed enjoy minimum variance of all linear unbiased estimates, there is no upper limit
bound on the variance of the estimates and the presence of multi co linearity may produce
large variances. As a result, one can visualize that, under the condition of multi co
linearity, a huge price is paid for unbiasedness property that one achieves by using
ordinary least squares. One way to alleviate this problem is to drop the requirement that
the estimate ? be unbiased. Biased estimation is used to attain a substantial reduction in
variance with accomplished increase in stability of the regression coefficients become
biased and , simply put, if one is successful, the reduction in variance is of greater
magnitude than he bias induced in the estimates.
2.1.1 Ridge regression
Consider the standard model for multiple linear regression, Y=X?+?, where X is
the matrix of predictors and Y is the matrix of the response. ? is the regression co
efficient matrix and is at this point unknown. The usual procedure of determining the
values is called the GaussMarkov linear functions.
13
Let B be the estimate of any vector ?. The residual sum of squares can be written
as,
YXXXYY
XYXYF
''2'''
)()'()(
???
????
?+=
??==
The estimate has been expressed as the difference between the observed and
fitted values. In an ideal case we want the value to be minimum; hence to find the
minimum, we differentiate the above equation and equate it to zero.
YXXX
YXXX
YXXXF
')'(
''
0'')(
1?=
=
=?=??
?
?
???
This method is good if and only if the determinant of the X?X matrix is nearly
one. However the case may not always be true as the determinant of the X?X matrix tends
to move toward zero if the columns of the X matrix are related to each other. In a lot of
engineering applications, some of the columns are derived functions of the other. In other
words, the factors that significantly contribute to the response can be derived functions of
each other. In such cases the regular solution to multiple linear regression,
YXXX ')'( 1
^ ?
=? would fail. There have been a lot of techniques developed over the
years to circumvent the resulting numerical snag. Since the determinant is approaching
zero, the method fails the coefficient tends to infinity losing the actual meaning and
failing to explain the actual significance of the variable.
The predominant methods are using Principal component Analysis and Ridge
Regression. The principal component method transforms the predictors into their
14
principal components and hence reduces the dimensions of the predictors and hence nulls
the effect of the corelation. The method is effective for curve fitting and for low
dimension data. It loses its accountability when the size of the dataset is large. It is more
a curve fitting tool than a prediction tool. Ridge regression on the other side uses a small
positive bias to keep the X?X from tending to infinity. The method is effective no matter
how big of a dataset we have.
For data where X matrix is not specified to be near co linearity, the dispersion is
expressed as,
12 )'()( ?= XXD ??
The trace of the dispersion matrix is the total variance, thus,
?
=
=
s
i i
TrD
1
2 1)(
??? (1)
Where the ?i are the nonzero eigen values of X?X,
It is clear that if one or more of the Eigen values are low, the variance inflation is
going to be high.
The ridge estimator as suggested by Hoerl and Kennard as a possible remedy for
the inflation is obtained by adding the scalar matrix kI to X?X in the least square
estimator. Thus the regression equation takes the form
YXkIXX ')'( 1
^ ?
+=?
Upon calculating the dispersion and variance for the above equation as we did
before, we get,
15
?
= +
=
s
i i
i
kTV 1 2
2^
)()( ?
???
It is clear that variance inflation will be lesser in the eqn (2) in the event if low
Eigen values.
The value of the ridge estimator ?k? was originally obtained by finding the point
on the ellipsoid centered at the LS estimator ?. The hyper ellipsoid is formed by the
residual sum of squares. The objective as we understand is to reduce the residual sum of
squares when there is inflation in the actual values. Let B be any estimate of the actual
vector ?. The residual sum of squares in that case will be.
)(
)(')'()()'(
)()'()(
min
^^^^
B
BXXBXYXY
XBYXBYBF
??
????
?
+=
??+??=
??==
As we understand, ? has inflated to B and hence ?(B) is the residual that has
added because of the inflation. Hence we try to reduce that term.
)(')'()(
^^ ???
??== BXXBFB
The ridge trace can be shown to be following a path through sum of the squares so
that for a fixed ? a single value of B is chosen and that is the one with the minimum
length.
Minimize B?B
Subject to 0
^^
)(')'( ??? =?? BXXB
Solving it using a lagrangian multiplier ?k?,
16
Minimize, ])'(')')[(1(' 0
^^ ???
???+= BXXBkBBF
Where (1/k) is the lagrangian multiplier,
0])'(2)'(2)[1(2
^
=?+= ??? XXBXXkBBF
Hence it reduces to
B= [X?X+kI]1X?Y [A.E Hoerl, 1970]
The value of ?k? is chosen such that k > 0 and then ?o is computed. In terms of ?*,
the residual sum of squares becomes,
??g4666kg4667 = g3435Y ? X?g3552?g3439g4593g3435Y ? X?g3552?g3439 = ?g2923g2919g2924 + kg2870?g3552?g4593g4666Xg4593Xg4667g2879g2869?g3552?
If the squared length of the regression vector B is fixed at R2, then ?g3552? is the value
of B that gives a minimum sum of squares. Hence ?g3552? is the value of B that minimizes the
function,
F= (YXB)?(YXB) + (1/k)(B?B R2)
The equation B= [X?X+kI]1X?Y is used to perform Ridge Regression, and as
seen it is just a modification of the original Multiple Linear Regression equation,
YXXX ')'( 1
^ ?
=? , except that we introduce a small positive bias term, ?k? which is
added to the diagonal of the variancecovariance portion of the Regression equation .In
essence, the original data is retained as we observe that there is no change to the second
part of the equation , rather the variance which is a derived property of the actual data
undergoes a small bias addition. The entire flow of the Ridge Regression process can be
explained by the flow chart shown in fig(3).
17
Figure 3 Ridge Process
The process is started like how a regular regression process is started by forming a
set of Predictor(X) and Response Variables (Y).An initial range of the Bias Parameters
?k? is chosen . The equation (5) is solved with the range of bias and in each attempt the
stability is looked for. The stability of the bias can be determined by a few methods
mentioned later. Care has to be taken to make sure that the model is neither overbiased
18
nor underbiased. A perfect bias will be at a point where both the Regression coefficients
and the VIF values remain stable or show minimal change upon further biasing. The
chosen k is taken and the results of the equation (5) for the chosen k value will be the
results of the Ridge Regression process.
The overall adequacy of the model is tested using ANOVA table. Small P value
of the ANOVA table rejects the null hypothesis proving the overall adequacy of the
model. Individual T tests on the coefficients of regression of variables should yield very
small P values indicating the statistical significance of all the predictor variables.
The individual T test values of variables are then used for conducting individual T
test on the coefficients of regression of original variables. The test statistic follows a
students? T distribution with (nk1) degrees of freedom. The P values of individual T
tests given by the ?p? values table are < 0.05 proving the statistical significance of
individual regression coefficients of original predictor variables at a 95 % confidence.
2.1.2 Principal Component Regression
Principal components models have been used for dealing with multicollinearity
and producing stable and meaningful estimates for regression coefficients [Fritts et al
1971]. Methodology for developing a Principal Component Regression Model is presented here:
Matrix Notation for the model:
}{}]{X[}y{ ?+?=
19
Where,
??
?
?
?
??
?
?
?
?
??
?
?
?
??
?
?
?
?
=
n
2
1
y
.
.
.
y
y
}y{ , sets_data_n
iablesvar_predictor_k
x...xx1
.......
.......
.......
x...xx1
x...xx1
X
knn2n1
2k2212
1k2111
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=
44444 844444 76
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=?
n
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.
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.}{
and
??
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.
.}{
Multicollinearity of predictor variables may cause large variance and covariance
of individual regression coefficients, high standard error of individual regression
coefficients in spite of high R2 values, instable regression models fluctuating in
magnitude and sign of regression coefficients for small changes in the specification, and
wider confidence intervals of regression coefficients. Previously the problem of multi
collinearity has been overcome by removing one of the variables which resulted in loss of
some influential parameters. The principal components technique determines a linear
transformation for transforming the set of X predictor variables into new set Z predictor
variables known as the principal components. The set of new Z variables are uncorrelated
20
with each other and together account for much of variation in X. The principal
components correspond to the principal axes of the ellipsoid formed by scatter of simple
points in the n dimensional space having X as a basis. The principal component
transformation is thus a rotation from the original x coordinate system to the system
defined by the principal axes of this ellipsoid. The principal component transformation is
used to rank the new orthogonal principal components in the order of their importance.
Multiple linear regression is then performed with the original response variable
and reduced set of principal components. The principal components estimators are then
transformed back to original predictor variables using the same linear transformation.
Since the ordinary least square method is used on principal components, which are pair
wise independent, the new set of predictor coefficients are more reliable. The Pearson?s
Corelation matrix is calculated to check for the multicolinearity in the matrix X. And
the Eigen values are used in transforming the original predictor variables in the new Z
variables. Scree plots, Eigen values and proportion of total variance explained by each
principal component are then used to eliminate the least important principal components.
The Equation for calculation of the Eigen values and the Eigen vector is:
]V])[I[]C([ ??
0]I[]X[]X[ *T* =??
Where ? is the eigen value and V is the eigen vector matrix. The original set of
predictors has been transformed (matrix A) to a new set of predictor variables (matrix Z)
called the principal components. The principal component matrix Z contains exactly the
same information as the original centered and scaled matrix A, except that the data are
arranged into a set of new variables which are completely uncorrelated with one another
21
and which can be ordered or ranked with respect to the magnitude of their Eigen values
(Draper and Smith 1981, Myers 1986).
jZ =[
*
1x
*
2x ??..
*
3x ]
321
j_with_associated_Vector_Eigen
kj
j2
j1
V
.
.
.
V
V
?
??
?
?
?
??
?
?
?
?
??
?
?
?
??
?
?
?
?
MLR has been performed with the transformed predictor variables and the
original response variable. The coefficients obtained as a result of this regression model
are stored in a variable named alpha. Matrix notation for the same is given as:
1*k
*k*kT
1*k }{]V[}{ ?=?
The Principal Components have been transformed back to the Original variables.
To eliminate the principal components the coefficients are transformed back to the
original ones by using the reverse transformation.
1*kk*k1*k }{]V[}{ ?=?
The individual T test values of principal components regression components are
then used for conducting individual T test on the coefficients of regression of original
variables. The test statistic proposed by Mansfield et al.[1997] and Gunst et al. [1980] for
obtaining the significance of coefficients of regression of original variables is given in the
equation below:
22
2
1
1
21
,
??
?
??
? ?
?
??
?
??
=
?
=
?
l
m
jmm
pcj
vMSE
bt
?
Where bj,pc is the coefficient of regression of the jth principal component, MSE is
the mean square error of the regression model with l principal components as its predictor
variables, vjm is the jth element of the Eigen vector vm and ?m is its corresponding Eigen
value. M takes the values from 1 to l, where l is the number of principal components in
the model. The test statistic follows a students? T distribution with (nk1) degrees of
freedom.
2.2 Physics of Failure Based Models
Manson and Coffin [1965, 1954] developed an equation that related plastic strain
??p, with number of cycles to failure N. Goldmann [1969] analyzed a controlled collapse
joint with spherical dimensions for developing an equation that related the plastic strain
of a joint with relative thermal expansion coefficients of chip to substrate, distance from
chip neutral point to substrate, height of the solder, volume of solder, radius of the cross
section under consideration and exponent from plastic shear stress strain relationship. The
plastic strain obtained from Goldmann formulation can be substituted in Coffin Manson
equation for predicting the number of cycles for fatigue failure. Norris and Landzberg
[1969] studied the effect of cycling frequency and maximum temperature of cycling on
fatigue failure of solder joints and added an empirical correction factor for time
dependent and temperature dependent effects for the thermal fatigue model.
Solomon [1986] analyzed the fatigue failure of 60Sn/40Pb solder for various
temperatures and developed an isothermal low cycle fatigue equation that correlated the
23
number of cycles to failure with applied shear strain range. Solomon also studied the
influence of frequency, and temperature changes and added corrections that account for
temperature changes, cycling wave shape and joint geometries.
Engelmaier [1990] developed a surface mount solder joint reliability prediction
model containing all the parameters influencing the shear fatigue life of a solder joint due
to shear displacement caused by thermal expansion mismatch between component and
substrate. Engelmaier developed separate equation for stiff solder joints and compliant
solder joints. The parameters of the model include effective solder joint area, solder joint
height, diagonal flexural stiffness, distance from neutral point and thermal coefficient
mismatch thermal cycling conditions, degree of completeness of stress relaxation and
slope of weibull distribution.
Knecht and Fox [1991] developed a strain based model using creep shear strain as
damage metric to determine the number of cycles to failure. The creep shear strain
included creep of component due to matrix creep alone ignoring the plastic work. The
equation was applicable to both 60Sn40Pb and 63Sn37Pb solder joints.
Vandevelde [1998] developed thermomechanical models for evaluating the
solder joint forces and stresses. Barker et al [2002] synthesized the Vandevelde models
for calculating the solder joint shear forces in ceramic and plastic ball grid array
packages. Clech [1996] developed a solder reliability solutions model for leadless and
leaded eutectic solder assemblies and extended it to area array and CSP packages. Clech
obtained the inelastic strain energy density from area of solder joint hysteresis loop and
developed a prediction equation correlating inelastic strain energy density with number of
cycles to failure.
24
2.3 Experimental Methods
Temperature cycling is a widely method for solder joint reliability predictions. In
this method the component is exposed to a series of low and high temperatures
accelerating the failure modes caused by cyclic stresses. The thermal cycling uses a
single air chamber in which the temperature ramp can be controlled carefully. Thermal
shock tests like thermal cycling are used for accelerated life testing of solder joints.
Thermal shock testing is a liquidliquid test in which two liquid chambers at different
temperatures are used. Thermal shock tests generate very high ramp rates.
Master, et al. [1998] conducted accelerated thermal cycling tests on CBGA
packages for various body size and assembly parameters to study the effect of package
thickness and card pad design on reliability of the package. Master, et al. [1995] studied
the effect of column length on fatigue life of solder joint for two different thermal
profiles using accelerated thermal cycling tests. Gerke, et al. [1995] studied the reliability
of high I/O CBGA packages used in computer environment using accelerated thermal
cycling tests for two different thermal profiles. Kang [2004] evaluated the thermal fatigue
life and failure mechanism of SnAgCu solder joints with reduced Ag contents for a
CBGA package. Hong [1998] predicted the mean fatigue life of CBGA packages with
lead free (Sb5Sn95, Ag3.5Sn96.5, Zn9Sn91) solder fillets and found the lead free
joints outperform the leaded ones. Ingallas [1998] conducted accelerated thermal cycling
tests on CCGA packages for two different ball pitch, to study the effect of solder ball
pitch on solder joint reliability of the package .Zhang, et al. [2001] evaluated the
reliability of SnCu0.7, SnAg3.8Cu0.7 and SnAg3.5 solder joints on both NiP and Cu
under bump metallurgies for flipchip application. Peng, et al. [2004] analyzed the
25
sensitivity of reliability of flip chip package to solder joint geometric parameters such as
standoff height, lower and upper contact angles, and solder joint profile using
accelerated thermal cycling tests. Wang, et al. [2001] assessed the reliability of flip chip
packages with no flow underfills using liquid to liquid thermal shock tests. Hou, et al.
[2001] conducted liquid to liquid thermal shock tests for reliability assessment of flip
chip packages with SnAgAu joints. He found the leaded solder joints perform better than
the lead free ones. Teo, et al. [2000] conduated accelerated thermal cycling tests for
investigating the effect of under bump metallurgy solder joint reliability of flip chip
packages. Braun, et al. [2005] studied the high temperature potential of flip chip
assemblies for automotive applications. Darveaux, et al [2000] studied the impact of
design and material choice on solder joint fatigue life of various BGA packages including
PBGA, FlexBGA, tape array BGA and mBGA.
Moire interferometry is an optical method which provides whole field contour
maps of inplane displacements with sensitivity as low as 0.417?m [Tunga 2004]. Moire
Interferometry technique has been increasing employed in mapping thermally induced
deformation of electronic packages. Meng [1997] applied this technique for solder joint
reliability prediction of BGA, CSP and flip chip packages. He subjected the packages to a
temperature cycling and extracted the accumulated thermal deformations for reliability
predictions. Zhu [1997] studied the reliability of OMPAC BGA and a flip chip BGA
using moir? interferometry technique. Zhu also studied the effect of bonding,
encapsulation, soldering and geometry on the reliability of both the packages and using
the same technique.
26
Chapter 3
Extended Life Prediction Model
Life prediction models have been developed over the years trying to assess the
influence of the different geometrical, material and thermocycling parameters on the life
of an electronic package. In this study the influence of silver content on packages based
on SAC alloys have been investigated.
3.1 Scope of Data:
The dataset includes accelerated test reliability data for a variety of packaging
architectures including, of plastic ballgrid array (PBGA), flipchip ballgrid array (FC
BGA), chiparray ballgrid array (CABGA) devices, MCMPBGA, HiCTE ball grid
array, FlipChip, ceramic ballgrid array (CBGA), metal lead frame (MLF), and quadflat
packs (QFP). The electronic components have been assembled on copper core and no
core printed circuit boards. Data gathered on test assemblies from nine temperature cycle
conditions including, TC1 (40 to 95oC, 30 min dwell), TC2 (55 to 125oC, 30 min
dwell), TC3 (3 to 100oC, 30 min dwell), TC4 (20 to 60oC, 30 min dwell), TC5 (20 to
80oC, 30 min dwell), TC6 (0 to 100oC, 15 min dwell), TC7 (0 to 100oC, 10 min dwell),
TC8 (55 to 125oC, 15 min dwell), TC9 (40 to 125oC, 15 min dwell). The chambers
were profiled with fullload, and the temperatures measured at various locations on the
test boards in the stack, to ensure that the packages experience uniform temperature
27
exposure. All packages are daisychained and the resistance monitored to identify
failures.
3.2 Model Development
Along with silver content, the solder ball configuration parameters such as ball
pitch, ball count, ball height, ball diameter and cycle conditions such as dwell time and
delta T have been considered .An assortment of packages such as CBGAs, PBGAs, flips
chips based on a variety of SAC alloys with a set of different silver contents were
considered for the analysis. Initial model diagnosis revealed that the data set had
correlated variables and hence was causing Variance inflation.
From the table below, it is clear that there are a few variables with high co
relation coefficients and hence the regression will yield high variance inflation and in
accurate regression coefficients. Below is the table of Pearson?s correlation coefficient,
the correlated variables have coefficient values of more than 0.5.
28
Table 3 Pearson?s Correlation Matrix
Die
Length
Ball
Count
Ball
pitch
Ball
Dia Ball H
% of
AG
Dwell
Time Delta T
Die
Length 1.00 0.86 0.61 0.19 0.42 0.55 0.51 0.66
Ball
Count 0.86 1.00 0.53 0.09 0.43 0.69 0.37 0.62
Ball
pitch 0.61 0.53 1.00 0.03 0.88 0.75 0.37 0.60
Ball Dia 0.19 0.09 0.03 1.00 0.03 0.11 0.05 0.07
Ball H 0.42 0.43 0.88 0.03 1.00 0.64 0.38 0.49
% AG 0.55 0.69 0.75 0.11 0.64 1.00 0.12 0.50
Dwell
Time 0.51 0.37 0.37 0.05 0.38 0.12 1.00 0.50
Del T 0.66 0.62 0.60 0.07 0.49 0.50 0.50 1.00
29
3.3 Selection of Ridge Parameter ?K?:
The regular multiple linear regression model with the above data will yield a
model with inaccuracies and high variance inflations. Below is the result of the multiple
linear regression performed on the above data.
Table 4 MLR Results
Variable Estimate Error T P VIF
Intercept 24.51 1.77 13.84 0.00 0.00
die_length 0.83 0.22 3.71 0.00 9.63
ball_count 0.29 0.17 1.72 0.09 8.51
ball_dia 1.13 0.46 2.46 0.02 12.65
ball_h 1.37 0.24 5.62 0.00 1.46
ball_pitch 0.41 0.21 1.93 0.06 6.81
Ag 0.77 0.18 4.25 0.00 5.02
dwell_time 0.27 0.09 3.04 0.00 1.85
delta_T 2.71 0.29 9.46 0.00 2.29
30
It can be seen that despite the fact that the VIF is high for a few variables, the
corresponding ?p? values a really low explaining the importance of the variable to the
model and so those variables cannot be dropped off.
Ridge regression is applied to the problem to circumvent the numerical snag. A
small positive bias k is introduced to try and reduce the variance and hence the MSE.
Different bias values from 0 to 0.1 in increments of 0.001 are tried to see if the regression
coefficients and the variance stabilize. In case no stability is observed, the upper limit of
the bias is increased until a range which records the stability of the parameters is reached
at. The biased models are made by using the k value in the equation (5).The process is
carried out as mentioned in fig (1).The regression coefficients and the VIF are recorded
in each case. These values are observed to see if there is any stability. Stabilization of
ridge parameter can be determined in a lot of ways. In most cases the requirements of the
model play a vital role. Since the objective is prediction we have to make sure the model
chosen to be with stable bias has good prediction accuracy and complied with the
physical interpretations of the case .The most common method is observing the ?ridge
plot?. The ridge plot is plot of the bias parameter on the Xaxis against parameters like
?VIF? or the ? coefficients themselves. An ideal bias parameter is determined based on
where the plots seem to stabilize. In some cases the VIF and coefficients stabilize at
different points. In those cases we try to choose the bias in such a way the model is
neither ?overbiased? nor ?underbiased?. The tradeoff can be based on prediction
accuracy of the model in each case. Below is a ridge plot of the model for assessing the
influence of silver content based on the parameters mentioned above.
31
Figure 4 Ridge Plot (based on VIF)
0
2
4
6
8
10
12
14
0 0.02 0.04 0.06 0.08 0.1
VI
F
Bias parameter 'k'
DieLengthMM BallCount BallDiaMM BallHgtMM
BallPitchMM PercAg DwellTimMIN DeltaT
32
Figure 5 Ridge Plot 2 (based on Coefficients)
3
2.5
2
1.5
1
0.5
0
0.5
1
1.5
2
0 0.02 0.04 0.06 0.08 0.1
Co
eff
ici
en
ts
Bias parameter 'k'
DieLengthMM BallCount BallDiaMM BallHgtMM
BallPitchMM PercAg DwellTimMIN DeltaT
33
In the initial attempt to regress the data, it was seen that a few variables had high
variance inflations. These variables are seen to stabilize with increase in bias parameter.
A bias at which a stability of both the coefficients and the VIF is seen is chosen to the
biasing parameter. In the above case, it is seen that the VIF of all the variables attain
stability at about 0.06 and after. Upon closely observing the second plot, we see that all
the variables stabilize at about 0.076. Hence the bias parameter is chosen to be 0.076. The
model corresponding to k=0.076 is given below:
Table 5 RR Results
parameter Constant Die Length Ball Count Ball Pitch
? 21.151 0.788 0.183 0.509
Ball Dia Ball Height % of Ag Dwell Time Delta T
1.2646 0.2818 0.5358 0.258 2.11
Hence we have,
)(11.2)(2585.0)(%5358.0)(2818.0)(2646.1
)(50908.0)(18294.0)(788.0151.21)_(
TLnTLnofAgLnHLnDLn
PLnCLnLLnlifecharLn
DBB
BBD
???+++
???=
(6)
The model reduces to,
11.22585.053.028.0264.1
50908.018294.0788.0
)()()(%)()(
)()()(151.21
??
???
?
=
TTofAgHD
PCLCharLife
DBB
BBD
(7)
34
Where, LD is the Die Length, CB is the ball count, PB is the ball pitch, DB is the
ball diameter, HB is ball height, TD is the dwell time and ?T is the temperature cycle
range.
As an inference, we have, die length, ball count, ball pitch, dwell time and Delta T
have negative dependencies on life and parameters like ball height, % of AG and ball
diameter have positive influence on life. The inferences are further assessed and
confirmed when the model is validated. An Analysis of variance is performed to verify
that the results of the Regression process are significant.
ANOVA
Table 6 ANOVA
Source DF
Sum of
squares
Mean
Square F P
Model 8 64.8702 8.1088 34.2 0.00010
Error 83 19.68 0.2371
Total 91 84.5504
3.4 Model Validation:
The Validation is done using data outside of the data set used for regression. A
few data are set aside from the original dataset for the purpose of validation. Since the
objective of the regression is prediction, a validation done by predicting the life of
packages that were not included in the regression will give a better validation the model.
A plot with the predicted and actual life of package would give an idea of the reliability
of the model.
35
3.4.1 Dwell Time:
Dwell time is a critical contributor to the life of the solder ball and as the model
suggests, the increase in dwell time decreases life. The plot below reconfirms the same.
Figure 6 Validation of Dwell Time
3.4.2 Percentage of Silver:
The model suggests that the increase in the silver content increases life. Two
similar packages with the only difference being silver content (2.1 and 2.3) are
considered and the plot shows that there is a increase in life both actual and predicted
with increase in Silver content.
0
200
400
600
800
1000
1200
1400
1600
20 40
N1
%
Li
fe
Dwell Time(min)
Dwell Time
Ridge
Actual
36
Figure 7 Validation of Silver Content
3.4.3 Delta T:
Temperature difference is the most significant parameter in thermocycling. The
model predicts square negative influence on life. The plot below shows that there is a
significant decrease in life with increase in Delta T.
0
200
400
600
800
1000
1200
1400
2.1 2.3
N1
%
Li
fe
PercAg
% of Silver
37
Figure 8 Validation of Delta T
3.4.4 Ball Diameter:
The solder ball diameter has a positive coefficient and the plot shows increase in
life with increase in Diameter. Hence we infer that considering two packages with similar
material and Geometric properties except for the ball diameter, we can say that the
package with the higher ball diameter will have better life amongst the two.
0
200
400
600
800
1000
1200
100 165
N1
%
Li
fe
Delta T
Delta T
Ridge
Actual
38
Figure 9 Validation of Ball Diameter
3.4 Model Prediction:
The model is used to predict life and the model predictions are plotted against the
actual life to see the accuracy of the model. The dotted lines represent the 90% interval.
As seen from the plot, most of the model predictions fall in the interval.
0
500
1000
1500
2000
2500
1.40 1.52
N1
%
Li
fe
Ball Diameter(mm)
Ball Diameter
Ridge
Actual
39
Figure 10 Model Prediction for Silver content model
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000 1000 3000 5000 7000 9000
P
r
e
d
i
c
t
e
d
Actual
40
Chapter 4
Norris Landzberg Constants Development
The NorrisLandzberg Equation is based on the Coffin Mansion Equation and the
Goldmann Equation. It provides a way of calculating the acceleration factor for
Controlled Collapse Interconnections [Norris 1969]. The equation is given by:
Where,
AF is the Acceleration factor.
NU and NA are the lives of the packages fU and fA are the frequencies
TA and TU are the temperature excursions
Tmax is the maximum temperature of the cycle in Kelvin
This Equation is often used in the form [Lau 1997]
The Equation can be transformed by computing the natural Log format as follows:
( )max
231
TTTffNNAF
U
A
A
U
A
U ???
?
?
???
?
?
?
???
?
???
?==
???
?
???
?
???
?
???
? ?
???
?
???
?
?
?
???
?
???
?==
AUU
A
A
U
A
u
TTT
T
f
f
N
NAF
max,max,
231 11
1414exp
( ) ??
?
?
???
? ?+
???
?
???
?
?
?+
???
?
???
?=
AUU
A
A
U
TTCT
TLnC
f
fLnCAFLn
max,max,
21
113
41
Now we model the above equation into a regression model with ratio of cyclic
frequencies, Temperature cycle magnitude and the difference of inverse of maximum
temperatures as the independent variables and the Acceleration factor as the response
variable.
Due to the presence of Multicolinearity Principal Component Regression is
implemented. Regression of the transformed Principal Components against the
Acceleration Factor is given in the table below:
Table 7 PCR Results (Transformed)
Predictor Coef SE Coef T P
Constant 0.7448 0.1161 6.4123 0
Z1 3589.0768 1354.5949 2.6496 0.0095
Z2 285.8296 107.7056 2.6538 0.0094
Z3 2802.1627 1057.2824 2.6503 0.0095
The ANOVA table is used to check the presence of a linear relationship between
the predictor variables and the response variables.
Table 8 ANOVA for PCR NLZ
Source DF SS MS F P
Regression 3 2.136 0.712 5.82 0.001
Residual Error 90 11.0016 0.1222
Total 93 13.1375
To get the relationship between the original variables and the response variable,
we need to back transform the Principal Components using the same back transformation.
Regression results for the same are:
42
Table 9 PCR Results Transformed back
Predictor Coef SE Coef T P
Constant 0.7448 0.1161 6.4123 0
Ln(Fu/Fa) 0.3035 0.1145 2.6496 0.0095
Ln(Delta Tu / Delta Ta) 2.3149 0.8722 2.6538 0.0094
(1/Tu1/Ta) 4562.3767 1721.45 2.6503 0.0095
The regression equation is given by:
???
?
???
? ?+
???
?
???
?+
???
?
???
?+=
AUA
U
A
U
TTLnTDelta
TDeltaLn
F
FLnAF 11*3767.4562
_
_*3149.2*3035.07448.0
Writing the equation in the form of the NL equation:
The PCR model that is shown above has been shown to predict the acceleration
factor of PBGAs with SAC305 alloys with good prediction accuracy. The same three
variables are taken for regression and with the same dataset using ridge regression.
4.1 Ridge Regression
An initial attempt to model the case using Multiple Linear regression was made
and the model was found to have very high values for regression coefficient and high
MSE and VIF due to a high variance. y'X)X'X(? 1?=? . A biasing parameter k was
introduced to the least squares formula and a significant reduction of variance and a more
meaningful coefficient matrix was seen. The value of ?k? was varied with small
increments to choose the best value of it.
???
?
???
?
???
?
???
? ?
???
?
???
?
?
?
???
?
???
?==
AUU
A
A
U
A
u
TTT
T
f
f
N
NAF
max,max,
31.23.0 11
4562exp
43
Figure 11 Ridge Plot  VIF
0
500
1000
1500
2000
2500
0 0.001 0.002 0.003 0.004
VI
F
Bias parameter 'k'
fRatio DeltaTratio Tmax
44
Figure 12 Ridge Plot Regression Coefficients
At one point of the biasing parameter, the ridge plots become stable and the VIF
also stabilizes. That model is checked to see if it fits accurately. A trial with k values
between 0 and 0. 003 with increments of 0.0001 is done and in each case, the k is
plugged into the equation ?+=? ? )X'X()kIX'X(? 1R and the corresponding regression co
18
16
14
12
10
8
6
4
2
0
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035
Coe
ffi
cie
nts
Bias parameter 'k'
fRatio DeltaTratio
30000
25000
20000
15000
10000
5000
0
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035
Co
ef
fec
ien
ts
Bias parameter 'k'
NLZ Ridge Plot
tmax
45
efficients are calculated. In each case, the corresponding variation inflation factor is
calculated. With each trial of the regression and increment of ?k?, the stability of the VIF
and the regression coefficients are closely watched for. If the case does not stabilize
within the given range of values for k, the range is increased and the same procedure is
followed to move toward a perfect biasing factor. The above graph, also called the ?ridge
trace? plots the stabilization of the regression coefficients with the increase in bias.
Based on the three plots, it is seen that all of them stabilize just about when k =0.0017.
Taking the regression coeffiecients at k=0.0017, the Norris Landzberg equation,
Table 10 NLZ Results (Ridge)
Constant F Ratio Delta T
Ratio
T Max
0.74 0.201 2.086 3802
???
?
???
?
???
?
???
? ?
???
?
???
?
?
?
???
?
???
?==
AUU
A
A
U
A
u
TTT
T
f
f
N
NAF
max,max,
086.2201.0 11
3802exp
(11)
ANOVA Table
Table 11 ANOVA NLZ (Ridge)
Source DF
Sum of
squares
Mean
Square F P
Model 3 1.313 0.437 5.36 0.002
Error 84 6.856 0.08162
Total 87 8.1698
4.2 Model Comparison
4.3 Model Validation
The Validation is done using data outside of the data set used for regression. A
few data are set aside from the original dataset for the purpose of validation. Since t
objective of the regression is prediction, a validation done by predicting the life of
packages that were not included in the regression will give a better validation the model.
A plot with the predicted and actual life of package would give an idea of
of the model.
46
he
the reliability
47
Table 12 Validation dataset NLZ
Package No. 1 2 3 4 5
Frequencies 0.01667 0.01667 0.01667 0.01667 0.03333
0.01667 0.0333 0.01667 0.01667 0.01667
Delta T 135 135 165 135 165
180 165 180 180 180
Tmax 368 368 368 368 398
398 398 398 398 398
Figure 13 Model Validation for NLZ (PCR Vs Ridge)
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5
Ac
cel
era
tio
n F
ac
tor
Package
PCR Vs Ridge
Actual
PCR
Ridge
48
4.3.1 Delta T
The temperature excursion or DeltaT is as known the most critical parameter in
thermocycling. The model predicts that the increase in Delta T decreases the
Acceleration factor. The plot shows the same trend.
Figure 14 Validation of DeltaT (NLZ)
0
0.5
1
1.5
2
0.82 0.92
DeltaT
Actual
Ridge
49
4.3.2 Frequency Ratio:
The increase in frequency ratio decreases the acceleration factor as explained by
the plot.
Figure 15 Validation of f Ratio (NLZ)
0
0.5
1
1.5
2
2.5
0.50 1
AF
f Ratio
f Ratio
Actual
Ridge
50
Chapter 5
Goldmann Constants Development
The Goldmann Model [1969] relates the coefficients of thermal expansion,
distance from chip neutral point to interconnections, temperature excursion of the cycle,
volume of the solder, radius and height of the solder ball, with the thermomechanical
reliability of components. The Goldmann Model [1969] has the form,
Where Nf is the number of cycles to failure, KT is the Constant in applied Coffin
Manson Equation, ?u is the ultimate shear strength of the critical joint interface, i.e., chip
pad interface or landsubstrate interface, rf is the radius of critical interface, h is the
solder joint height, A is the constant in stressstrain relationship ? = A??, V is the solder
volume of joint, ? is the shear deformation of joint, d is the distance of the solder joint
from neutral point of the chip, ?rel is the relative coefficient of thermal expansion, ?T is
change in temperature. The equation has been rearranged as follows,
51
Since KT, ?u and A are material and damage constants, these have been combined
into one constant, C. The equation has been modified as follows,
Substituting m = 1.9, ? = 0.58 into the equation,
The equation has been transformed by computing a natural logarithm of equation.
lng3435g1840g3033g3439 = lng4666g1829g4667 + g4672g3040g3081 g4673 + g1865. lng4666?g4667 + g1865. lng4666g1856g4667 + g1865. lng4666g2009g3045g3032g3039g4667 + g1865. lng4666?g1846g4667
5.1 Principal Component Regression
The critical parameters like Difference in coefficients of thermal expansion,
Distance from chip neutral point to interconnections, Temperature excursion of the cycle,
Volume of the solder, radius and height of the solder ball, are included in the Goldmann
equation. Using these parameters as predictor variables, we model the Goldmann`s
Equation in the form of a log transformed Principal Component Regression model for
PBGAs assembled on Copper Core BGAs
52
The principal components matrix Z is obtained using the transformation:
]V[*]X[]Z[ =
MLR is performed with the transformed predictor variables and the original
response variable. The coefficients obtained as a result of this regression model are stored
in a variable named alpha. Matrix notation for the same is given as:
1*k
*k*kT
1*k }{]V[}{ ?=?
Regressing the transformed Z variables against the N1% life of the packages, we
get the following results.
Table 13 Transformed Z variable regression for Goldmann`s model of Cu Core Assemblies
Predictor Coef SE Coef T P
Constant 2.651 4.014 0.66 0.511
Z1 0.8412 0.2914 2.89 0.005
Z2 1.3919 0.1823 7.64 0
Z3 1.3075 0.1682 7.77 0
Z4 0.4962 0.1579 3.14 0.002
Z5 0.1626 0.1114 1.46 0.148
The overall adequacy of the model has been tested using ANOVA table. Small P
value of the ANOVA table rejects the null hypothesis proving the overall adequacy of the
model. Individual T tests on the coefficients of regression of principal components
yielded very small P values indicating the statistical significance of all the five variables
53
ANOVA
Table 14 ANOVA Goldmann PCR
Table 15 Transforming Z back to Original Variables in the Goldmann`s Model for Copper Core
Predictor Coef
SE
Coef T P
Constant 2.651 4.014 0.66 0.511
IntTerm 0.0495 0.0171 2.89 0.005
Ln(BallhgtMM) 0.4121 0.054 7.64 0
Ln(HalfDLengthMM) 0.3705 0.0476 7.77 0
Ln(CTEppmC) 1.3721 0.4369 3.14 0.002
Ln(DeltaT) 1.56 1.068 1.46 0.148
We write the model in equation format to compare the values of constants
obtained from the PCR model with standard values for Cu Core Assemblies. Following
are the two models:
Goldmann`s Model:
( ) Ch1hrV)T()L(N
152.0
32.1
152.0
2
222
rel =??
?
?
???
?
???????????????
??
???
Source DF
Sum of
squares
Mean
Square F P
Model 5 11.5452 2.3090 15.08 0.0001
Error 98 15.003 0.1530
Total 103 26.5482
54
Statistical form based on PCR for Goldmann`s Model:
5.2 Ridge Regression:
The initial bias range was set to be 0 and 0.1 with intervals of 0.001. The stability
of bias is looked for in the VIF and regression coefficient ridge plot.
Figure 16 Ridge plot (coefficients)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.02 0.04 0.06 0.08 0.1 0.12
Co
ef
fic
ien
ts
Bias parameter 'k'
Goldmann Ridge Plot
Int Term BallHgtMM HalfDLengthMM CTEppmC DeltaT
?
?
?
?
?
?
?
?
???
?
?
???
?= ??? 56.13721.13705.04121.00495.02 Tdh
V
hrCN
rel
f
f ?
?
55
Stabilization of VIF
Figure 17 Ridge plot (coefficients)
Upon close examination of the results and of the plot, stability is attained at
different point for different variables but at k =0.062, all the variables are stable. The
results at this particular bias factor are taken to be best bias. The results are as follows:
Table 16 Ridge Results (Goldmann Model)
Predictor C Int
term
Ln
(BallHgtMM)
Ln(Half
DLengthMM)
Ln
(CTEppmC)
Ln(DeltaT)
Coeff 1.64 0.050 0.402 0.31 1.09 1.584
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.02 0.04 0.06 0.08 0.1 0.12
VI
F
Bias Parameter 'k'
Goldmann Ridge Plot
Int Term BallHgtMM HalfDLengthMM CTEppmC DeltaT
ANOVA
Upon plugging in the above values on to equation and taking anti
5.3 Model Comparison
The above equations
created using accelerated data of SAC alloy using two different methods and show close
resemblance. Model validation would reveal which model would work better.
Source
Model
Error
Total
?
?
?
?
= ?? 3174.009.1%1 dCN rel?
56
Table 17 ANOVA  Ridge (Goldmann)
log, we end up with,
are in close comparison. The equations 15 and 16 were
DF
Sum of
squares
Mean
Square F P
5 11.5452 2.3090 15.08 0.0001
98 15.003 0.1530
103 26.5482
?
?
?
?
???
?
???
?? ? 4.005.0258.1 h
V
hrT f?
57
5.4 Model Validation
The Validation is done using data outside of the data set used for regression. A
plot with the predicted and actual life of package would give an idea of the reliability of
the model.
Table 18 Model Validation Dataset (Goldmann)
Package
No
Ball
Height
Diagonal
Length Diff in CTE ?T
1 3.31 0.36 3.95 3.50E06 135
2 5.52 0.36 3.95 3.50E06 180
3 2.12 0.19 3.10 5.00E06 180
4 5.52 0.36 3.95 3.50E06 135
5 2.12 0.19 3.10 5.00E06 135
???
?
???
?
V
hrf2?
58
Figure 18 Model Validation (PCR Vs Ridge)
5.4 Model Validation (Parameters)
The Validation is done using data outside of the data set used for regression. A
plot with the predicted and actual life of package would give an idea of the reliability of
the model.
0
200
400
600
800
1000
1200
1 2 3 4 5
Lif
e(cy
cle
s)
Package
PCR Vs Ridge
PCR Prediction
Actual life
Ridge Prediction
59
5.4.1 Half Diagonal Length
Figure 19 Effect of Die Length
The distance of the solder ball from the neutral axis of the chip increases the
stress induced in the solder ball. The increase in diagonal length means more number of
solder balls away from the neutral axis. Hence the half diagonal length has negative
influence on life. The plot shows the same trend.
0
100
200
300
400
500
600
700
800
4.9497 5.3033
N1
%
Li
fe
Half DLengthMM
Actual
Predicted
60
5.4.2 Delta T
Figure 20 Delta T Validation
The temperature difference between the minimum and maximum temperatures is
the most significant variable on a thermal cycle. The temperature excursion (Delta T) has
square dependency on life. The plot shows the same trend. The packages looked at have
the same dimensions and test conditions but undergo different thermal cycle. The
package undergoing lesser temperature excursion has better life.
0
100
200
300
400
500
600
700
800
900
1000
135 180
N1
%
Li
fe
DeltaT
Predicted
Actual
61
5.5 Model Predictions
The model is used to predict life and the model predictions are plotted against the
actual life to see the accuracy of the model. The dotted lines represent the 90% interval.
As seen from the plot, most of the model predictions fall in the interval.
Figure 21 Goldmann Prediction
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200 1400
Pr
ed
ict
ed
L
ife
Actual Life
Goldmann Prediction
62
Chapter 6
Summary and Conclusions
Statisticsbased modeling methodologies have been presented in this work.
The methods have been used for development of NorrisLandzberg Acceleration
Factors and Goldmann Constants for areaarray packages with Sn3Ag0.5Cu solder
alloy interconnects. The methodology is based on accelerated test data. Testboards
with various package architectures, tested under different temperature ranges, dwell
times, maximum temperature, and minimum temperatures have been included in the
dataset. The time to one percent failure and characteristic life of the weibull distribution
has been used for the model development. The models developed have been validated
with experimental data in a number of different ways. Life has been predicted for a
completely different dataset and the error in the model predictions quantified. In
addition, change in thermo mechanical fatigue life versus individual parameter
variations has been studied for a number of test cases. The model predictions in
each case have been correlated with experimental data and Weibull distributions
presented for each case. The presented approach provides a method for institutional
learning based on databases of accelerated test data developed for product specific
applications. The closed form models are a time effective solution for doing trade
offs and the thermo mechanical reliability assessment of the areaarray packages on
PCB assemblies subjected to extreme environments. The developed methodology also
63
allows the user to understand the relative impact of the various geometric parameters,
material properties and thermal environment on the thermomechanical reliability of the
different configurations of area array devices with leaded as well as leadfree solder
joints. The influence of silver content on the life of SAC alloy based packages has been
assessed and the model has been validated. A wide range of SAC alloys like SAC
105,305,405, 387 and other not very common SAC alloys have also been used for the
model, making the model dependable. The convergence between experimental results
and the model predictions with higher order of accuracy than achieved by any first
order closed form models has been demonstrated, which develops the confidence for
the application of the models for comparing the reliability of the different BGA
packages for various parametric variations. The current approach allows the user to
analyze independent as well as coupled effects of the various parameters on the
package reliability under harsh environment.
64
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