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TU/e Mechanical Engineering
Computational and Experimental Mechanics
Warpage of Printed Circuit Boards Bachelor Final Project
MT05.19
L.A.A. Beex
Tutor: dr.ir. P.J.G. Schreurs
Eindhoven, May 18th 2005
1
Table of Contents
1. Introduction ........................................................................................................................................2 2. Model ...................................................................................................................................................3
2.1 Balance Laws..................................................................................................................................3 2.2 Geometry of a PCB.........................................................................................................................4 2.3 Materials ........................................................................................................................................7 2.4 Final Model ....................................................................................................................................9
3. Experiment........................................................................................................................................ 12 3.1 Measuring Device......................................................................................................................... 12 3.2 Experiment.................................................................................................................................... 13
4. Results ............................................................................................................................................... 15 4.1 Comparison between Experimental Data and Numerical Analysis .............................................. 15 4.2 Variation of Different Parameters................................................................................................ 19
5. Conclusion......................................................................................................................................... 23 Symbols ................................................................................................................................................. 24 Literature .............................................................................................................................................. 25 Acknowledgements ............................................................................................................................... 26
2
1. Introduction
In electronic devices the components, like chips, are mounted on a Printed Circuit
Board (PCB) by solder joints. The components are connected with each other with
copper wires, integrated in the PCB. The chips are often attached to the PCB with
pins, but due to miniaturisation an increased use of solder balls, that are much smaller
than pins, can be seen (see Figure 1.1). Melted solder paste is used during the solder
reflow process to attach the chips to the PCB. During this process the temperature can
increase to 2600 Celsius.
When the chips are operating they generate heat. This heat is conducted by the solder
balls to the PCB so this will locally get very hot. The PCB will locally expand and it
will warp, because it has a different coefficient of thermal expansion than the chips
and because it is a laminate of fiberglass-reinforced epoxy and thin copper layers that
both have different coefficients of thermal expansion. Due to the warpage, the stresses
in the solder balls will increase and eventually they may break; a frequently occurring
problem in electronics. It is also possible that the PCB gets damaged by the warpage;
for example delamination can occur.
Warpage is an important factor for the lifetime of electronic devices and it is
necessary to know and predict it. Mostly this is done with experiments (see Chapter
3). The problem is that making a prototype PCB and doing the experiments takes a lot
of time. Another disadvantage is that the PCBs in these tests can only have a uniform
temperature. Also they can’t be tested with chips soldered on it. The aim is to make an
accurate finite element model which describes the behavior of a PCB. This is
especially important to simulate the solder reflow process and cycling loads. Cycling
loads are important because they will occur in daily use of the electronic device.
Warpage will occur every time it is switched on and eventually the solder balls may
break by fatigue.
This report describes how the model is made and how it can be used for variation of
different parameters. Firstly the model will be described on the basis of the balance
laws that are used in the model, the geometric properties and material properties of
PCBs. Eventually there are two models created, that are compared to each other. An
experiment will be considered to have a reference for the numerical analyses. In this
way it is possible to check if the model is accurate enough. After that, variation of
parameters will be discussed. Finally the most important points will be summarised in
the conclusion.
Figure 1.1: A Printed Circuit Board in detail. Notice the thin copper lines
and the solder balls on top of the PCB.
3
2. Model
A PCB with chips in an electronic device has a complex geometry and is subjected to
thermo mechanical loads that are a function of position and time. Therefore it is
necessary to make a simplified model that can be analysed with a finite element
method. A simplification of the problem is pictured in Figure 2.1. The chips are
considered as blocks. Via solder balls they are attached to the PCB which is
considered to be a thin plate, in fact a laminate built up from several layers with
different material properties. In Figure 2.1 it is fixed at the four corners but there are a
number of other ways to attach it.
If the chips are operating they generate heat that is conducted to the PCB which will
warp because it has a different coefficient of thermal expansion than the chips and
because the different layers have different coefficients of thermal expansion. All the
heat leaves the PCB via the supports, because it is assumed that they have the same
temperature as the surrounding and because the heat transfer to the air is ignored.
Figure 2.1:
Simplification of a PCB
with chips. In blue the
chips, in brown the
PCB and in black the
supports.
There are three issues that are important to model this problem. Firstly it is important
to use the right balance laws together with proper initial and boundary conditions.
Secondly the geometric properties of a PCB are important. Finally it is relevant to
know which materials are used in a PCB and chips because the different material and
thermal properties are necessary for the numerical analysis. These three issues are
described in different sections.
2.1 Balance Laws
If a PCB is not uniformly heated, e.g. by chips, generating heat, two balance laws are
relevant for the analysis. Firstly it is necessary to calculate the heat that is conducted
in the PCB. Therefore we use the law of balance of energy:
RDTkTcp ρσρ ++∇⋅∇= :)(& [2.1.1]
In the model of the PCB with chips there is no heat source, therefore the term ρR
equals zero. It is also assumed that there is no plastic strain and thus no dissipation, so
the term σ:D equals zero. To solve the equation thermal boundary conditions are
needed. Every point on an outside surface or edge needs a thermal boundary
condition. If a point is mechanically connected to the electronic device, it is assumed
that this point has the same temperature as the surrounding (room temperature). In this
case a fixed temperature is prescribed. This is called a Dirichlet boundary condition or
essential boundary condition.
4
All the other points, on an outside surface or edge and for which there isn’t a fixed
temperature prescribed, also need a thermal boundary condition. It is assumed that
there is no outward energy flux. The heat transfer to the air is ignored. In real
applications however this is substantial part of the total heat transfer. It is mostly
increased by the use of a ventilator that blows cold air over the PCB. If the heat
transfer is prescribed and equals zero, the boundary condition is called a
homogeneous Neumann boundary condition. This is a natural boundary condition.
The law of balance of energy is a partial differential equation in space and time, so we
have to formulate also thermal initial conditions. In all the cases in this report the
initial temperature is room temperature. Therefore the thermal initial conditions are
200 Celsius. When the chips generate heat, this is simulated by giving them
immediately a temperature of 1000 Celsius.
The PCB will therefore locally expand and stresses will be generated. The law of
balance of momentum is necessary to calculate deformation and stress state:
vq &ρρσ =+⋅∇ [2.1.2]
The term qρ equals zero because the gravitation forces are relatively very small and
thus neglected. The term v&ρ also equals zero because there are no accelerations in a
PCB. To solve this equation mechanical boundary conditions are necessary. For
different situations there are different mechanical boundary conditions, which will be
explained in section 4.2. If a point is fixed, it is assumed in this study that every
degree of freedom is zero.
Both these equations have to be solved simultaneously, because the solutions of the
two equations depend on each other. This can’t be done analytically so a software
program has to be used. The software program that is used to analyse this problem is
Marc/Mentat. This is a combination of the two programs Marc and Mentat. Mentat is
the program that is used to create the model and Marc is the program used for
calculations. In Mentat the PCB and chips have to be divided in elements. It is
necessary that there are enough elements because otherwise the numerical analysis
isn’t accurate enough. If there are too many elements however, the numerical analysis
will take a lot of time, so an optimum has to be reached. The time has to be divided in
increments. These time-increments have to be small enough because otherwise large
steps in the solution may occur. On the other hand if the time-increments are too small
the numerical analysis will take a lot of time. So for the time-increments it is also
necessary that an optimum has to be reached. In every numerical analysis there are 50
time steps taken to be sure that the heat transfer is in steady-state. Marc/Mentat is able
to solve the two balance laws simultaneously with the function ‘coupled’. To solve
them we need to know which materials are used in PCBs and chips. In section 2.3
they will be described.
2.2 Geometry of a PCB
PCBs come in different sizes. In cell phones for example they are just a couple of
centimeters, but in computers they are much bigger. PCBs can have different numbers
of copper layers. In this study the number of copper layers is one of the parameters for
variation. The copper layers are mostly 18 or 35 micrometer thick. In this study all the
copper layers are 35 micrometers.
5
PCBs are thin plates that are exposed to bending and warpage. In Marc/Mentat they
can be modeled with shells or bricks. These two elements have to be compared with
each other to know which one can best be used.
Element 75 can best be used if we use shells. In Figure 2.2.1 element 75 is pictured. It
is a four-node element. Every node has six degrees of freedom (three displacements
and three rotations) and is therefore very good to describe bending.
If we use bricks to model the PCB, element 7 can best be used (see Figure 2.2.2).
Element 7 has 8 nodes and every node has three degrees of freedom. The
displacements are the only degrees of freedom. Therefore element 7 doesn’t describe
bending well. However it is possible to describe bending with element 7, but in that
case enough layers of element 7 must be used over the thickness. The numerical
analysis will therefore take a lot of time. If the model is made with shells, there is no
need to use more than one layer, because shells describe the bending very well. The
numerical analysis will be relatively fast in that case. Also the use of composites
(laminates) in Marc/Mentat is a lot easier for shells. Therefore the model of the PCB
is made with shells.
Figure 2.2.1: Element 75 in Marc/Mentat. Figure 2.2.2: Element 7 in Marc/Mentat.
For small deformations of shells the linear bending theory can be used. On the hand of
Figure 2.2.3 the linear bending theory can easily be explained.
Figure 2.2.3: Displacement components of out-of-mid-plane
points: P and P0 in undeformed and deformed state.
The points P and P0 are shown in undeformed and deformed state. If the mid-plane
bends, the angle φ becomes bigger than zero. The angle φ can be described in terms of
w, the displacement of a mid-plane-point in z-direction:
0>∂
∂=
x
wϕ [2.2.1]
6
The curvature can be described with the following formula:
01
2
2
<∂
∂−=
∂
∂−==
xx
w
Rx
xx
ϕκ [2.2.2]
For a point out of the mid-plane, the linear strain components can be expressed in the
mid-plane strains and curvatures:
xxxxxx zκεε += 0 [2.2.3]
yyyyyy zκεε += 0 [2.2.4]
xyxyxy zκγγ += 0 [2.2.5]
We can apply the linear bending theory on laminates. The difference with the normal
linear bending theory is that every ply (k) has different material properties. In Figure
2.2.4 part of a laminate is pictured.
Figure 2.2.4: Schematic picture of a laminate.
The chips are modeled with bricks because they are considered as blocks instead of
thin plates. For the chips bending isn’t as important as for the PCB.
However shells have a thickness, it isn’t visible in Marc/Mentat. Only the mid-plane
of the shells is modeled in Marc and shown in Mentat (see Figure 2.2.1). That is why
it can be preferable to place the bricks not directly on top of the shells. In this case the
bricks are placed just above the mid-plane, at a height of half the thickness of the
PCB. After that, the bricks are attached to the mid-plane with ties. A tie links all the
degrees of freedom and the temperature of one node to another one.
An alternative is that the chips are placed directly on the mid-plane. Therefore it isn’t
necessary to tie the bricks to the mid-plane. In this case the model is a bit extended.
An advantage of this model is that you can even model the solder balls and the
adhesive. A disadvantage of this model is that the chips are placed in the PCB. The
results of the comparison between the two models can be found in section 2.4.
It is also important to know how PCBs are supported. In electronic devices PCBs are
sometimes fixed at all the edges. In section 4.2 different kinds of fixations are tested.
When all the four edges are fixed, the temperature is also fixed on these edges, which
means that all the heat that the chips generate leaves the PCB via these supports.
Mostly a part of the heat leaves the PCB via the supports and the other part leaves it
via the air that is blown over it, but in our model the heat transfer to air is ignored. It
is also possible that a PCB is fixed on three edges, so that it can be relatively easily
removed. This is often the case in personal computers. In this case it doesn’t only
affect the mechanical boundary conditions but also the thermal boundary conditions.
In the experiment the PCB is not fixed but simply supported on two bars in an oven.
In this case it is necessary to describe mechanical boundary conditions that simulate
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the placement on the two bars. The PCB will be uniformly heated because it is placed
in an oven. Therefore there is no heat conduction in the PCB itself. In this case there
is no need to calculate the heat conduction and to formulate the thermal boundary
conditions.
2.3 Materials
PCBs are made of two different materials: FR-4 (fiberglass-reinforced epoxy) and
copper (metal). The layers of copper between the FR-4 connect the electronic
components with each other. They are very thin, mostly 18 or 35 micrometer. PCBs
are laminates of layers FR-4 and copper. Laminates can easily be modeled in
Marc/Mentat. Firstly the two different materials have to be formulated and then you
can create a laminate, as shown in Figure 2.3.1.
Figure 2.3.1: Laminate structure of a PCB with three layers of copper and four layers of FR-4, as modelled in Marc/Mentat. In
red the layers of copper and in blue the layers of FR-4. Notice that the layers of copper are very thin compared to the the layers
of FR-4. This aren’t the real colors of the materials.
Copper is a metal and is isotropic. In Table 2.3.1 the relevant material properties of
copper are shown as used in the model. They aren’t temperature dependent.
Young’s modulus E [GPa] 110
Poissons’s ratio ν [-] 0,34
Shear modulus G [GPa] 41,0
Coefficient of thermal expansion α [0 Celsius
-1] 17e-6
Heat conduction coefficient k [W/m*K] 398
Specific heat cp [J/Kg*K] 386
Mass density ρ [kg/m3] 8930
Table 2.3.1: Relevant material properties of copper as used in the model. Marc/Mentat calculates the shear modulus G itself
with.
FR-4 is a fiberglass-reinforced epoxy. This means that FR-4 is a polymer with a
fiberglass woven through the matrix structure of the polymer. This is shown in Figure
2.3.2. The fiberglass is much stronger than the polymer. The fiberglass is oriented in
the PCB-plane, so FR-4 is in the PCB-plane much stronger than in the thickness-
direction. FR-4 is therefore orthotropic. The relevant material properties of FR-4 are
shown in Table 2.3.2. They are temperature dependent because FR-4 has a glass-
transition temperature of 1300 Celsius that only affects the Young’s and shear
modulus.
8
Figure 2.3.2: Cross section of FR-4. Light
gray is the matrix structure of the polymer
and dark gray is the woven fiberglass.
300 C 95
0 C 125
0 C 150
0 C 270
0 C
Young’s modulus E11 [GPa] 22,4 20,7 19,3 17,9 16,0
Young’s modulus E22 [GPa] 22,4 20,7 19,3 17,9 16,0
Young’s modulus E33 [GPa] 1,6 1,2 1,0 0,6 0,45
Poisson’s ratio ν 12 [-] 0,02 0,02 0,02 0,02 0,02
Poisson’s ratio ν 23 [-] 0,02 0,02 0,02 0,02 0,02
Poisson’s ratio ν 31 [-] 0,143 0,143 0,143 0,143 0,143
Shear modulus G12 [GPa] 11,0 10,1 9,5 8,8 7,8
Shear modulus G23 [GPa] 11,0 10,1 9,5 8,8 7,8
Shear modulus G31 [GPa] 0,70 0,52 0,44 0,26 0,20
Coefficient of thermal expansion α 11 [0 Celsius
-1] 20e-6 20e-6 20e-6 20e-6 20e-6
Coefficient of thermal expansion α 22 [0 Celsius
-1] 20e-6 20e-6 20e-6 20e-6 20e-6
Coefficient of thermal expansion α 33 [0 Celsius
-1] 86,5e-6 86,5e-6 86,5e-6 86,5e-6 86,5e-6
Heat conduction coefficient k [W/m*K] 0,03 0,03 0,03 0,03 0,03
Specific heat cp [J/Kg*K] 1800 1800 1800 1800 1800
Mass density ρ [kg/m3] 1500 1500 1500 1500 1500
Table 2.3.2: Relevant material properties of FR-4 as used in the model.
It is also needed to know the material properties of a chip. In Figure 2.3.4 a chip is
pictured. It consists for the largest part of resin. The chip is therefore modeled as a
block of resin. The relevant material parameters of the resin are shown in Table 2.3.3.
They can be considered as temperature independent because we assume that the glass
transition temperature of resin will never be reached.
Figure 2.3.4: Internal structure of a chip.
Table 2.3.3: Relevant material properties of
resin as used in the model.
In one model of a chip Sn63Pb37 is used to model the solder joints, that are attached
to the PCB, and the adhesive that is used to attach the chip to the PCB. The relevant
material parameters of Sn63Pb37 are shown in Table 2.3.4 and those of the adhesive
Young’s modulus E [GPa] 20,0
Poissons’s ratio ν [-] 0,15
Shear modulus G [GPa] 8,7
Coefficient of thermal expansion α [0 Celsius
-1] 15e-6
Heat conduction coefficient k [W/m*K] 0,03
Specific heat cp [J/Kg*K] 1800
Mass density ρ [kg/m3] 1500
9
are shown in Table 2.3.5. Sn63Pb37 has a melting point of 1830 Celsius, but because
this temperature is never reached in a normal situation, the material properties can be
considered temperature independent for this study.
Young’s modulus E [MPa] 30,0
Poissons’s ratio ν [-] 0,4
Shear modulus G [MPa] 10,7
Coefficient of thermal expansion α [0 Celsius
-1] 25e-6
Heat conduction coefficient k [W/m*K] 50
Specific heat cp [J/Kg*K] 178
Mass density ρ [kg/m3] 8340
Table 2.3.4: Relevant material properties of Sn63Pb37 as used in the model.
Young’s modulus E [GPa] 20,0
Poissons’s ratio ν [-] 0,3
Shear modulus G [MPa] 7,7
Coefficient of thermal expansion α [0 Celsius
-1] 15e-6
Heat conduction coefficient k [W/m*K] 0,03
Specific heat cp [J/Kg*K] 1800
Mass density ρ [kg/m3] 1500
Table 2.3.5: Relevant material properties of the adhesive as used in the model.
2.4 Final Model
In Figure 2.4.1 a PCB is shown which is used for the variation of different parameters.
The variation itself will be considered in section 4.2. The dimensions of this PCB are
200×150×1,55 millimeter. It has two layers of FR-4 and one layer of copper in the
middle. This laminate is shown in red. In blue this same laminate is shown, but it has
an extra copper layer of 35 micrometer on top. The chips can be attached to this extra
layer.
Figure 2.4.1: Final model of the PCB. All the geometries are shells.
The laminate has on layer of copper of 35 micrometer in the middle.
This is shown in red. In blue the laminate with an extra layer of copper
of 35 micrometer on top to attach the chips to the PCB. Nodes aren’t
shown.
For the chips two models are created. One model (from now on it will be called model
with ties) consists out of bricks, which represent the resin. They are attached to the
mid-plane of the PCB via ties (see Figure 2.4.2). The other model (model without
ties) consists only out of bricks. In this model the bricks are directly attached to the
mid-plane of the PCB. They represent the solder joints, the resin and the adhesive
between the chip and the PCB (see Figure 2.4.3).
Figure 2.4.2: The chip with ties (in red). In brown the bricks are
pictured which represent the resin.
10
Figure 2.4.3: The chip without ties. In pink the bricks that
represent resin, in red the adhesive and in blue the solder balls.
In Figure 2.4.4 the final model of the PCB with tied chips is shown. On the right side
there are small chips placed, but they are in principal the same as the large chips.
Figure 2.4.4: The final model of the PCB with chips with
ties. In pink the shell elements of the PCB and in brown
the brick elements which represent the resin of the chips.
Now we have to compare the two models with each other. For the comparison all the
edges of the PCB are fixed in. This implies that all the nodes on the edges have a
fixed temperature of 200 Celsius. The chips have a temperature of 100
0 Celsius. At the
start of the analysis the PCB and the chips have a temperature of 200 Celsius. In
Figure 2.4.5 the result is shown for the model with ties. Only the displacement in the
thickness-direction (z-direction) matters because the other displacements are
relatively small.
Figure 2.4.5: The result of the model with ties is shown. The chips are 1000 Celsius and the edges are 200 Celsius. The colours
represent displacements in z-direction. The maximum displacement in z-direction is 3,74*10-5 meter (yellow) and the minimum
displacement is -9,09*10-7 meter (blue).
To compare the two models with each other the displacements in z-direction of the
midline in length-direction are pictured in Figure 2.4.6. The z-displacements for the
model without ties are bigger than for the other model. This can be the consequence
of the fact that the chips are directly placed on top of the mid-plane, which means that
the chips are placed in the PCB. Unfortunately it can’t be said which model represents
the reality best, because there are no experimental data about this PCB. In Figure
2.4.7 the results are shown for three different situations of the two models.
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-1
0
1
2
3
4
5
6x 10
-5
arclength [m]
dis
pla
cem
ent
in z
-direction [
m]
model with ties
model without ties
Figure 2.4.6: The displacement of the PCB in z-direction as a function for the midline in the length-direction for the model with
and without ties.
Figure 2.4.7: In the first picture the model with ties is now only fixed at three edges. In the second picture it is fixed at one edge.
The undeformed model is shown with yellow lines. In the last picture the model without ties is used but there are now only four
chips placed on the PCB. The different colours represent displacements in z-direction.
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3. Experiment
An experiment is carried out to check the accuracy of the model. The experiment is
done at Philips Applied Technologies in Eindhoven. In this experiment two PCBs
without chips are heated in an oven. A video camera, interfaced to a computer,
measures the z-displacement. The two PCBs are also modeled and analysed in
Marc/Mentat. The numerical analysis is compared with the experimental data to check
if the model is good enough. In this chapter the measuring device and the experiments
will be discussed. The results of the experiment and the numerical analysis will be
discussed in section 4.1.
3.1 Measuring Device
In the experiment two PCBs are placed in an oven and are uniformly heated. A video
camera interfaced to a computer measures the z-displacement. The measuring device
is based on the Shadow Moiré Principle (see Figure 3.1.1). A light source shines
through the grating on the sample at an angle of about 450. If this sample isn’t bend,
all the light that shines on the sample is observed by the video camera that is placed
right above the sample. But if the sample has a curvature, not all the light that shines
on the sample can pass the grating and thus a series of dark and light Moiré fringes is
produced. These fringes are detected by the video camera. The video camera is
interfaced with a computer with Fringe Analysis Software that is able to analyse the
image of the fringes. The measuring device can only measure displacements in the z-
direction, because other displacements don’t produce series of dark and light Moiré
fringes.
Figure 3.1.1: Schematic picture of the Shadow Moiré Principle. A light source shines through the grating on the sample. In this
picture only a part of the light is observed by the video camera due to the warpage of the sample.
The measuring device that is used is called the Akrometrix PS88+ (see Figure 3.1.2).
It can detect height differences of 5 micrometer. The maximum temperature of the
oven is 3000 Celsius. The temperature is measured with the same computer that
analyses the images of the fringes. In this way the correct temperature can be linked to
the image. The surface of the sample has to be continuous: abrupt steps in the surface
can influence the measurement negatively. Therefore only PCBs without chips can be
tested. The samples are often painted white because the video camera of the
13
Akrometrix isn’t able to detect small contrast differences. The sample is placed on
two bars. The result of one measurement is a 3D-surface-plot of the z-displacement
and the z-displacement as a function of the diagonals.
Figure 3.1.2: The Akrometrix PS88+. The black box bottom left is the oven. Above the oven the video camera (black) is placed.
From right the light source shines on the oven. Bottom right the computer with Fringe analysis software is placed.
3.2 Experiment
In the experiment there are two PCBs tested and modeled in Marc/Mentat. Though the
PCB is placed on two bars in the oven, it still needs mechanical boundary conditions
in Marc/Mentat. Because the PCBs are placed in an oven they are uniformly heated.
Therefore there will be no heat conduction and thus it isn’t necessary to calculate it.
During the experiment every 15 seconds a measurement is done. Both PCBs are
heated twice. The first time, they are heated to 1000 Celsius with a velocity of 20
0 per
minute. The temperature of 1000 Celsius is maintained for 20 minutes. This is a
normal situation in an electronic device when it is operating. The second time, they
are heated to 2600 Celsius with a velocity of 20
0 per minute. The temperature of 260
0
is maintained for one minute. This situation simulates the solder reflow process.
In Figure 3.2.1 the large PCB and the model of the large PCB made in Marc/Mentat
are pictured. Its dimensions are 200×150×1,55 millimeter. The surface that is
measured is 180×145 millimeter. It isn’t possible to measure the whole surface
because the device isn’t able to measure the abrupt steps at the edges. The PCB isn’t a
laminate because it has no copper layer. Only the small yellow lines on top of the
PCB are copper layers of 35 micrometer. Two samples of the large PCB are tested.
The small PCB that is tested is pictured in Figure 3.2.2. This one is also not a laminate
because it has no layer of copper. Only on top it has small lines and surfaces of copper
layer of 35 micrometer. The measured surface is 145×145 millimetre. The results are
discussed in section 4.1. Only one sample of the small PCB is tested.
14
Figure 3.2.1: The large PCB (left) and the model of the large PCB in Marc/Mentat (right). Green and blue is FR-4 and yellow
and red is FR-4 with a copper layer of 35 micrometer on top. The dimensions are 200×150×1,55 millimetre.
Figure 3.2.2: The small PCB (left) and the model of the small PCB in Marc/Mentat (right). Green and blue is FR-4 and yellow
and red is FR-4 with a copper layer of 35 micrometer on top. The dimensions are 150×150×1,55 millimetre.
15
4. Results
This chapter contains two sections. In the first section the results of the experiment
will be pictured and compared to the numerical analysis of the experiment. In the
second section the results of three parameter variations will be discussed.
4.1 Comparison between Experimental Data and Numerical Analysis
In this section the models made in Marc/Mentat will be compared to the measured
data of the real PCBs. This is necessary to see if the models describe reality well. Of
the large PCB two samples are tested; they will be called sample A and sample C. The
sample of the small PCB will be called sample B. The three samples are tested at 960
and 2600 Celsius.
Firstly it is important to know the warpage of the samples at room temperature. In
Figure 4.1.1 the three samples are pictured. As you can see they already are warped at
room temperature. However the models made in Marc/Mentat are perfectly straight at
room temperature.
Figure 4.1.1: The displacements in z-direction for the three samples (A, C and B) are given in a 3D-figure (left). The
temperatures are for A, C and B respectively 340, 280 and 300 Celsius (almost room temperature). The legends at the three 3D-
surface-plots are in micron. The graphs on the right are the z-displacements of the two diagonals for each sample.
16
In Figure 4.1.2 the results of the samples A and C and the numerical results of the
model are shown. The temperature is 960 Celsius. As you can see the general form of
the 3D-surface-plot of the model is almost the same as the 3D-surface-plots of the two
samples. The values of the z-displacements however are not the same. Not the
absolute values of the z-displacements matter, but only the relative values; the
difference between the maximum and minimum z-displacement. As you can see the
two samples show more warp than the model. The relative z-displacements of the
model are in a range of about 130 microns, while the relative z-displacements of the
samples are in a range of about 500 microns. A possible explanation for this
difference is that the samples are already warped at room temperature.
Figure 4.1.2: The displacements in z-direction for the two samples (A and C) and the model of sample A and C are given in a
3D-figure (left). The legend at the 3D-surface-plot of the model is in meter, the other legends are in micron. The temperature is
960 Celsius. The graphs on the right are the z-displacements of the two diagonals for each sample and the model.
The results of sample B and the numerical results of the model are shown in Figure
4.1.3. The temperature is 960 Celsius. As you can see the general form of the 3D-
surface-plot of the model is almost the same as the 3D-surface-plot of the sample. The
values of the z-displacements are also almost the same. The relative z-displacements
of the model are in a range of about 300 microns and the relative z-displacements of
the samples are also in a range of about 300 microns.
17
Figure 4.1.3: The displacements in z-direction for sample B and the model are given in a 3D-figure (left). The legend at the 3D-
surface-plot of the model is in meter, the other legend is in micron. The temperature is 960 Celsius. The graphs on the right are
the z-displacements of the two diagonals for the sample and the model.
Now the models have to be checked for temperatures above the glass-transition
temperature. The temperature in the next tests is 2600 Celsius. In Figure 4.1.4 the
results of the samples A and C and the numerical results of the model are shown. As
you can see in the 3D-surface-plot of sample A some wrong measurements are made,
but this isn’t a real problem because the general form is still visible and the z-
displacements as function of the diagonal can still be plotted. As you can see the 3D-
surface-plot of the model doesn’t look like the one of the samples. Also the z-
displacements of the diagonals of the samples and the model don’t match. The relative
z-displacements of the model are much bigger than those of the samples.
In Figure 4.1.5 the results of sample B and the numerical results of the model are
shown. The temperature is 2620 Celsius in the test and in the model. The 3D-surface-
plot of the model doesn’t look like the sample at all. Also the z-displacements of the
diagonals of the samples and the model don’t match. The relative z-displacements of
the model are much bigger than those of the samples. A possible explanation for this
aberration can be the fact that the coefficient of thermal expansion in the models isn’t
temperature dependent.
The model is not accurate for temperatures above the glass-transition temperature of
FR-4, because the models don’t describe the behavior of all the samples above the
glass-transition temperature well. Therefore the model can’t be used to simulate for
example the solder reflow process. However if the temperature in a simulation doesn’t
reach the glass-transition temperature the simulation can be considered as accurate
enough. Thus cycling loads for example can be tested, because in that case the
temperature doesn’t reach the glass transition temperature of FR-4.
18
Figure 4.1.4: The displacements in z-direction for the two samples (A and C) and the model of sample A and C are given in a
3D-figure (left). The legend at the 3D-surface-plot of the model is in meter, the other legends are in micron. The temperature is
2600 Celsius. The graphs on the right are the z-displacements of the two diagonals for the samples and the model.
Figure 4.1.5: The displacements in z-direction for sample B and the model are given in a 3D-figure (left). The legend at the 3D-
surface-plot of the model is in meter, the other legend is in micron. The temperature is 2620 Celsius. The graphs on the right are
the z-displacements of the two diagonals for the sample and the model.
19
4.2 Variation of Different Parameters
A lot of parameter variations can be researched with the two models. In this section
three parameter variations will be considered. The first parameter is how a PCB can
best be fixed so that the warpage is minimized. The second parameter is the number
of copper layers. For these two parameters the model with ties is used. The model
without ties is used for the last one. In that case four chips are placed on the PCB;
three large chips and a small one. The question is how the chips can best be placed so
that the warpage is minimized. In all cases the fixed edges have a temperature of 200
Celsius and the chips are 1000 Celsius.
The first parameter that is investigated is how a PCB can best be fixed. The model
with ties is used for the PCB with chips in the next analyses (see Figure 2.4.4). There
are five situations tested (see Figure 4.2.1). These situations are all possible ways to
fix a PCB in electronic devices. There are more fixations possible but it was needed to
make a selection.
Figure 4.2.1: Five different kinds of fixations of a PCB. In black the edges and points that are fixed in (all the degrees of freedom
are fixed). The temperature on the black edges and points is fixed at 200 Celsius. The chips are 1000 Celsius. In pink the PCB and
in brown the chips..
The results are shown in Figure 4.2.2. The z-displacements are plotted for the midline
in length-direction. They are the smallest for situation A and for situation C they are
the largest. This doesn’t automatically mean that situation A is the best and C the
poorest. In this study it is namely assumed that the loads on the solder balls are
determined by the curvature of the PCB on the place of every chip. The assumption
that the loads on the solder balls are determined by the curvature is in fact a
simplification of a very complex problem. So it is necessary to calculate the curvature
with Formula 2.2.2. These are only curvatures for chips that are placed almost at the
midline in length-direction. It is assumed they have the largest curvature. The results
of the curvatures for the different situations are shown in Table 4.2.1. There is no
difference made between the directions of the curvatures. This means that there is no
distinction between tensile or pressure loads on the solder balls.
From Figure 4.2.2 it is clear that situation C and E have the largest z-displacements.
In these situations only one edge of the PCB is fixed. Also it is remarkable that the z-
displacements of the midline in situation D are almost all negative. This is an effect of
the fact that in the beginning of the simulation only the chips are heated. Therefore
only the chips will expand. The z-displacements are now positive because only the
chips expand. But when the simulation is in steady-state, the PCB is also heated. The
PCB has a bigger coefficient of thermal expansion than the chips and because the
mechanical boundary conditions in this situation allow it, the PCB will bend in the
negative z-direction.
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5
0
5
10
15
20x 10
-4
arclength [m]
dis
pla
cem
ent
in z
-direction [
m]
situation A
situation B
situation C
situation D
situation E
Figure 4.2.2: The displacement of the PCB in z-direction as a function for the midline in the length-direction for situation A, B,
C, D and E.
Situation A B C D E
Chip 1 0,1494 0,0957 0,2072 0,1734 0,1793
Chip 2 0,1488 0,0841 0,1671 0,1352 0,1683
Chip 3 0,1387 0,0441 0,1593 0,1078 0,1716
Chip 4 0,1391 0,0004 0,1569 0,1010 0,1754
Chip 5 0,1438 0,0680 0,1605 0,1199 0,1795
Chip 6 0,1562 0,1317 0,1657 0,1586 0,1833
Chip 7 0,0769 0,0228 0,0016 0,0520 0,0148
Table 4.2.1: The absolute values of the curvatures in meter-1 of every chip on the midline for the five situations. In red the
maximum values.
The maximum curvature is qualifying for the lifetime of the PCB, because the solder
joints at the place of the maximum curvature will be the first to break. According to
Table 4.2.1 the fixation in situation B is best for the solder balls because the
maximum curvature is the smallest. But one can’t say that this always is the best
option to fix a PCB, because these results are only calculated for thermal loads. For
instance situation C and E are often used in electronic devices because in these
situations the mechanical properties of the PCB are better. For instance, if you drop
the electronic device to the floor, the PCB will react better if it is fixed as in situations
C and E.
The number of layers of copper is another parameter that is investigated. For this test
the model with ties is used in situation A, but with different numbers of copper layers.
In Figure 4.2.3 the results are pictured. As you can see, the differences between the z-
21
displacements are very small. To determine which number of copper layers is best, it
is necessary to calculate the curvatures of the PCB under every chip (see Table 4.2.2).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5
0
0.5
1
1.5
2
2.5
3
3.5x 10
-5
arclength [m]
dis
pla
cem
ent
in z
-direction [
m]
no layer Cu
1 layer Cu
2 layers Cu
3 layers Cu
4 layers Cu
Figure 4.2.3: The displacement of the PCB itself in z-direction as a function for the midline in the length-direction for 0, 1, 2, 3
and 4 layers of copper.
# layers of Cu 0 1 2 3 4
Chip 1 0,1529 0,1494 0,1421 0,1354 0,1295
Chip 2 0,1522 0,1488 0,1420 0,1358 0,1302
Chip 3 0,1387 0,1387 0,1345 0,1298 0,1250
Chip 4 0,1393 0,1391 0,1345 0,1297 0,1247
Chip 5 0,1437 0,1438 0,1393 0,1344 0,1294
Chip 6 0,1556 0,1562 0,1517 0,1469 0,1420
Chip 7 0,1387 0,0769 0,0469 0,0300 0,0198
Table 4.2.2: The absolute values of the curvatures in meter-1 of every chip on the midline for the five different composites. In red
the maximum values.
As you can see in Table 4.2.2 the PCB with four layers of copper turns out to be the
best. Also for these results it holds that there is only looked at the thermal loads. For
another fixation the test should be done again, because it isn’t sure that these results
are valid for every fixation. Only in situation A four layers of copper are best. It
seems that there is a connection between the number of copper layers and the
maximum curvature; namely if there are more copper layers in the PCB the maximum
curvature decreases. This can be an effect of the fact that copper has a higher E-
modulus than FR-4. So if there are more copper layers in the composite, the overall
strength of the composite increases and there will be less warpage in the PCB.
The last parameter is how four chips can best be placed on a PCB. For this test the
model without ties is used in situation A. Three large chips and one small chip are
22
placed on the PCB. To find the optimal locations for the chips, so that the warpage is
minimized, seven different configurations are modeled (see Figure 4.2.4).
Figure 4.2.4: The seven different configurations. In black the four edges that are fixed in (all the degrees of freedom are fixed).
The temperature on the black edges is fixed at 200 Celsius. The chip have a temperature of 1000 Celsius In pink and blue the
PCB and in brown the chips.
In this case it isn’t very useful to picture the z-displacements of the midline of the
PCBs. Therefore only the curvatures of the chips are given in Table 4.2.3.
Configuration I II III IV V VI VII
Chip 1 0,1441 0,1440 0,1385 0,1391 0,1381 0,1354 0,1372
Chip 2 0,1310 0,1403 0,1369 0,1210 0,1381 0,1435 0,1336
Chip 3 0,1305 0,1442 0,1238 0,1213 0,1230 0,1576 0,1365
Chip 4 0,1170 0,1049 0,0934 0,1086 0,1075 0,0868 0,1074
Table 4.2.3: The absolute values of the curvatures in meter-1 of every chip for the seven different configurations. In red the
maximum values.
Because the maximum curvature is qualifying for the best configuration, number VII
is the best. Its maximum curvature is namely the smallest. In this test however, only
thermal loads are taken into account. So for this test it is also possible that another
configuration can be better if the application of the PCB differs.
In this section it is made clear that the model can be used to optimize PCBs without
the use of the real PCBs. This saves time and money, because the PCBs that need to
be investigated don’t have to be bought, produced and tested.
23
5. Conclusion
A PCB with chips that generate heat is a very complex problem considering geometry
and thermo mechanical loading. Therefore a simplified model had to be made. Two
balance laws have been used for the analysis. Heat conduction occurs because the
PCB is locally heated. This can be calculated with the law of balance of energy. It will
also locally expand and thus stresses will occur. The law of balance of momentum is
necessary to analyse stresses. A software program had to be used to calculate these
two laws simultaneously.
In the model the PCB is considered as a thin plate. It is a laminate built up from FR-4
(fiberglass epoxy) and thin layers of copper. The chips are considered as blocks. They
have a different coefficient of thermal expansion than the PCB and thus warpage will
occur. Warpage will also occur because the different materials in the PCB have
different coefficients of thermal expansion. To describe this it is needed that bending
in the PCB is well described. Therefore it is modeled with shell elements. The chips
are considered as blocks and modeled with bricks. Eventually two models are made
for the chips, but because no experimental data are known for PCBs with chips it
can’t be determined which model is the best.
An experiment of two PCBs without chips is done to determine whether the model
describes the behavior of the PCBs well enough. They are tested twice; once at a
temperature below the glass-transition temperature of FR-4 and once above. For the
temperature below the glass-transition temperature the model was accurate enough,
but for the higher temperature the results of the numerical analysis didn’t match the
results of the experiment. The model can therefore only be used to simulate tests
below the glass-transition temperature of FR-4.
Eventually three parameter variations are investigated. Firstly five kinds of fixation
are tested. The second variation parameter was the number of copper layers in the
laminate. The last test was useful to determine in what way four chips have to placed
on a PCB to minimize warpage.
Though the model can’t be used for simulations above the glass-transition
temperature of FR-4, it can be very useful to simulate the behavior of PCBs in
applications. In this way it can be used for a lot of variation parameters to find the
best situation that gives the least warpage.
To improve the model it has to be adapted so that it is also accurate for temperatures
above the glass-transition temperature. In this way the solder reflow process can also
be simulated. It can also be useful to test the behavior of the chips at different
temperatures because in this study there was no data about the behavior of the chips.
In this way it can be determined if the model with chips or without chips is best.
24
Symbols
Symbols Name Unit
ρ Density kg/m3
cp Specific heat J/(Kg*K)
T Temperature 0 Celsius
∇ Gradient operator -
k Heat conduction coefficient W/(m*K)
σ Stress tensor Pa
D Rate-of-deformation tensor s-1
R Heat generation per unit mass W/kg
q Body force N
v& Acceleration m/s2
E Young’s modulus Pa
G Shear modulus Pa
ν Poisson’s ratio -
α Coefficient of thermal expansion 0 Celsius
-1
25
Literature
[1] Brekelmans, W.A.M, Balken en Platen, Eindhoven University of Technology,
1996.
[2] Bailey, C., Wheeler, D., Cross, M., An Integrated Modeling Approach to
Solder Joint Formation, IEEE Transactions on Components and Packaging
Technology, Vol. 22, No. 4, December 1999.
[3] Xie, W., Sitaraman, W.K., Numerical Study of Interfacial Delamination In A
System-On-Packaging (SOP) Integrated Substrate under Thermal Loading,
Georgia Institute of Technology, Atlanta, 2000.
[4] Zhang, Z., Sitaraman, W.K., Wong, C.P., FEM Modeling of Temperature
Distribution of a Flip-Chip No-Flow Underfill Package During Solder Reflow
Process, IEEE Transactions on Components and Packaging Technology, Vol.
27, No. 1, January 2004.
[5] Miura, H., Nishimura, A., Kawai, S., Structural Effect of IC Plastic Package
on Residual Stress in Silicon Chips.
26
Acknowledgements
Firstly I would like to thank dr. ir. P.J.G. Schreurs for his guidance and time,
especially for the fact that you checked my report over and over again.
Secondly I would like to thank J. de Vries of Philips Applied Technologies for his
contribution to this study.
Finally I thank T. van der Ackerveken for his time carrying out my experiment and
delivering the results so perfectly.