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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

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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.

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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.

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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.

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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]

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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



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



Coefficient of thermal expansion α 11 [0 Celsius

-1] 20e-6 20e-6 20e-6 20e-6 20e-6


-1] 20e-6 20e-6 20e-6 20e-6 20e-6


-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


Shear modulus G [GPa] 8,7


-1] 15e-6

Heat conduction coefficient k [W/m*K] 0,03



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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


Shear modulus G [MPa] 10,7


-1] 25e-6

Heat conduction coefficient k [W/m*K] 50



Table 2.3.4: Relevant material properties of Sn63Pb37 as used in the model.

Young’s modulus E [GPa] 20,0


Shear modulus G [MPa] 7,7


-1] 15e-6

Heat conduction coefficient k [W/m*K] 0,03



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.

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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.

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