PRINCIPAL COMPONENT REGRESSION MODELS FOR THERMO
MECHANICAL RELIABILITY OF PLASTIC BALL GRID
ARRAYS ON CUCORE AND NO CUCORE PCB
ASSEMBLIES IN HARSH ENVIRONMENTS
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
__________________________________________
Aniket Shirgaokar
Certificate of Approval:
____________________________ __________________________
Jeffrey C. Suhling Pradeep Lall, Chair
Quina Distinguished Professor Thomas Walter Professor
Mechanical Engineering Mechanical Engineering
___________________________ ___________________________
Hyejin Shin George T. Flowers
Assistant Professor Dean
Mathematics and Statistics Graduate School
PRINCIPAL COMPONENT REGRESSION MODELS FOR THERMO
MECHANICAL RELIABILITY OF PLASTIC BALL GRID
ARRAYS ON CUCORE AND NO CUCORE PCB
ASSEMBLIES IN HARSH ENVIRONMENTS
Aniket Shirgaokar
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirement for the
Degree of
Master of Science
Auburn, Alabama
August 10, 2009
iii
PRINCIPAL COMPONENT REGRESSION MODELS FOR THERMO
MECHANICAL RELIABILITY OF PLASTIC BALL GRID
ARRAYS ON CUCORE AND NO CUCORE PCB
ASSEMBLIES IN HARSH ENVIRONMENTS
Aniket Shirgaokar
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of individuals or institutions at their expense.
The author reserves all publication rights.
___________________________
Signature of Author
___________________________
Date of graduation
? [2009]
Copyright
Aniket Shirgaokar
All rights reserved
iv
VITA
Aniket Shirgaokar, son of Mr. Jeevan Shirgaokar and Smt. Anuja Shirgaokar was
born on March 29, 1985 in Aurangabad, Maharashtra, India. He graduated in 2006 with a
Bachelor of Engineering degree in Production Engineering from Shivaji University,
Maharashtra, India. In the pursuit of enhancing his academic qualification he joined the
M.S. Program at Auburn University in the Department of Mechanical Engineering in
Spring, 2007. Since then, he has worked for Center for Advanced Vehicle Electronics
(CAVE) as a Graduate Research Assistant in the area of harsh environment electronic
packaging reliability.
v
THESIS ABSTRACT
PRINCIPAL COMPONENT REGRESSION MODELS FOR THERMO
MECHANICAL RELIABILITY OF PLASTIC BALL GRID
ARRAYS ON CUCORE AND NO CUCORE PCB
ASSEMBLIES IN HARSH ENVIRONMENTS
Aniket Shirgaokar
M.S. Mechanical Engineering, August 10, 2009
(B.E. Production Engineering, Shivaji University, India, 2006)
113 Typed Pages
Directed by Pradeep Lall
In the current work, Goldmann constants and NorrisLandzberg acceleration
factors have been developed for eutectic Tin Lead and Lead free solders (SAC 305) with
the help of statistical tools including Principal Component Regression for reliability
prediction and part selection of Plastic Ball grid array packages. Two types of PCB
assemblies including PCBs with integral copper core and PCBs with no integral copper
core have been tested. The models have been developed based on thermomechanical
reliability data acquired on packages subjected to several different thermal cycling
conditions. The thermal cycling conditions differ in temperature range, dwell times,
maximum temperature, minimum temperature to enable development of constants needed
for life prediction and assessment of acceleration factors.
vi
Goldmann constants and the NorrisLandzberg acceleration factors have been
benchmarked against previously published values. In addition, model predictors 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, solder
volume, diagonal length of chip, etc. The predicted and measured acceleration factors
have also been computed and correlated. The correlations achieved are of a good
accuracy for different parameters examined. Statistics based log transformed models have
been presented to show their power dependencies. Box ? Tidwell power law modeling
has been demonstrated. The presented methodology is valuable in development of
fatigue damage constants for the application specific accelerated test datasets and
provide a method to develop institutional learning based on prior accelerated test data.
vii
ACKNOWLEDGEMENTS
The author would like to thank his advisor Dr. Pradeep Lall, and committee
members Dr. Jeffrey Suhling and Dr. Hyejin Shin for their invaluable guidance and help
during the course of this study. The author acknowledges and extends gratitude for
financial support received from the NSF Center for Advanced Vehicle and Extreme
Environment Electronics (CAVE3).
Author would like to express his deep gratitude and gratefulness to his father Mr.
Jeevan Shirgaokar for being a constant source of inspiration and motivation, mother Mrs.
Anuja Shirgaokar, brother Mr. Sanyam Shirgaokar for their enduring love and immense
moral support. The author would like to thank Mr. Ganesh Hariharan, Mr. Jonathan
Drake and Mr. Timothy Moore for passing on their knowledge and guidance in data
collection and analysis. The author wishes to acknowledge his colleagues Mr. Prashant
Gupta, Mr. Chandan Bhat, Mr. Robert Hinshaw, Ms. Madhura Hande, Ms Deepti
Iyangar, Mr. Sandeep Shantaram, Mr Rahul Vaidya, Mr. Vikrant More, Mr. Mahendra
Harsha, Mr. Mandar Kulkarni, Mr. Dineshkumar Aurunachaklam, Mr. Arjun Angral and
Mr. Ryan Lowe for their friendship, help and all the stimulating discussions.
viii
Style manual or journal used Guide to Preparation and Submission of Thesis and
Dissertations
Computer software used Microsoft Office 2003, Minitab 13.1, Ansys 10.0,
Matlab R2007a, WinSmith Weibull 3.0, SAS 9.1
ix
TABLE OF CONTENTS
LIST OF FIGURES
x
LIST OF TABLES
xii
CHAPTER 1 INTRODUCTION 1
CHAPTER 2 LITERATURE REVIEW 7
CHAPTER 3 DATASET 14
CHAPTER 4 APPROACH AND PROCEDURE FOR PCR 25
CHAPTER 5 PCR ON CU CORE ASSEMBLIES 31
CHAPTER 6 PCR ON NO CU CORE ASSEMBLIES 41
CHAPTER 7 POWER LAW DEPENDENCY OF PREDICTOR VARIABLES 48
CHAPTER 8 STATISTICAL FORM OF NORRIS LANDZBERG`S MODEL 55
CHAPTER 9 STATISTICAL FORM OF GOLDMANN`S MODEL
60
CHAPTER 10 MODEL VALIDATION 72
CHAPTER 11 CONCLUSION
78
BIBLIOGRAPHY
80
APPENDIX LIST OF SYMBOLS 98
x
LIST OF FIGURES
1.1: Solder joint fatigue failure due to thermal cycling 2
3.1: Individual Packages tested for ATC 16
3.2: Representative List of Different Package Architectures 17
3.3: Front Side of test board CCA 091099 (TC2: 55C to 125C) 19
3.4: Back Side of test board CCA 091099 (TC2: 55C to 125C) 20
3.5: Front Side of test board CCA 136144 (TC3: 3C to 100C) 21
3.6: Back Side of Test Board CCA 136144 (TC3: 3C to 100C) 22
3.7: Front Side of test board CCA 145154 (TC4: 20C to 60C) 23
3.8: Back Side of test board CCA 145154 (TC4: 20C to 60C) 24
4.1: Flow Chart for Modeling Methodology 26
5.1: Contribution of each Principal Component 34
5.2: Residual Analysis for PCR on Cu Core Assemblies 38
5.3: Plot of Studentized residuals Vs Normal Quantiles 39
5.4: Plot of Actual Vs the Predicted Life for the PCR Model for Cu
Core Assemblies 40
6.1: Contribution of each Principal Component for PCR of Cu Core Assemblies 42
6.2: Analysis of Residuals for PCR on No Cu Core Assemblies 45
6.3: Plot of studentized residuals Vs Normal Quantiles 46
6.4: Plot of Actual Vs the Predicted Life for the PCR Model for No Cu Core
xi
Assemblies 47
7.1: Model Adequacy Checking for Interaction effect model 53
7.2: Plot of Mean Square Error Vs Power Transposed 54
8.1: Model Adequacy checking for NL Model 59
9.1: Different predictor variables in the Goldmann`s model 61
9.2: Model Adequacy Checking for Goldmann model on Cu Core Assemblies 67
9.3: Model Adequacy Checking for No Cu Core Goldmann Model 70
10.1: Effect of Delta T on N1% Life of the Packages assembled on Cu Core
PCBs 73
10.2: Effect of Solder Volume on Life of the Package assembled on Cu Core
PCBs 74
10.3: Effect of Die to body ratio on the life of Package assembled on No Cu
Core PCBs 75
10.4: Effect of Half Diagonal Length on Life of Package Assembled on Cu Core
PCBs for Goldmanns Model 76
10.5: Effect of cyclic frequency on Acceleration Factor 77
10.6: Effect of temperature cycle magnitude on Acceleration Factor 77
xii
LIST OF TABLES
3.1 Scope of the Test Dataset 15
3.2: Thermal Cycling Conditions 18
5.1: Checking the VIF values 32
5.2: Pearson?s Correlation Matrix 33
5.3: Transformed Z variable regression for Cu Core Assemblies 36
5.4: ANOVA table for Cu Core Assemblies 36
5.5: Transforming Z back to Original Variables in the Cu Core Assemblies 37
5.6: Shapiro Wilk Test 38
6.1: Transformed Z variable regression for PCR on No Cu Core Assemblies 43
6.2: Analysis of Variance for PCR on No Cu Core Assemblies 44
6.3: Transforming Z back to Original Variables in the NL Model for Cu Core
Assemblies 44
6.4: Results for the Shapiro Wilk test on No Cu Core Assemblies 46
7.1: Comparison of Power Law Dependence values 50
7.2: PCR Model for Cu Core Assemblies with the Interaction Effect between
Delta T and Half Diagonal Length 51
7.3: ANOVA Table for Interaction Effect Model 52
7.4: Transforming the Z`s Back to the Original Variables in the Interaction
Effect Model 52
xiii
8.1: Transformed Z variable regression for NL model 56
8.2: ANOVA Table for Z transformed Variables of NL model 57
8.3: Transforming Z back to Original Variables in the NL Model for Cu Core
Assemblies 57
9.1: Transformed Z variable regression for Goldmann`s model of Cu Core
Assemblies 63
9.2: ANOVA Table for Z transformed Variables of Cu Core Assemblies. 64
9.3: Transforming Z back to Original Variables in the Goldmann`s Model for
Cu Core Assemblies 65
9.4: Regression of Z transformed variables in Goldmann`s equation
against N1% life for No Cu Core Dataset 68
9.5: ANOVA Table for No Cu Core Assemblies 68
Table 9.6: Transforming back to the original variables in the NL equation
for No Core Assemblies 69
1
CHAPTER 1
INTRODUCTION
The increasing pressure for developing small, reliable and cheap packages on the
microelectronics industry have lead to the use of area array packages. After their wide
spread use in the commercial field, PBGAs are now implemented in aerospace and
military applications [Ghaffarian 2005]. Considering the various factors like geometric
parameters, material properties, thermal cycling conditions which govern the reliability of
electronic packages, statistical models have been developed for the data obtained by
accelerated life cycling of different boards with Cu core and No core PCB substrates.
Principal component regression models are used for life prediction of these packages
which are subjected to harsh environments.
It is very important to understand the underlying physics and the mechanical
failure theories which govern the failure of the solder joint. 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 has to be taken into
consideration while evaluating the fatigue behavior of solder interconnections under
accelerated conditions. Previously researchers have studied the behavior of the solder and
developed life predictions for Eutectic Tin Lead solder. With the Electronic industry
2
going Lead free, there have been many challenges for the researchers to predict
the behavior of the solder and thus their failure.
L
h
At Stress Free Temperature (T)
At Temperature T
2
(T
2
T)
Silicon
PCB
Figure 1.1: Solder joint fatigue failure due to thermal cycling
When the package under goes thermal cycling, may it be an accelerated one or
one in the field, the PCB which has a higher coefficient of thermal expansion heats up
and expands more than the silicon. When the temperature decreases, due to cessation of
the operation or environment, the PCB will contract faster. The expansion and
contraction introduces shear strains and shear stresses in the solder joint. High shear
stress can cause delamination of various interfaces like UBM/intermetallic,
solder/underfill etc. Apart from delamination, the repeated heating and cooling can
3
eventually cause fatigue of the solder joints. The high shear stresses would enhance the
fatigue initiation making solder interconnect more susceptible to such fatigue failures as
shown in Figure 1.1 [Singh 2006] represents the same. Hence evaluation of stresses at the
joints has become critical to predict the reliability of the assembly.
The Classical Coffin Manson?s Equation which related the plastic strain that
develops due to the difference in coefficient of thermal expansion is given in the equation
1.1 below:
C)(N
n
p
??? Eqn 1.1
Where,
p
?? is the plastic strain,
N is number of cycles to failure,
n is empirical constant observed to be 2 for nearly all metals,
C is the proportionality factor.
4
Goldmann developed his form of the Coffin Manson which is given in Equation 1.2
below
Goldmann`s Equation:
m
m
12
fu
Tf
1
V
h
A
rT
KN
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
???
Eqn 1.2
Where,
T
K is a constant which is a function only of parameters of the testing cycle,
u
T is ultimate shear strength of the critical interface,
r
f
is the radius of the critical interface,
A and ? are constants in the stress strain relationship,
h is the height of the solder joint,
V is the volume of the solder joint,
? is the shear deformation of the joint
m is an empirical constant.
5
The Norris Landzberg Model is given in the following equation 1.3 below:
??
max
2
U
A
3
1
A
U
A
U
T
T
T
f
f
N
N
AF ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?? Eqn 1.3
Where,
AF is the Acceleration factor.
Subscript U stands for useconditions and Subscript A is used for acceleratedtest
conditions
N
U
and N
A
are the lives of the packages f
U
and f
A
are the frequencies
T
A
and T
U
are the temperature excursions
T
max
is the maximum temperature of the cycle in Kelvin
The Equation is often in used in the form [Lau 1997] given by Equation 1.4 below
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
Amax,Umax,
2
U
A
3
1
A
U
A
u
T
1
T
1
1414exp
T
T
f
f
N
N
AF Eqn 1.4
Principal Component Regression is used to formulate these equations in the
statistical model. In case of the Goldmann Equation, the Number of cycles to failure is
taken as the response variable and the terms on the right hand side of the equation like the
ultimate shear strength of the critical interface, height of the solder joint, volume of the
solder joint are taken as the response variables. In case of the Norris Landzberg`s model
the Acceleration Factor which is the ratio of the lives of the package is taken as the
response variable and the parameters on the right hand side of the equation like the ratio
of the frequencies and the ratio of temperature excursions are taken as the predictor
variables.
6
Principal Component Regression is a method to overcome the multicolinearity in
a regression model by transforming the original predictor variables to a new dataset with
the help of Eigen vectors and then transforming the original variables back after the
regression is done. It will be discussed in details in the further chapters.
7
CHAPTER 2
LITERATURE REVIEW
2.1 Experimental Techniques
Various experimental tests, such as Accelerated Thermal Cycling (ATC), Thermal
Shock, HAST (highly accelerated stress test) and vibration test, have been used by the
researchers to analyze the solder joint fatigue life for qualifying the components for
different applications. ATC exposes the packages to a series of low and high
temperatures usually in a single air chamber in which the temperature ramp can be
controlled carefully. Thus accelerating the failure modes caused by cyclic stresses.
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.
Darveaux, et al. [2000] conducted several board level thermal cycle reliability
tests, the packages used included FlexBGA, Tape Array Ball Grid Array, PBGA and
MicroBGA. He tested wide range of package and board variables and reported findings
about life of the package by changing dies size, package size thickness of test boards etc.
He also reported 1.6X acceleration factor between 40?C to 125?C and 0?C to 100?C
temperature cycling ranges.
Mercado, et al. [2000] conducted test on flip chip PBGA package for FSRAM
(Fast Static RAM) application in order to analyze the effect of pad size and substrate
thickness on the solder joint reliability. It was reported that C5 solder joints with larger
8
solder pad and thicker substrates demonstrated higher reliability. Hung, et al. [2000]
investigated the effect of chip size, surface finish, Au plating thickness, epoxy thickness,
polyimide thickness and underfilling on the interconnect thermal cyclic fatigue life by
conducting experimental test on FlexBGA packages. Chip size, polyimide thickness and
underfilling were found to have significant impacts on the joint fatigue life, epoxy
thickness was found to have little effect on the joint fatigue life.
Suhling, et.al. [2004] presented research on the thermal cycling reliability of lead
free solder joints for use in the automotive industry. Four solder compounds were tested:
95.5Sn3.8Ag0.7Cu and three variations of lead free SAC solder that incorporate small
additions of bismuth and indium to enhance fatigue resistance. These solder joint
compounds were thermally cycled under two test conditions: 40 C to 125 C, and 40 C
to 150 C. Results from this study showed that the eutectic SAC alloy 95.5Sn3.8Ag0.7Cu
gave comparable reliability results to standard 63Sn37Pb solder alloy during the 40 C to
125 C temperature condition, but differed greatly, demonstrating much lower reliability
relative to the 63Sn37Pb alloy, when subjected to the more harsh 40 C to 150 C
temperature range. It was also shown that adding trace amounts of bismuth and indium
can enhance the 40 to 150 C thermal cycling fatigue resistance relative to
95.5Sn3.8Ag0.7Cu.
9
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. 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
number of cycles to failure with applied shear strain range. He 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
10
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.
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.
11
2.3 Statistical Analysis
Researchers have used different statistical methods for the analysis of the
experimental test failure data, the most common being regression analysis and Weibull
distribution. Clech, et al. [1994] presented statistical analysis of thermomechanical wear
out failure data from 26 accelerated tests and tested the goodnessoffit using two and
three parameter Weibull and lognormal distributions. It was concluded that the three
parameter Weibull treatment provides more accurate reliability projections and failure
free time prediction, potentially qualifying component assemblies that would be rated
marginal or unacceptable based on conservative two parameter Weibull or lognormal
analysis.
Stoyanov [2002] used a design of experiments and response surface modeling
methodology for building a quadratic equation that related underfill modulus, underfill
CTE, stand off height and substrate thickness with number of cycles to failure for a flip
chip package. The data for model building was collected from a finite element analysis of
a flip chip package. Residual analysis, analysis of variance and statistical efficiency
measure were used for validating the models. Taguchi optimization technique was used
by Lai [2005] for optimizing the thermomechanical reliability of a package on package
for various design parameters. The package parameters considered for optimization
included die thickness, package size, mold thickness, substrate thickness and solder joint
stand off.
Muncy, et al. [2003, 2004] conducted thermal reliability test including airtoair
thermal cycling (AATC) and liquidtoliquid thermal shock (LLTS) on various
configurations of flipchip on board (FCOB) packages. The failure data was then
12
analyzed using multiple linear regression and ANOVA (analysis of variance) to
determine the parameters that had influence on the reliability performance of the
components in accelerated life testing, the input parameters investigated included,
substrate metallization, substrate mask opening area versus the UBM area of the flip chip
bump, die size, perimeter or area array flip chip interconnect pattern, underfill material,
location of the die on the test board, frequency of cycling, number of I/O, and percent
area voiding. A model based on regression analysis was developed in order to quantify
the effect of process and design decisions on the reliability of a flip chip on board
assembly.
Perkins [2004] developed a multiple linear regression based polynomial equation
for correlating fatigue life of a ceramic package with its design parameters. A data matrix
was formulated using a full factorial design of simulation study for the five design
parameters including substrate size, substrate thickness, CTE mismatch between substrate
and board, board thickness and solder ball pitch with two levels each. Simulations were
run for each data point using a finite element analysis and the fatigue life was extracted.
Interactions between the predictor variables were studied and a regression model with
both main terms and interaction terms was built.
Iyer [2005] correlated the reliability of a flip chip package with its properties of
underfill and flux using a regression and back propagation neural networks based models.
Data from accelerated life testing of flip chip package with 95 different underfill flux
combinations was used for model building. The underfill parameters for model building
included modulus of elasticity, coefficient of thermal expansion, glass transition
temperature and filler content. The flux parameters studied include acid number and
13
viscosity. The regression models and the neural network models were validated using a
test data set and the neural networks model was found to outperform the regression model
owing to minimum residual mean square errors.
Singh [2006] developed multivariate regression based models for life prediction
of BGA packages. The input data for model building was collected from published
literature and accelerated test reliability database based on the harsh environment testing
of BGA packages by the researchers at the NSF Center for Advanced Vehicle Electronics
(CAVE). The predictor variables considered for model building included die, die to body
ratio, ball count, ball diameter, solder mask definition, printed circuit board surface finish
printed circuit board thickness, encapsulant mold compound filler content and deltaT.
Dummy variables were used for categorical variables like borad finish, encapsulant mold
compound filler content and solder mask definition. Linear, modified linear and non
linear models were developed using regression analysis and analysis of variance and
validated with experimental data.
Hariharan [2007] developed MLR and PCR model for Predicting the reliability of
various Ball Grid Array Packages including FlexBGA, CBGA, CCGA and Flip Chip
Packages. He also demonstrated the power law dependencies of the various parameters in
the regression model with Box Tidwell Power law modeling.
14
CHAPTER 3
DATASET AND THERMAL CYCLING CONDITIONS
The table 3.1 gives a brief idea of the scope of the packages and the range of the
data which was tested for accelerated life and the failure data was utilized for statistical
analysis. The database is fairly diverse in terms of materials and geometry parameters.
The dataset used for model building has been accumulated from an extensive accelerated
test reliability database of plastic ballgrid array (PBGA) and chiparray ballgrid array
(CABGA) devices based on the harsh environment accelerated test database developed
by the researchers at the NSF Center for Advanced Vehicle Electronics. Each data point
in the database is based on the WeibullParameters including the time to onepercent
failure, characteristic life, and the shape parameter for the area array devices of a given
configuration tested under harsh thermal cycling or thermal shock conditions. The
material properties and the geometric parameters investigated include die thickness, die
size, die to body ratio, substrate thickness, ball count, ball pitch, board finish, solder joint
height, solder joint volume, bump size, weight of the package and printed circuit board
thickness.
15
Table 3.1 Scope of the Test Dataset
Pa
ck
ag
e t
yp
e
Ar
ray
typ
e
I/O
Pit
ch
(m
m)
I/O
Co
un
t
Ra
ng
e
So
lde
r
all
oy
Pa
ck
ag
e s
ize
(m
m)
Di
e
Siz
e
Ra
ng
e
Pa
ck
ag
e
to
Di
e
siz
e
rat
io
Full
Perimeter PBGA
Mixed
0.5 
1.00 49  900
Pbfree
SAC
305
7.0  31.0
Full
Perimeter
FCPBGA
0.81.00 532  1508
Pbfree
SAC
305
23.0 
40.0
Full
Perimeter MCMPBGA
1.00 128  324 22.0
Full
HiTCE CBA
1.27 360
Pbfree
SAC
305
25.0
Full
CBGA
1.27 483 29.0
Full
Perimeter CSP
0.5  0.8 132  228
Pbfree
SAC
305
7.0  12.0
Full
Perimeter Flip Chip
0.25 
0.45 48  317
Pbfree
SAC
306
5.08 
6.35
Perimeter Micro  Lead Frame
0.40 
0.65 44  100
Pbfree
SAC
307
9.0  12.0
QFP /
LQFP Perimeter 0.4  0.5 100  176
Pbfree
SAC
308
14.0 
20.0
4.00 
24.0
1.00 
3.94
16
The figure 3.1 and Figure 3.2 shows the individual packages which were mounted two
different types of boards viz. PCB with integral Copper Core and PCB without integral
Copper Core for thermal cycling.
Figure 3.1 Individual Packages tested for ATC
17
Figure 3.2: Representative List of Different Package Architectures
Table 3.2 shows the temperature ranges, dwell times, and ramp rates for the four
thermal cycling profiles labeled as TC1, TC1, TC3 and TC4.
18
Table 3.2: Thermal Cycling Conditions
Profile
Low Temp
(oC)
High Temp
(oC)
Low Dwell
(min)
High Dwell
(min)
Ramp Rate
(oC/min)
TC1 40 95 30 30 3
TC2 55 125 30 30 3
TC3 3 100 30 30 3
TC4 20 60 30 30 3
TC5 20 80 30 30 3
TC6 0 100 15 15 3
TC7 0 100 10 10 3
TC8 55 125 15 15 3
TC9 40 125 15 15 3
19
Pictures of the boards which were subjected to thermal cycles are shown in
Figures 3.3 to Figure 3.8 below:
Figure 3.3 Front Side of test board CCA 091099 (TC2: 55C to 125C).
20
Figure 3.4 Back Side of test board CCA 091099 (TC2: 55C to 125C).
21
Figure 3.5 Front Side of test board CCA 136144 (TC3: 3C to 100C)
22
Figure 3.6 Back Side of Test Board CCA 136144 (TC3: 3C to 100C).
23
Figure 3.7 Front Side of test board CCA 145154 (TC4: 20C to 60C).
24
Figure 3.8 Back Side of test board CCA 145154 (TC4: 20C to 60C).
25
CHAPTER 4
APPROACH AND PROCEDURE FOR PRINCIPAL COMPONENT
REGRESSION
Multiple linear regression methods assume the predictor variables to be
independent of each other. Linearly dependent variables result in multicollinearity,
instability and variability of the regression coefficients [Cook et al. 1977]. Principal
components models have been used for dealing with multicollinearity and producing
stable and meaningful estimates for regression coefficients [Fritts et al 1971]. The Figure
4.1 shows the modeling methodology and procedure for developing the PCR models. The
different parameters like part architecture and geometry, thermal cycling environment
have been used to formulate the mission requirements using the different statistical
techniques like Principal Component Regression, Box Tidwell Transformation. Models
have been validated using the other reliability database by comparing the results with
failure mechanics models. The effects of the output design parameters and acceleration
factors have been presented.
26
Figure 4.1 Flow Chart for Modeling Methodology
Harsh Environment Part Architecture
Mission requirements
Box Tidwell
Models
Empirical models and correlation with
FEM and underlying failure mechanics
Principal Component
Regression models
Closed form
Mathematical models
Interaction effects& tradeoffs package design
parameters, thermal conditions subjected, package
reliability
Reliability
Database
OUTPUT DESIGN PARAMETERS
Package size, Die size, Solder ball Composition, Ball Pitch, Ball
Height, Pad Size and Configuration, Under fill Composition,
Substrate Thickness and Material, etc
Assessment of acceleration
factors
Geometry
27
Methodology for developing a Principal Component Regression Model is
presented here:
Matrix Notation for the model is given in the Eqn 4.1 below:
}{}]{X[}y{ ?+?= Eqn 4.1
Where,
??
?
?
?
??
?
?
?
?
??
?
?
?
??
?
?
?
?
=
n
2
1
y
.
.
.
y
y
}y{ , sets_data_n
iablesvar_predictor_k
x...xx1
.......
.......
.......
x...xx1
x...xx1
X
knn2n1
2k2212
1k2111
??
?
?
?
??
?
?
?
?
??
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
=
horizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketext upcurlybracketrighthorizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketext upcurlybracketmidupcurlybracketleft
??
?
?
?
??
?
?
?
?
??
?
?
?
??
?
?
?
?
?
?
?
=?
n
1
0
.
.
.}{
and
??
?
?
?
??
?
?
?
?
??
?
?
?
??
?
?
?
?
?
?
?
=?
n
1
1
.
.
.}{
Multicollinearity of predictor variables may cause, large variance and co
variance 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
28
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
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 given in
the Eqn 4.2 below:
]V])[I[]C([ ?? Eqn 4.2
0]I[]X[]X[ *T* =?? Eqn 4.3
29
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
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 ]
dncurlybracketrightdncurlybracketmiddncurlybracketleft
j_with_associated_Vector_Eigen
kj
j2
j1
V
.
.
.
V
V
?
??
?
?
?
??
?
?
?
?
??
?
?
?
??
?
?
?
?
Eqn 4.4
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 by the
Equation 4.5:
1*k
*k*kT
1*k }{]V[}{ ?=? Eqn 4.5
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 given in the Equation 4.6 below.
1*kk*k1*k }{]V[}{ ?=? Eqn 4.6
The overall adequacy of the model is tested using ANOVA table. Small P value
of the ANOVA table rejects the null hypothesis and proves the overall adequacy of the
30
model. Individual T tests on the coefficients of regression of principal components
yielded very small P values indicate the statistical significance of all the predictor
variables.
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 4.7 below:
2
1
l
1m
2
jm
1
m
pc,j
vMSE
bt
??
?
??
? ?
?
??
?
? ??
=
?
=
?
Eqn 4.7
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. The P values of individual T tests retaining values < 0.05 prove the statistical
significance of individual regression coefficients of original predictor variables at a 95 %
confidence.
31
CHAPTER 5
PRINCIPAL COMPONENT REGRESSION ON COPPER CORE ASSEMBLIES
A superset of predictor variables including Area of the chip, Board finish, Die
length, Die to body ratio, Ball count, Ball Pitch, Solder ball diameter, Weight of the
package, Solder ball height, Solder Volume, Package pad area, and Thermal cycling
conditions has been created. The predictor variables have then been checked for being
correlated to each other since independence of predictor variables is one of the most
important assumptions of a linear regression model. Predictor variables with very strong
correlation for e.g. die length and area of the die, which have a correlation factor of
almost 1 have been tackled by eliminating one of the two as they convey more or less the
same information from analysis point of view. Predictor variables that are needed for
model building are then selected through stepwise regression and method of best subsets
using the following criteria: Maximization of Coefficient of determination R2,
Maximization of Adjusted R2 and Minimization of Residual Errors. Predictor variables
which contribute significantly with a confidence level of 95 % and more are retained in
the model. The procedure for Principal component regression which is discussed in
Chapter 4 in details is then followed to construct the model.
A check for determining the presence of multicolinearity was done. The
Pearson?s corelation matrix and the Variance Inflation Factors have been used to gauge
32
the intensity of the multicolinearity. The VIF values in the table 5.1 below are
more than 10 and confirm the presence of Multicolinearity in the model.
Table 5.1 Checking the VIF values
Predictor Coef SE Coef T P VIF
Constant 24370 5479 4.45 0
BrdFinis 66.69 28.6 2.33 0.026 1.1
DieLenMM 227.75 57.26 3.98 0 89.2
DieToBod 254.6 264 0.96 0.342 4.6
BallCoun 3.314 2.315 1.43 0.162 28
BallPtch 4745 1296 3.66 0.001 167.7
BallHgtM 10628 2412 4.41 0 246.2
SdrVol 0.08249 0.0892 0.92 0.362 5.7
1/TmeanK 5855583 1481712 3.95 0 36.2
DeltaT 20.564 3.363 6.12 0 37.1
The Pearson`s correlation matrix in the Table 5.2 below also shows many values
greater than 0.8 which suggest the same.
33
Table 5.2 Pearson?s Correlation Matrix
BF DLmm DTB BC BaPtmm PPdDmm PWtgm BHgtmm SdrVol DeltaT
BF
1 0.01

0.01

0.01 0.02 0.02 0.00 0.00 0.00 0.01
DL
mm

0.01 1 0.64 0.89 0.78 0.78 0.92 0.84 0.71 0.05
DT
B

0.01 0.64 1 0.67 0.14 0.14 0.63 0.20 0.19 0.09
BC

0.01 0.89 0.68 1 0.62 0.62 0.99 0.63 0.52 0.08
Ba
Pt
mm
0.01 0.78 0.14 0.62 1 1 0.72 0.98 0.86 0.01
PP
dD
mm
0.01 0.78 0.14 0.62 1 1 0.72 0.98 0.86 0.01
PW
tgm

0.00 0.92 0.63 0.99 0.72 0.72 1 0.72 0.59 0.08
BH
gtm
m
0.00 0.84 0.20 0.63 0.98 0.98 0.72 1 0.85 0.00
Sd
rV
ol

0.00 0.71 0.19 0.52 0.86 0.86 0.59 0.86 1 0.03
De
lta
T

0.01 0.05

0.09

0.08 0.02 0.01 0.08 0.00 0.00 1
34
Figure 5.1 is the plot of Principal component on Xaxis and the Cumulative %
contribution of the Eigen value on the Yaxis
Figure 5.1 Contribution of each Principal Component
The variable selection was done based on the stepwise regression procedure and
the partial Ftests which help in selecting the variables which contribute significantly to
the linear regression model. One of the tests for ball pitch is demonstrated below:
Partial F test:
Hypothesis: H0: 05 =? , where 5? is the slope for ball count
Test statistic:
ducedRe
ducedRe,sRe
ducedReFull
ducedRe,sReFull,sRe
0
Df
SS
DfDf
SSSS
F ?
?
= Eqn 5.1
35
93
2856834
9394
28568342866013
?=
298808.064516.307189179 ==
As F0 = 0.2988 < F0.05,1,93 = 3.94,
We Accept H0: 04 =? which implies that the parameter ball count does not
contribute significantly to form a linear model. Similarly, all the other variables viz.
board finish, area of the chip, Solder Volume and Solder Ball Diameter which fail to
contribute significantly to form a linear regression model have been eliminated. A
regression model with the rest of the predictor variables is then developed. Results and
discussion of the same has been presented below.
The regression equation obtained by regressing the Z predictor variables against
the N1% life of the package is given by Eqn 5.2 below:
7Z*1.3826Z*1.8475Z*38.303
4Z*5.10423Z*9.7662Z*2.10751Z*4.3903.2859%1N
+?+
?+??= Eqn 5.2
The Table 5.3 shown below represents a detailed result of the regression of
principal components against the N1% life of the packages. The Pvalues of all the
predictors are less than 0.05 suggesting the statistical significance of the 7 predictors with
95% confidence.
36
Table 5.3 : Transformed Z variable regression for Cu Core Assemblies
Predictor Coef SE Coef T P
Constant 2859.3 237.7 12.03 0
Z1 390.4 100.6 3.88 0
Z2 1075.2 201.9 5.33 0
Z3 766.9 196.5 3.9 0
Z4 1042.5 270.1 3.86 0
Z5 303.38 62.8 4.83 0
Z6 847.1 176.6 4.8 0
Z7 382.1 80.86 4.73 0
The Table 5.4 below represents the Analysis of Variance used initially to prove
that the predictors have a linear relationship with the response variable N1%
Table 5.4 : ANOVA table for Cu Core Assemblies
Source DF SS MS F P
Regression 7 5156023 736575 32.56 0
Residual Error 90 2035939 22622
Total 97 7191962
Regression equation for original variables is given by Eqn 5.3 below
DeltaTDEGC*95.6PkgWtGM*49.312QMMPkgPdAreaS
*55.1301PitchMM*53.632atioDietoBodyR*61.1319
DiagLenMM*66.55MMChipAreaSQ*17.606.2859%1N
??
+??
?+=
Eqn 5.3
37
The Table 5.5 below gives a detailed result of the regression between the transformed
original variables and N1%
Table 5.5: Transforming Z back to Original Variables in the Cu Core Assemblies
Predictors Coeff SE
(a0, fk) (bk) Coeff
T Value
PValue
Constant 2859.06 237.66 12.03 0
ChipAreaSQMM 6.17 1.59 3.88 0
DiagLengthMM 55.66 10.44 5.33 0
DietoBodyRatio 1319.61 338.36 3.9 0
PitchMM. 632.53 163.87 3.86 0
PkgPdAreaSQMM 1301.55 269.47 4.83 0
PkgWgtGM 312.49 65.102 4.8 0
DeltaTDEGC 6.95 1.47 4.73 0
38
Model Adequacy Checking:
Figure 5.2: Residual Analysis for PCR on Cu Core Assemblies
From the above Figure 5.2, the plot of residuals Vs Fits we do not observe any
specific pattern which implies that the linearity assumptions are met.
The plot does not show any signs of the scatter increasing with the fitted values which
implies that the constant variance assumptions are satisfied.
Normality Test:
The Shapiro ?Wilk test was performed to check if the normality assumptions are
satisfied. A Pvalue of 0.3076 which is > 0.05 confirms the normality assumptions the
dataset. The results for the same are shown in Table 5.6 below
Table 5.6: Shapiro Wilk Test
Test Test Statistic (W) Pvalue
ShapiroWilk 0.985305 0.3076
39
The Figure 5.3, plot of studentized residuals Vs Normal Quantiles also produces
points close to a straight line suggesting that the normality assumptions are met same.
Figure 5.3 Plot of Studentized residuals Vs Normal Quantiles
Figure 5.3: Plot of Studentized residuals Vs Normal Quantiles
40
Model Validation:
Figure 5.4 below shows the correlation of the actual N1% life obtained from the
experimentation and predicted N1% life obtained from the PCR Model.
Figure 5.4 Plot of Actual Vs the Predicted Life for the PCR Model for Cu Core
Assemblies
41
CHAPTER 6
PRINCIPAL COMPONENT REGRESSION ON NO COPPER CORE
ASSEMBLIES
An approach similar to the one discussed in Chapter 7 for the Ball Grid Array
Packages on copper core Assemblies is used for the assemblies with no Copper core
PCBs. A log transformation is done on all the predictor variables to have a better fit to the
dataset. Principal Component Regression is used to overcome the Multicolinearity which
exists between the predictor variables. Different Predictor variables like Area of the chip,
Chip to Package Ratio, Ball Count, Board finish, Die length, Die to body ratio, Ball
Pitch, Solder ball diameter, Weight of the package, Package Pad Diameter, Delta T and
Solder Volume have been selected as input variables in the model. The original Matrix X
of predictor variable has been transformed to a new matrix Z by multiplying it with the
Eigen vector matrix of the correlation coefficients. The contribution of the individual
variables has been checked for a 95 % level of significance and only those variables
which contribute significantly to form a linear model have been retained. The figure
below shows the contribution of individual principal components to the model. The main
aim for implementation of Principal Components here is to overcome the multi
colinearity and not dimensional reduction. The following Figure 6.1 is the plot of %
cumulative contribution of each Eigen value:
42
0
10
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5 6
Principal component number
Cu
mu
lat
ive
%
va
ria
tio
n
Figure 6.1: Contribution of each Principal Component for PCR of Cu Core
Assemblies
Stepwise regression led to the elimination of variables which do not contribute
significantly to form a linear regression model with 95 % confidence. The variables
which got eliminated in this process are: Board finish, Solder Ball diameter, Package
Weight, Ball Pitch and Package length.
Regression equation for Z predictor variables:
The regression equation obtained by regressing the Z predictor variables against
the Log transformed N1% life of the package is given by equation 6.1 below:
6Z*2462.25Z*7727.04Z*685.1
3Z*1781.32Z*857.41Z*856.2492.24%1LnN
+++
??+= Eqn 6.1
43
The Table 6.1 shown below represents a detailed result of the regression of 6
principal components against the Ln N1% life of the packages. The Pvalues of all the
predictors are less than 0.05 suggesting the statistical significance of the 6 predictors with
95% confidence.
Table 6.1: Transformed Z variable regression for PCR on No Cu Core Assemblies
Predictor Coef SE Coef T P
Constant 24.492 2.714 9.03 0
Z1 2.856 0.7961 3.59 0.001
Z2 4.5857 0.8566 5.35 0
Z3 3.1781 0.5787 5.49 0
Z4 1.685 0.433 3.89 0
Z5 0.7727 0.3517 2.2 0.033
Z6 2.2462 0.5403 4.16 0
The Table 6.2 below represents the Analysis of Variance used initially to prove
that the predictors have a linear relationship with the response variable N1%. The Pvalue
of < 0.05 suggests that at least one predictor has a significant linear relationship with the
response variable.
44
Table 6.2: Analysis of Variance for PCR on No Cu Core Assemblies
Source DF SS MS F P
Regression 6 19.51 3.22 21.32 0
Residual Error 51 7.77 0.15
Total 57 27.29
Table 6.3: Transforming Z back to Original Variables in the NL Model for Cu
Core Assemblies
Predictor Coef SE Coef T P
Constant 24.49 2.71 9.03 0
LnChipAreaSQMM 1.23 0.34 3.59 0.001
LnChipToPkgRatio 0.038 0.0071 5.35 0
LnBallCount 0.095 0.017 5.49 0
LnPkgPadDiaMM 6.54 1.68 3.89 0
LnDeltaTDEG C 1.79 0.815 2.2 0.033
LnSolderVolCUMM 0.38 0.091 4.16 0
The Principal Components are then transformed back to the original variables
using the same back transformation. Table 6.3 above gives a detailed result of the
regression between the log transformed original predictors and log N1% Life of the
package.
45
Regression equation for original predictor variables is given in the Equation 6.2 below:
MMLnSdrVolCU*38.0
GCLnDeltaTDE*79.1MMLnPkgPdDia*54.6tLnBallCoun*095.0
gRatioLnChipToPk*038.0SQMMLnChipArea*23.149.24%1LnN
?
?++
+?=
Eqn 6.2
Model Adequacy Checking:
Figure 6.2 Analysis of Residuals for PCR on No Cu Core Assemblies
From the above Figure 6.2, Plot of residuals Vs Fits we do not observe any
specific pattern which implies that the linearity assumptions are met.
The plot does not show any signs of the scatter increasing with the fitted values which
implies that the constant variance assumptions are satisfied.
46
Normality test:
The Shapiro?Wilk test was performed to check if the normality assumptions are satisfied.
A Pvalue of 0.4838 which is > 0.05 confirms the normality assumptions. Table 6.4
below gives details of the test.
Table 6.4 Results for the Shapiro Wilk test on No Cu Core Assemblies
Test Test Statistic (W) Pvalue
ShapiroWilk 0.980468 0.4838
Figure 6.3 Plot of studentized residuals Vs Normal Quantiles
47
The figure 6.3 above, plot of studentized residuals Vs Normal Quantiles also
produces points close to a straight line suggesting that the normality assumptions are met
same.
Model Validation:
Figure 6.4 below shows the correlation of the actual N1% life obtained from the
experimentation and predicted N1% life obtained from the PCR Model on No Cu Core
Assemblies
.
Figure 6.4 Plot of Actual Vs the Predicted Life for the PCR Model for No Cu Core
Assemblies
48
CHAPTER 7
POWER LAW DEPENDENCY OF PREDICTOR VARIABLES:
Power law relationship of predictor variables with N1% life have been developed
for various area array packages including PBGAs, flip chip BGA and CABGA packages.
These power law relationships form the basis of reliability models in determining the
appropriate family of transformations for linearizing the predictor variables for building
robust multiple linear regression models that describe the data more efficiently. The
power law relationship also help determining the appropriate transformation of predictor
variables for coping with multicollinearity, non normality and hetroskedasticity. The
power law dependence of predictor variables have been obtained using BoxTidwell
power law modeling and compared with traditional failure mechanics values.
BOX TIDWELL POWER LAW MODELLING:
BoxTidwell power law model attempts to model the power law dependence
between predictor variable and a response variable. The relationship is expressed as an
equation that predicts a response variable from a function of predictor variables and
parameters. The parameter is adjusted so that residual sum of squares is minimized. The
prediction equation is of the form given by the equation 7.1 below
( )?
=
?= n
1k
k0%1
kfaN Eqn 7.1
49
Where, parameter %1N on the left hand side of the equation represents the 1 percent
failures of threeparameter Weibull distribution for the PBGA packages when subjected
to accelerated thermomechanical stresses. The parameters on the right hand side of the
equation are the predictor variables or the various parameters that influence the reliability
of the package and the parameter ?k is the power law value obtained from box Tidwell
method.
The BoxTidwell method has been used to identify a transformation from the
family of power transformations on predictor variables. Box, et. al. [1962] described an
analytical procedure for determining the form of the transformation on regressor
variables, so that the relation between the response and the transformed regressor
variables can be determined. Assume that the response variable t, is related to a power of
the regressor,
( ) ( ) ??+?=???= 1010 ,,ftE Eqn 7.2
Where,
?
?? =? ??=?
?
0,xln
0,x ,
and ?o , ?1, ? are unknown parameters. Suppose that ?o is the initial guess of the constant
?. Usually the first guess is 10 =? , so that xx 00 ==? ? , or that no transformation at all
is applied in the first iteration. Expanding about the initial uses in Taylor series,
0
0d
),(df)(),(f)t(E ,0
01,0
?=?
?=??
?
?
?
???
?
?
??????+???= +
0
0
2
,0
22
0
d
),(fd
!2
)(
?=?
?=??
?
?
?
???
?
?
?????? +
50
0
0
3
,0
33
0
d
),(fd
!3
)(
?=?
?=??
?
?
?
???
?
?
?????? + ???.. +
0
0
n
,0
nn
0
d
),(fd
!n
)(
?=?
?=??
?
?
?
???
?
?
?????? Eqn 7.3
and ignoring terms of higher than first order gives the Equation 7.2 below,
0
0d
),(df)(),(f)t(E ,0
01,0
?=?
?=??
?
?
?
???
?
?
??????+???=
0
0d
),(df)1(x ,0
10
?=?
?=??
?
?
?
???
?
?
?????+?+?= Eqn 7.4
Now if the terms in braces in Equation 7.3 were known, it could be treated as an
additional regressor variable, and it would be possible to estimate the parameters ?o , ?1,
and ? by method of least squares. This way the value necessary to linearize the regressor
variable can be determined.
This procedure has been carried out for both the Copper core as well as no core
PBGAs for each of its predictor variable and the results are tabulated and compared with
power law dependence values obtained from failure mechanics method. The power law
dependence values obtained from BoxTidwell method are found be very close to the
power law dependence values obtained from failure mechanics models. Table 7.1 below
shows the comparison of these values:
Table 7.1: Comparison of Power Law Dependence values
Parameter BoxTidwell A B C
Cu Core
PBGAs
No Core
PBGAs
Die Length 2.7 1.2 2 2.3 2
Delta T 1.6 7.8 2.3 2 2
51
INTERACTION EFFECT MODEL:
Predictor variables for model building have been selected by developing a super
set of variables that are known to influence the characteristic life of an area array package
and then selecting the potentially important variables using stepwise regression and
method of best subsets. Coefficient of multiple determination, adjusted R2, residual mean
squares and induced bias has been used as criteria for variable selection. Coefficient of
multiple determinations (R2 which measures the overall adequacy of the regression model
and variables that create a significant increase in coefficient of multiple determination are
retained in the model. As coefficient of multiple determination increases marginally for
every newly added variable, adjusted R2 has been used for studying the overall adequacy
of the model and variables that create significant increase in adjusted R2 are retained in
the model. A PCR model with the interaction term between Delta T and Half Diagonal
Length along with the original predictor variables has been developed. The
shows the results for regression between the transformed Z variables as
predictors and N1% life as the response variable
Table 7.2: PCR Model for Cu Core Assemblies with the Interaction Effect between
Delta T and Half Diagonal Length
Predictor Coef SE Coef T P
Constant 2081.1 423.3 4.92 0
Z1 954.5 392.9 2.43 0.017
Z2 3908 1617 2.42 0.018
Z3 4587 1833 2.5 0.014
Z4 5355 2227 2.4 0.018
Z5 3189 1464 2.18 0.032
Z6 84.39 50.62 1.67 0.099
Z7 1362.6 863.9 1.58 0.118
Z8 2906 1389 2.09 0.039
52
The Analysis of Variance given below is used to check if a linear relationship
exists between the response variable and at lease one of the predictor variables.
Table 7.3: ANOVA Table for Interaction Effect Model
Source DF SS MS F P
Regression 8 4541750 567719 18.08 0
Residual Error 91 2858166 31408
Total 99 7399916
To establish the relationship between the Response variable and the original
predictor variables, the Principal components have to be back transformed using the same
back transformation which was used to convert them into Principal components. The
table below shows the relation between the response variable and the original predictor
variables.
Table 7.4: Transforming the Z`s Back to the Original Variables in the Interaction
Effect Model
Predictor variable
PCR
Coeffs.
S.E.
Coeffs
T
Statistic
P
Value
Constant 2081 422.96 4.92 0
HalfdiaglenMM 68.1 28.02 2.43 0.017
DieToBodyRatio 642.31 265.41 2.42 0.018
BallCount 0.5569 0.22 2.5 0.014
PkgPdArSQMM 1671.2 696.33 2.4 0.018
PkgWtGM 2.1949 1.00 2.18 0.032
Delta T DegC 8.4376 5.05 1.67 0.099
Halfdialen*
DeltaTMMoC 0.2965 0.18 1.58 0.118
SdrVolCUMM. 9107.4 4357.6 2.09 0.039
53
The regression equation is given below:
.
3
o
SdrVolMM*4.9107
DeltaT*HalfDiaLen*2965.0CDeltaT*44.8PkgWtGM
*1949.2MPkgPdArSQM*2.1671BallCount*5569.0
atioDieToBodyR*31.624nMMHalfDiagLe*1.682081%1N
?
+?
++?
??=
Eqn 7.5
Residual Model Diagnostics:
Figure 7.1 Model Adequacy Checking for Interaction effect model
From the Figure 7.1, the Plot of residuals Vs Normal Quantiles shows almost
straight line. The histogram is also more like a bell shape suggesting that the normality
assumptions are met. From the plot of residuals Vs Fits we do not observe any specific
pattern which implies that the linearity assumptions are met. The plot does not show any
54
signs of the scatter increasing with the fitted values which implies that the constant
variance assumptions are satisfied.
A Box ? Tidwell Transformation was done on the interaction term as the predictor
variable and the N1% Life of the package as the response variable to estimate the Power
of the interaction term. The power retained by using SAS is 1.42 whereas the Classical
models (NorrisLandzberg`s and Goldmann`s Equation) suggest a power transformation
of 2.
53000
54000
55000
56000
57000
58000
59000
60000
61000
2.5 2 1.5 1 0.5 0
Power
MS
E
rro
r
Figure 7.2: Plot of Mean Square Error Vs Power Transposed
The figure 7.2 above shows the change in values of the Mean Square Residual
with the change in Power of the response variable. The value of the mean square error is
lowest at the power transformation value of about 1.5 which is consistent with our value
of 1.42
55
CHAPTER 8
STATISTICAL FORM OF THE NORRIS LANDZBERGS MODEL
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, Landzberg 1969]. The equation 8.1 below
represents the same
)T(TTffNNAF max
2
U
A3
1
A
U
A
U ???
?
?
???
?
?
?
???
?
???
?== Eqn 8.1
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 given below [Lau 1997]
???
?
???
?
???
?
???
? ?
???
?
???
?
?
?
???
?
???
?==
Amax,Umax,
2
U
A3
1
A
U
A
u
T
1
T
11414exp
T
T
f
f
N
NAF Eqn 8.2
The Equation can be transformed by computing the natural Log format as follows:
( ) ??
?
?
???
? ?+
???
?
???
?
?
?+
???
?
???
?=
Amax,Umax,U
A
2
A
U
1 T
1
T
13C
T
TLnC
f
fLnCAFLn Eqn 8.3
56
This Model was initially developed by Norris and Landzberg [1969] of IBM for
controlled collapse chip interconnects for 595 SnPb solder composition on ceramic
substrate which had silverpalladium paste, tinned with 1090 SnPb solder deposition.
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 predictor variables and the Acceleration factor as the
response variable. The Solder composition used for this model is lead free SAC 305.
Due to the presence of Multicolinearity Principal Component Regression is
implemented. Regression results of the transformed Principal Components against the
Acceleration Factor are given in the Table 8.1 below:
Table 8.1: Transformed Z variable regression for NL model
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 8.2 below is used to check the presence of a linear
relationship between the predictor variables and any response variables. Pvalue less than
0.05 confirm the presence of a linear relationship between the response variable and
atlease one predictor variable.
57
Table 8.2: ANOVA Table for Z transformed Variables of NL model
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 given in the Table 8.3 below,
Table 8.3: Transforming Z back to Original Variables in the NL Model for Cu
Core Assemblies
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 Ta / Delta Tu) 2.3149 0.8722 2.6538 0.0094
(1/Tu1/Ta) 4562.3767 1721.45 2.6503 0.0095
The regression is given as follows
???
?
???
? ?+
???
?
???
?+
???
?
???
?+=
AUA
U
A
U
T
1
T
1Ln*3767.4562
T_Delta
T_DeltaLn*3149.2
F
FLn*3035.07448.0AF
Eqn 8.4
58
The NL model is given by [Lau 1997]:
???
?
???
?
???
?
???
? ?
???
?
???
?
?
?
???
?
???
?==
Amax,Umax,
2
U
A3
1
A
U
A
u
T
1
T
11414exp
T
T
f
f
N
NAF
Eqn 8.5
Writing the equation in the form of the NL equation:
???
?
???
?
???
?
???
? ?
???
?
???
?
?
?
???
?
???
?==
Amax,Umax,
31.2
U
A
3.0
A
U
A
u
T
1
T
14562exp
T
T
f
f
N
NAF
Eqn 8.6
The differences in the values of the constants are justified by the difference in the
solder joint composition of the two models. The original model [Norris, Landzberg 1969]
was developed for 595 SnPb Solder on ceramic substrates whereas, the model which we
have developed is for Lead Free SAC 305 solder composition for Plastic substrates. The
type of PCB in our study has an integral copper core which may also be one of the factors
for the difference in the values of the constants retained.
59
Figure 8.1 Model Adequacy checking for NL Model
From the Figure 8.1, the Plot of residuals Vs Normal Quantiles shows almost
straight line. The histogram is also more like a bell shape suggesting that the normality
assumptions are met. From the plot of residuals Vs Fits we do not observe any specific
pattern which implies that the linearity assumptions are met. The plot does not show any
signs of the scatter increasing with the fitted values which implies that the constant
variance assumptions are satisfied.
60
CHAPTER 9
STATISTICAL FORM OF GOLDMANN`S MODEL
L.S. Goldmann of IBM presented his work in mechanical reliability of controlled
collapse solder joints in May 1969. His main emphasis was on design variability and how
the shape and dimensions of solder joint and chip affect reliability. He presented a
systematic technique to optimize pad dimensions. His life prediction equation is
developed based on the Coffin Manson equation. He used the local shear strain as the
determinant parameter. 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
equation.
T..d
1
V
h
A
rKN
rel
mm12
fu
Tf
??=?
?????????
?
?
???
?
???
?
???
?
???
?
???
? ??= ??+
[Goldmann 1969] Eqn 9.1
Where,
Nf is number of cycles to failure,
u? is the ultimate shear strength of the critical interface.
rel? is the relative thermal expansion of the chip to substrate,
d is the distance from chip neutral point to interconnection,
T? is the temperature excursion of the cycle,
61
V is the volume of solder joint,
r is the radius of cross section under consideration,
h is the height of solder,
A and? are constants from plastic shear stressshear strain relationship
m is empirical constant in Coffin Manson Equation
The equation is rearranged as per our convenience and the values of the exponents
for 595 SnPb solder are given by Equation 9.2 below:
( ) ( ) ( ) ( ) ??
?
?
?
?
?
?
???
?
???
? ???= ??? 9.1275.32f9.19.19.1
relf hV
hrTdCN Eqn 9.2
The Figure below represents all the terms involved in the Goldmann`s Equation:
Solder Joint
Passivation
Printed Circuit Board
Copper Pad
Solder Mask Silicon Chip
h
d
Volume of Solder Joint (V)
Critical c/sradius (r)
Figure 9.1: Different predictor variables in the Goldmann`s model
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 Cu Core PCB:
62
A Log transformed X matrix is created using the original predictor variables.
The X matrix is given by:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
??
??
=
3559.02145.09053.49708.15627.12
.........................
1117.04721.01930.52911.2061.12
1117.02833.09053.42911.22061.12
1117.04721.09053.42911.22061.12
]x[
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([ ?? Eqn 9.3
0]I[]C[ =?? , or Eqn 9.4
0]I[]X[]X[ *T* =?? Eqn 9.5
Where ? the Eigen value and V is is the matrix of Eigen vectors.
The transformation matrix V of Eigen vectors of the correlation matrix is given by:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
??
?
?
???
=
6157.01005.01319.02966.0711.0
339.0613.05233.00182.04849.0
0384.06396.07386.02095.00074.0
5197.00527.02981.0797.00549.0
4841.04497.02728.04823.05062.0
]V[
63
The principal component matrix Z contains exactly the same information as the
original matrix, except that the data are arranged into a set of new variables which are
completely uncorrelated with one another and which can be ordered or ranked with
respect to the magnitude of their Eigen values (Draper and Smith 1981, Myers 1986).
The principal components matrix Z is obtained using the transformation:
]V[*]X[]Z[ = Eqn 9.6
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[}{ ?=? Eqn 9.7
Regressing the transformed Z variables against the N1% life of the packages, we get the
following results as shown in Table 9.1
Table 9.1 Transformed Z variable regression for Goldmann`s model of Cu Core
Assemblies
Predictor Coef SE Coef T P
Constant 17.014 8.375 2.03 0.047
Z1 0.8251 0.4777 1.73 0.09
Z2 0.703 0.4837 1.45 0.152
Z3 1.8552 0.3743 4.96 0
Z4 1.167 0.193 6.05 0
Z5 0.8535 0.3332 2.56 0.013
The overall adequacy of the model has been tested using ANOVA table given by
Table 9.1 above. 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
64
principal components yielded very small P values indicating the statistical significance of
all the five variables.
The Table 9.2 below shows the Analysis of variance in the statistical form of the
Goldmann`s model
Table 9.2: ANOVA Table for Z transformed Variables of Cu Core Assemblies
Source DF SS MS F P
Regression 7 9.61 1.3721 21.77 0
Residual
Error 90 5.67 0.063
Total 97 15.27
In order to obtain the relationship between the N1% life and original predictor
variables the Z transformed variables are transformed back using the same back
transformation
1*kk*k1*k }{]V[}{ ?=? Eqn 9.8
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 9.9 below:
2
1
l
1m
2
jm
1
m
pc,j
vMSE
bt
??
?
??
? ?
?
??
?
? ??
=
?
=
?
Eqn 9.9
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
65
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. The P values of individual T tests given by Table 9.3 below are < 0.05 proving
the statistical significance of individual regression coefficients of original predictor
variables at a 95 % confidence.
Table 9.3: Transforming Z back to Original Variables in the Goldmann`s Model for
Cu Core Assemblies
Predictor Coef SE Coef T P
Constant 2.651 4.014 0.66 0.511
???
?
???
? ?
V
hrln 2f
0.0495 0.0171 2.89 0.005
ln(h) 0.4121 0.054 7.64 0
ln(d) 0.3705 0.0476 7.77 0
ln(?rel) 1.3721 0.4369 3.14 0.002
ln(?T) 1.56 1.068 1.46 0.148
The regression equation between the N1% Life and the original predictors is
given by equation 9.10 below:
CLnDeltaT*56.1
C/lPPMReLnAlpha*3721.1LenLnHalfDiag*3705.0
LnBallHt*4121.0VhrLn*0495.065.2Life%1N
o
o
2
f
?
??
+??
?
?
???
??+?=
Eqn 9.10
66
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:
( ) ( ) ( ) ( ) ??
?
?
?
?
?
?
???
?
???
? ???= ??? 9.1275.32f9.19.19.1
relf hV
hrTdCN Eqn 9.11
Statistical form based on PCR for Goldmann`s Model for Copper Core assemblies is
given by Equation 9.12 below:
?
?
?
?
?
?
?
?
???
?
???
? ???= ??? 41.005.02f56.137.037.1
rel%1 hV
hrTdCN Eqn 9.12
The differences in the values of the constants are justified by the difference in the
solder joint composition of the two models. The original model was developed for 595
SnPb Solder whereas, the model which we have developed is for Lead Free SAC 305
solder composition. The type of PCB in our study has a integral copper core which may
also be one of the factors for the difference in the values of the constants retained.
67
Figure 9.2 Model Adequacy Checking for Goldmann model on Cu Core Assemblies
From the above 9.2, the Plot of residuals Vs Normal Quantiles shows almost
straight line. The histogram is also more like a bell shape suggesting that the normality
assumptions are met. From the plot of residuals Vs Fits we do not observe any specific
pattern which implies that the linearity assumptions are met. The plot does not show any
signs of the scatter increasing with the fitted values which implies that the constant
variance assumptions are satisfied.
Results for No Cu Core Assemblies:
A model similar to one developed for the Cu Core Assemblies is also developed
for the No Cu Core Assemblies. 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
68
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 No Cu Core PCB:
The procedure for PCR described in Chapter 4 is used to develop the model. The
results for regression of the transformed Z variables and the Predictor variable are given
in the Table 9.4 below:
Table 9.4 Regression of Z variables against N1% life in Goldmanns Equation for No
Cu Core Dataset
Predictor Coef SE Coef T P
Constant 2.54 10.96 0.23 0.818
Z1 1.2882 0.8402 1.53 0.131
Z2 0.4325 0.6319 0.68 0.497
Z3 1.7433 0.4698 3.71 0
Z4 0.8542 0.2402 3.56 0.001
Z5 0.4093 0.2628 1.56 0.125
The overall adequacy of the model has been tested using ANOVA table given by
Table 9.5 below. 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.
Table 9.5: ANOVA Table for Goldmanns Equation on No Cu Core Assemblies
Source DF SS MS F P
Regression 5 14.5313 2.9063 11.01 0
Residual Error 55 14.5236 0.2641
Total 60 29.0549
69
The Principal Components are then transformed back to the original variables
using the same back transformation. Table 9.6 below gives the detailed results of the
regression between the log transformed original predictors and log N1% Life of the
package.
Table 9.6: Transforming back to the original variables in the Goldmann equation
for No Core Assemblies
Predictor Coef SE Coef T P
Constant 2.54 10.96 0.23 0.818
???
?
???
? ?
V
hrln 2f
0.3733 0.244 1.53 0.131
( )hln 0.3109 0.457 0.68 0.497
( )dln 1.2119 0.327 3.71 0
( )rel?ln 1.2825 0.36 3.56 0.001
( )T?ln 1.5592 0.999 1.56 0.125
The regression equation for the model with its original predictors is given in the
equation 9.13 below
CLnDeltaT*56.1
C/lPPMReLnAlpha*2825.1LenLnHalfDiag*2119.1
LnBallHt*3109.0VhrLn*3733.054.2Life%1N
o
o
2
f
?
??
+??
?
?
???
????=
Eqn 9.13
We write the model in equation format to compare the values of constants
obtained from the PCR model with standard values for No Cu Core Assemblies.
70
Following are the two models:
Goldmann`s Model:
( ) ChrV)T()L(N
152.0
12
222
rel =??
?
?
???
?
?????? ???
?
?+
??? Eqn 9.14
Statistical model based on PCR for Goldmann`s Model:
?
?
?
?
?
?
?
?
???
?
???
? ???= ???? 3.037.02f6.12.13.1
rel%1 hV
hrTdCN Eqn 9.15
The differences in the values of the constants are justified by the difference in the
solder joint composition of the two models. The original model was developed for 595
SnPb Solder where as, the model which we have developed is for Lead Free SAC 305
solder composition.
Now we check if the assumptions of the linear regression model are satisfied,
Figure 9.3 Model Adequacy Checking for No Cu Core Goldmann Model
71
From the above Figure 9.3, the Plot of residuals Vs Normal Quantiles shows
almost straight line. The histogram is also more like a bell shape suggesting that the
normality assumptions are met. From the plot of residuals Vs Fits we do not observe any
specific pattern which implies that the linearity assumptions are met. The plot does not
show any signs of the scatter increasing with the fitted values which implies that the
constant variance assumptions are satisfied.
72
CHAPTER 10
MODEL VALIDATION
In order to determine the effect of individual design parameters on the thermo
mechanical reliability of the Cu Core PBGAs, the life of various packages was studied
and the effect of each parameter was measured by keeping all other parameters at a
constant level and varying just the parameter under consideration. The effect of
individual parameter which is gauged by the sensitivity factor is of a great help to build
confidence in tradeoff decisions. Results obtained from the statistical analysis using the
Principal Component Regression models were used to predict the life of the packages.
The convergence of the predicted values of life with the experimental data has been
demonstrated in this section.
Delta T:
A negative sensitivity factor for Delta T from the PCR models implies that the
thermomechanical reliability of CuCore PBGA packages reduces with increase in the
temperature range of ATC. The life obtained from the experimental data and PCR models
have been plotted against temperature differences of 180 and 135 deg C. The predicted
values from the prediction model follow the experimental values quite accurately and
show the same trend, as in Figure 10.1.
73
Actual N1%
PCR Predicted N1%
Figure 10.1: Effect of Delta T on N1% Life of the Packages assembled on Cu Core
PCBs
Solder Volume:
A negative sensitivity factor for Solder Volume from the PCR models implies that
the thermomechanical reliability of CuCore PBGA packages reduces with increase in
the solder volume. The life obtained from the experimental data and PCR models have
been plotted against the Solder volumes of 1200 and 720 MM3. The predicted values
from the prediction model follow the experimental values quite accurately and show the
same trend represented in Figure 10.2 This trend is supported by failure mechanics theory
as, increasing the solder volume would make the solder joint very stiff leading to
increased stress conditions resulting in higher hysteresis loops with more dissipated
energy per cycle.
74
0
100
200
300
400
500
600
700
800
1200 720
Solder Volume
N 1
% Actual N1%
PCR Pred_N1%
Figure 10.2 Effect of Solder Volume on Life of the Package assembled on Cu Core
PCBs
Die to Body Ratio:
A negative sensitivity factor for Die to body ratio from the PCR models implies
that the thermomechanical reliability of No Core PBGA packages reduces with increase
in Die to body ratio. The life obtained from the experimental data and PCR models have
been plotted against the Die to body ratio of 0.5 and 0.7407. The predicted values from
the prediction model shown in Figure 10.3 follow the experimental values quite
accurately and show the same trend. This is also consistent with the failure mechanics
standpoint, as the Die to body ratio increases the solder balls in the vicinity of the die
shadow region undergo much higher strains and are bound to fail faster.
75
Figure 10.3: Effect of Die to body ratio on the life of Package assembled on No Cu
Core PCBs
Half Diagonal Length:
The thermomechanical reliability of packages generally decreases with increase
in the half diagonal length. This effect has been demonstrated for Goldmanns model and
Cu Core Assemblies used to develop the same. The predicted values from the prediction
model follow the experimental values quite accurately and show the same trend. This
trend is also consistent from the failure mechanics standpoint, as the solder joints with
larger die length are subjected to much higher strains due to the increased distance from
the neutral point, thus having lower reliability. The figure 10.4 represents shows the
variation in the life with the variation of half diagonal length.
76
Actual N1%
Goldmann Predicted N1%
Figure 10.4: Effect of Diagonal Length on Life of Package Assembled on Cu Core
PCBs for Goldmanns Model
Model Validation plots for Norris Landzbergs PCR model:
In this section, the effect of individual parameters on the acceleration factor
predicted by NL Model for SAC305 area array assemblies has been validated. The
acceleration factor varies with a 0.3power with increase in the ratio of frequencies.
Model predictions agree with the experimental data. In addition, acceleration factor has
been shown to vary with a 2.3power of the temperature cycle magnitudes. The figures
10.5 and 10.6 given below represent the same.
77
Actual AF
NLZ Predicted AF
Figure 10.5: Effect of cyclic frequency on Acceleration Factor
Actual AF
NLZ Predicted AF
Figure 10.6: Effect of temperature cycle magnitude on Acceleration Factor
78
CHAPTER 11
SUMMARY AND CONCLUSION
A perturbation modeling methodology based on multiple linear regression,
principal components regression and power law modeling has been presented in this
research. The method provides an extremely cost effective and time effective solution for
doing tradeoffs and the thermomechanical reliability assessment of various Plastic BGA
packages, CABGA, Flipchip BGA subjected to extreme environments. This
methodology also allows the user to understand the relative impact of the various
geometric parameters, material properties and thermal environment on the thermo
mechanical reliability of the different configurations of BGA packages with leaded as
well as leadfree solder joints.
The model predictions from both statistics and failure mechanics based models
have been validated with the actual ATC test failure data. 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. It is recommended to use these models
79
for analyzing the relative influence of the parametric variations on the thermo
mechanical reliability of the package instead of using them for absolute life calculations.
Power law relationship of predictor variables with 63 % characteristic life have
been developed for various area array Packages. Interaction effects between different
parameters which are often overlooked are also presented in this work. These power law
relationships form the basis of reliability models in determining the appropriate family of
transformations for linearizing the predictor variables for building robust multiple linear
regression models that describe the data more efficiently. The power law values show
good conformance with failure mechanics values for most of the variables. Advanced
power law models can then be developed by transforming each predictor variable with its
appropriate power law transformation and then conducting a linear regression analysis.
Such power law transformed linear regression models can describe the data more
efficiently and resulting in better prediction models. Also, the power law lamda values
can be used for adding correction factors to existing first order failure mechanics models
and building power law based models.
Development of the classical failure mechanics equations like the Norris
Landzberg`s and the Goldmann equation in the statistical form has been presented. Log
transformation and has been used to convert the original multiplicative model to additive
model and the power values of the various terms involved in these equations are
compared to the ones obtained by statistical PCR model.
80
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98
APPENDIX
List of Symbols
? Coefficient of Thermal Expansion
? Coefficient of regression
?T Temperature Cycle Magnitude
? Model random error
?
0
Predictor Variable after PowerLaw Transformation (?
0
=
0
x
?
)
1/TmeanK Inverse of the mean temperature in Kelvin
AF Acceleration Factor
[A] Matrix of Predictor Variables, of full column rank
1/TmeanK Inverse of the mean temperature in Kelvin
AlphaRelPPMC Difference in CTE between part and PCB in ppm/C
BGA Ball Grid Array
BallCount Number of solder balls in the package
BallDiaMM Diameter of the solder ball in millimeters
BallHtMM Height of the solder ball in millimeters
ChipAreaSQMM Area of the chip in Sq. millimeters
CABGA Chip array BGA
Coef Coefficient
Cu Copper
99
DeltaTdegC Temperature cycle range in degree centigrade
DieLengthMM Chip Length in millimeters
DietoBodyRatio Ratio of the length of the chip to the length of the package
ENIG Electroless Nickel Immersion Gold
f
u
frequency of temperature cycle under use conditions
f
a
frequency of temperature cycle under accelerated test conditions
h Solder Joint Height
HalfDiagLenMM Half Diagonal Length of chip in mm.
HASL Hot Air Solder Leveling
k number of predictors
m Empirical Constant in CoffinManson Equation
MS
res
Mean Square of residuals
n number of data points
N
U
Life under Use Conditions
N
A
Life under Accelerated Test Conditions
p number of variables
PitchMM Solder Ball Pitch in millimeters
Prefix Ln Natural logarithm
PBGA Plastic Ball Grid Array
PCB Printed Circuit Board
PCR Principal Component Regression
PkgPadDiaMM Diameter of the package pad in millimeters
PkgPdAreaSQMM Area of the Package Pad in sq. millimeters
100
PkgWtGM. Weight of the package in grams
R
2
Multiple coefficient of determination
2
j
R Adjusted R Square
s Standard Deviation
SolderVolCUMM Volume of the solder in cubic mm
SS
res
Sum of Squares of residuals
T
max,U
Maximum Use Temperature
T
max,A
Maximum Accelerated Test Temperature
?T
U
Use Temperature Excursion
?T
A
Accelerated test temperature Excursion
V Volume of Solder Joint
[V] The k x k eigenvector matrix consisting of normalized
eigenvectors
VIF Variance Inflation Factor
X Predictor Variable
[X] Scaled and Centered Predictor Variable Matrix
Y Regressor Variable
[Z] The n x k matrix of principal components