DETERMINING THE COEFFICIENT OF THERMAL EXPANSION OF
PRINTED WIRING BOARD COMPONENTS
by
Tong Wa Chao
TC 660H
Plan II Honors Program
The University of Texas at Austin
May, 1998
DETERMINING THE COEFFICIENT OF THERMAL EXPANSION OF
PRINTED WIRING BOARD COMPONENTS
by
Tong Wa Chao
TC 660H
Plan II Honors Program
The University of Texas at Austin
May, 1998
Kenneth M. Liechti, Ph.D.
Department of Aerospace Engineering and Engineering Mechanics
Supervising Professor
Mark E. Mear, Ph.D.
Department of Aerospace Engineering and Engineering Mechanics
Second Reader
ABSTRACT
The integrity and reliability of multi-layer electronic boards depend on the
strength and quality of the bond between the layers. High temperature circuit integration
processes (such as soldering) that exceed the glass transition temperature, Tg, of the
polymeric components and pre-existing flaws such as cracks, voids, and areas of poor
adhesion can produce delaminations whose growth can cause device failures.
Delamination’s negative effects on manufacturing yield make it important to understand
how it occurs and characterize the resistance to delamination. However, before fracture
experiments on a Printed Wiring Board material can be performed, it is necessary to
establish the material’s thermomechanical properties.
This work sought to achieve part of this objective through determining the
coefficient of thermal expansion of Printing Wiring Board Components. The challenges
arose from accurately measuring both the milli-inch-level displacements and the
changing temperatures in the measuring frame, which had a large thermal mass compared
to the specimen.
ACKNOWLEDGEMENTS
I would like to gratefully acknowledge my thesis advisor, Professor Kenneth
Liechti, for his guidance and support throughout this project and for providing me many
other valuable research opportunities that have enriched my undergraduate career. I
would also like to thank Professor Mark Mear, my undergraduate advisor, for reviewing
this thesis. More than that, I would like to thank him for offering me wise counsel during
my four years as an undergraduate.
I am grateful to Plan II for giving me the primary motivation to take on this
rewarding endeavor.
Among the graduate students who worked in the basement labs at the WRW
building, I would like to thank especially Greg Swadener and Alex Arzoumanidis for
their lab equipment tutorial and technical support. I would also like to thank Patrick
Klein for his opinions.
I would like to thank my parents, Chao Wa and Ho Heong-Peng, and my sister,
Chao San-San, for their ongoing encouragement and emotional support. Lastly but
certainly not the least, I would like to thank Louisa for giving me happiness.
TABLE OF CONTENTS
1.0 INTRODUCTION .................................................................................................................................. 1
1.1 MOTIVATION........................................................................................................................................... 1
1.2 BACKGROUND ........................................................................................................................................ 1
1.3 OBJECTIVES............................................................................................................................................ 2
1.4 LITERATURE REVIEW............................................................................................................................... 2
1.4.1 Vitreous Silica dilatometers........................................................................................................... 2
1.4.2 Twin-telemicroscope method ......................................................................................................... 3
1.4.3 Interferometers .............................................................................................................................. 3
1.4.4 Thermomechanical analyzers ........................................................................................................ 4
2.0 DESCRIPTION OF SPECIMENS........................................................................................................ 5
2.1 FUSED SILICA ......................................................................................................................................... 5
2.2 PRINTED WIRING BOARD ........................................................................................................................ 5
3.0 EXPERIMENTS..................................................................................................................................... 7
3.1 RATIONALE FOR CHOOSING THIS APPROACH ........................................................................................... 7
3.2 EXPERIMENTAL APPROACH ..................................................................................................................... 7
3.3 DISPLACEMENT TRANSDUCER DESCRIPTION ............................................................................................ 8
3.4 EXPERIMENTAL SET-UP .......................................................................................................................... 9
3.5 TEST PROCEDURE................................................................................................................................... 9
3.6 DATA REDUCTION ................................................................................................................................ 11
4.0 DISCUSSION OF RESULTS .............................................................................................................. 13
5.0 CONCLUSIONS................................................................................................................................... 15
TABLES ...................................................................................................................................................... 16
FIGURES .................................................................................................................................................... 18
REFERENCES ........................................................................................................................................... 26
APPENDIX ................................................................................................................................................. 27
A1 ERROR ANALYSIS ................................................................................................................................ 27
A2 RAW DATA .......................................................................................................................................... 28
A3 EQUIPMENT LIST ................................................................................................................................. 30
VITA..............................................................................................ERROR! BOOKMARK NOT DEFINED.
LIST OF TABLES
Table 1: Printed Wiring Board Coefficient of Thermal Expansion
Table 2: Fused Silica Expansivity Values
Table 3: Invar 36 Coefficient of Thermal Expansion
LIST OF FIGURES
Figure 1: Geometry of the Printed Wiring Board
Figure 2: Model Printed Wiring Board
Figure 3: Geometry of the Epoxy/Glass Weave Adhesive
Figure 4: Geometry of the Fused Silica Specimen Used for Calibration and its V-notch
Frame
Figure 5: Schematic of the Experimental Set-up
Figure 6: Description of the Capacitive Displacement transducer
Figure 7: Calibration of the Capacitive Displacement Transducer
Figure 8: Change in Displacement versus Change in Temperature
Figure 9: Change in Displacement versus Change in Temperature for the Printed Wiring
Board Material
Figure 10: Strategy for Calculation of the Printed Wiring Board Expansion
1.0 Introduction
1.1 Motivation
Microelectronic boards are usually made up of several layers of different
materials [1]. The integrity and reliability of multi-layer electronic boards depend on the
strength and quality of the bond between the layers. High temperature circuit integration
processes (such as soldering) that exceed the glass transition temperature, Tg, of the
polymeric components and pre-existing flaws such as cracks, voids, and areas of poor
adhesion can produce delaminations whose growth can cause device failures. Since
delamination affects manufacturing yield, it is important to understand how delamination
occurs and to characterize the resistance to delamination. Before fracture experiments on
a Printed Wiring Board material can be performed, it is necessary to establish its
thermomechanical properties, one of which is the coefficient of thermal expansion.
1.2 Background
This project was a follow-up of the coefficient of thermal expansion experiments
performed by an aerospace engineering undergraduate, Carlos Aquila, at the University
of Texas at Austin. He designed the prototype of the Invar measuring frame that I had
later modified and used for my experiments. The notes and data that Aquila left at the
Engineering Mechanics Labs at the University of Texas at Austin indicated that he
attempted to find the coefficient of thermal expansion of the Printed Wiring Board
material. He used an extensometer to measure strain, but subsequently concluded that it
was undesirable due to thermal effects. Later, he used a capacitive displacement
transducer that was mounted on an Invar frame for his measurements. I found it difficult
1
to verify his experimental results and calibrations because I could not find sufficient
record on his assumptions and initial conditions. His experimental set-up also needed
improvements. The Invar measuring frame did not have the proper electrical insulation
required by the capacitive displacement transducer, and the probe positioner (converted
from a micrometer) introduced significant calibration errors. Hence, it was my task to
improve the measuring device and rerun the coefficient of thermal expansion
experiments.
1.3 Objectives
The primary objective of this project was to determine the coefficient of thermal
expansion for a 3-inch long Printed Wiring Board Material. Before this was done, it was
necessary to calibrate a suitable experimental setup for measuring the milli-inch-level
thermal expansion of the Printed Wiring Board Material.
1.4 Literature Review
The following sections summarize four important methods developed for measuring
the coefficient of thermal expansion of a general group of solid materials.
1.4.1 Vitreous Silica dilatometers
In this method, a vitreous silica dilatometer of the single push-rod or tube type is
used to determine the change in the length of the specimen relative to that of a holder as a
function of temperature [2]. The temperature is controlled either over a series of steps or
at a slow constant heating or cooling rate over the entire range. Greater attention must be
2
paid to the change in temperature than change in displacement because very often the
temperature measured is not the temperature of the specimen if thermal equilibrium is not
attained [3].
1.4.2 Twin-telemicroscope method
This method is used for measuring the absolute expansion of large specimens at
high temperatures [3]. Best results are obtained when the two microscopes are rigidly
mounted to a bar of low-expansion material outside the furnace and the length change
measured with filar micrometer eyepieces. Telemicroscopes, or microscopes with relay
lenses, are needed due to the large working distance imposed by the furnace. Sharp
images of the gauge length can be produced by marks indented into flat specimen
surfaces at low temperatures or by knife edges at high temperatures. Furnace windows
may cause significant errors if their surfaces are not optically flat.
1.4.3 Interferometers
This method is based on the interference of monochromatic light reflected from
two surfaces that are separated by a specimen or by the combination of a specimen and a
reference material [3]. Typical configurations involve a specimen of known geometry
that is placed between two reflecting surfaces. Monochromatic light illuminates the
surfaces simultaneously to produce a fringe pattern. As the specimen expands or
contracts, the fringe pattern changes. This change is converted into change in
displacement [4].
3
1.4.4 Thermomechanical analyzers
To determine the coefficient of thermal expansion, this test method uses a
thermomechanical analyzer, which consists of a specimen holder and probe, a
displacement transducer, furnace, a temperature sensing element, a means of purging the
specimen environment with dry inert gas, a data recorder, and calipers [5]. Sometimes it
may also include a stepper motor for automation.
The approach that was chosen for this project will be described in Sections 3.1
and 3.2.
4
2.0 Description of Specimens
Two types of specimens were employed. The fused silica, with known coefficient
of thermal expansion, was used for calibration. The Printed Wiring Board is also
described below.
2.1 Fused Silica
The geometry of the low thermal expansion fused silica rod (Standard Reference
Material 739 from NIST) used for calibration of the whole measuring device is shown in
Figure 4. It was a circular rod, 3 inches long and 0.25 inch in diameter. The values of
expansion and expansivity relevant to this experiment are given in Table 2. The rod was
obtained as a single drawing from an ingot of fused silica that was at least 99.8 % weight
pure. It was annealed by soaking at 1373 K for 7 hours and cooling to 1173 K at 12 K
per hour [6].
2.2 Printed Wiring Board
Figure 2 displays a model of the Printed Wiring Board Material [1]. The laminate
was symmetric about the mid-plane (z = 0). The signal layers have copper lines that ran
in the x and y-directions (see Figure 2), and the ground and power layers were made of
copper sheet. The signal lines were 6 × 10−3 inches wide and had 24 × 10−3 inches of
spacing. These lines made up 20 % copper in each layer. The matrix material was made
with a difunctional epoxy with 1080 glass weave. The layer adhesive was a difunctional,
49 oC glass transition temperature epoxy system (ED 130 Resin with 1080 Glass Fabric),
5
and it was composed of an epoxy matrix with layers of glass weave throughout (Figure
3).
The geometry of the Printed Wiring Board specimen is shown in Figure 1. It was
a rectangular piece 3-inch long (same length as the fused silica), 0.5-inch wide, and
0.045-inch thick.
6
3.0 Experiments
3.1 Rationale for Choosing this Approach
In this experiment a high temperature capacitive displacement probe was mounted
on a low-expansion frame to measure the displacement of the Printed Wiring Board,
while a thermocouple was used to measure the temperature of the specimen. This
approach was the simplest compared to all the other methods considered Section 1.4. Yet
it was also the most suitable. Being only 3 inches long, the Printed Wiring Board was a
small specimen, and hence did not need the twin-telemicroscope method that was
intended for large specimens. Moreover, the Printed Wiring Board did not have a
coefficient of thermal expansion that was low enough to require the high-precision
interferometry method. The push-rod method was also undesirable because significant
errors might be introduced by the deformation of the soft Printed Wiring Board under the
load of the push-rod. Lastly, inconvenient accessibility to a thermomechanical analyzer
had made this method unappealing.
3.2 Experimental Approach
The specimen was heated in an oven and its change in expansion and temperature
over time was measured. A series of temperature ramps and soaks were employed.
Static equilibrium could be assumed because only those displacements at thermoequilibrium were considered. Thermo-equilibrium was determined with regard to the
Invar frame, because it had the largest thermal mass amongst the elements concerned. In
other words, displacements were used for calculation of coefficient of thermal expansion
only when the Invar frame had attained a steady soak temperature.
7
Like any experimental-setup, the measuring frame would introduce its own errors
if not calibrated properly. Although Invar has a very small coefficient of thermal
expansion (Table 3), the Invar frame’s expansion would cause significant errors in the
measurements if it was not considered. The thermal expansion and electronic drift of the
capacitive displacement transducer must also be taken into account. To get around the
overall offset caused by device errors, a material with known coefficient of thermal
expansion can be used for calibration. A fused silica rod with very small coefficients of
thermal expansion (Table 2) was employed to determine the offset caused by the whole
measuring device. Once this offset was determined, it was deducted from the total
expansion measured in the Printed Wiring Board experiments to get the material’s actual
expansion.
3.3 Displacement Transducer Description
Figure 6 shows the principles of the Capacitec capacitive displacement
transducer. The transducer head and the conductive target acted like a parallel plate
capacitor. Since capacitance was inversely proportional to the gap distance, a smart
linearization circuitry was used to translate the capacitance into voltage that was
proportional to the target displacement. This circuitry was embedded in a Capacitec
Series 3000 amplifier. A calibration curve for the displacement transducer can then be
obtained by plotting known distances made by a micrometer against the output voltage.
8
3.4 Experimental Set-Up
Figure 5 is a schematic of the experimental set-up. In the calibration experiment,
the fused silica specimen rested on a V-notched frame made of Al 6061 T6511, which in
turn was placed on an Invar frame. In the CTE experiment, the fused silica and the Vnotched frame was simply replaced by the Printed Wiring Board. The Printed Wiring
Board was electrically insulated from the Invar frame by a short strip of teflon.
The specimen’s (whether it was the fused silica rod or the Printed Wiring Board)
ends were bonded to the wall of the Invar frame and to a grounded steel target with
double-sided Scotch tapes. When the specimen slowly expanded under heat, the target
was pushed towards the capacitive displacement transducer, thus reducing the gap
distance between the target and the probe head. The change in gap distance was
measured by the probe, converted into voltage by the amplifier, and stored and plotted
using LabView data acquisition software. Simultaneously, a thermocouple measured the
Invar frame’s (the main thermal mass in the setup) temperature, which was also recorded
by the computer. A hole was drilled into the center of the Invar frame to allow the
insertion of the thermocouple.
3.5 Test Procedure
The test procedures described below are identical for both the calibration (with fused
silica) and Printed Wiring Board experiments.
1. Before any calibration or experiment, the Capacitec Series 3000 Amplifier and the
Capacitec capacitive displacement probe was allowed to warm up for 1 hour.
9
2. The amplifier was calibrated with the capacitive displacement probe, a calibration
wheel (or micrometer), and a voltmeter by following the instructions in a manual that
came with the amplifier. Recalibration was performed before each experiment if a
different probe was used for the amplifier, but not necessarily if the same probe was
used. The probe cable was insulated from the ground or other electrical equipment.
3. After the amplifier was tuned, the probe was calibrated with the calibration wheel and
voltmeter. This step was necessary before each experiment to ensure accuracy.
Voltages were recorded in 1 milli-inch increments over a range of 20 milli-inches. A
graph of voltage versus displacement was plotted to acquire the resolution of the
probe on the day the experiment was run.
4. Proper gap distance between the probe and the target was adjusted before the
experiment began. If the gap distance exceeded 20 milli-inches (~10 volts), the probe
would be out of range. On the other hand, if the gap distance was too short, there
would not be enough space for the specimen to expand for the desired temperature
range. In both the calibration and Printed Wiring Board experiments, the probe was
placed so that the initial gap distance voltage was around 9 volts.
5. The temperature profile for the furnace was built in steps of 10 oC from room
temperature up to 233 oC. Soak time at each temperature was set at 90 minutes to
allow the transients to die out and ensure a steady displacement.
6. Both displacement and temperature data were recorded using Labview software at a
rate of 1 sample per 20 seconds. Since the experiment duration was long, the low
sampling rate was selected to avoid computer memory problems.
10
3.6 Data Reduction
To calculate the coefficient of thermal expansion of the Printed Wiring Board,
three groups of data were used: the probe calibration curve, the fused silica calibration
curve, and the Printed Wiring Board expansion curve. The slope of the probe calibration
curve, k (volts/milli-inch), provided the inverse of the resolution of the probe. The
expansion, ∆L, from initial temperature To to temperature T was calculated as
∆L = (V-Vo)/k
(3.1)
where
V = voltage output of the probe at temperature T
Vo = voltage output of the probe at initial temperature To
The voltage values of the probe were taken into consideration only at stabilized
displacements and temperatures to ensure static and thermal equilibrium.
The same temperatures and soak times were employed for both the fused silica
calibration and the Printed Wiring Board experiment to guarantee proper correspondence.
The ∆L’s measured from the fused silica experiment were subtracted from the ∆L’s
measured from the Printed Wiring Board experiment. This subtraction removed the
undesired expansion of the Invar frame and the probe drift from the overall expansion
recorded in the Printed Wiring Board experiment, but had not yet taken into account of
the small expansion of the fused silica.
∆LPWBcorrected = ∆LPWBuncorrected – ∆Lfused silica
(3.2)
Lastly, the coefficient of thermal expansion of the Printed Wiring Board material,
approximated by (∆L/∆T)/Lo, was calculated by accounting for α fused silica, the average
expansivity of the fused silica over the relevant temperature range
(∆L/∆T)/Lo = (∆LPWBcorrected /∆T)/Lo + α fused silica
11
(3.3)
where
∆LPWBcorrected /∆T = slope of the curve of calibrated expansion versus temperature
change
Lo = length of the specimen at initial temperature To
Although the Eulerian definition of the coefficient of thermal expansion was (∆L/∆T)/L
(where L = Lo + ∆L), (∆L/∆T)/Lo was used because ∆L was so small that it could be
assumed that L ≈ Lo.
A pictorial explanation of this calibration strategy is shown in Figure 10.
12
4.0 Discussion of Results
In this section, results from the calibration of the probe, the calibration of the
Invar frame, and the coefficient of thermal expansion experiment are discussed.
A calibration curve of the capacitive displacement probe is shown in Figure 7.
The calibration was performed at room temperature, and the displacement corresponded
very linearly with the voltage over a range of 20 milli-inches as expected. The
resolution, given by the inverse of the slope, was 1.91 milli-inches/volt on the day the
Printed Wiring Board expansion experiment was performed.
Figure 8 shows the change in displacement versus change in temperature for both
the fused silica calibration experiment and the Printed Wiring Board experiment. The
initial temperature was set at 36.2 oC. Since the displacements showed unreasonable
jumps above 161 oC, those data were discarded. It was suspected that the probe was
faulty.
Both curves show linear trends, but there is a jump on each curve consistently at
approximately 80 oC. The magnitude of the jump, however, was not the same. At this
temperature, the y-intercepts of both curves seem to have been translated in the positive
direction, but the slopes are not altered significantly. In other words, a sudden, new bias
error seemed to have occurred after 80 oC. A “corrected” curve is also plotted on Figure
8 after subtracting the fused silica experiment displacements from the Printed Wiring
Board experiment displacements. This curve is plotted again on Figure 9 with two curve
fits, one before the jump and one after the jump. The slopes differ by about 4%.
The displacement jumps at one consistent temperature for different specimens
suggested that the measuring device was somehow flawed. It was suspected that these
13
errors were somehow related to thermal effects on the probe. Due to time constraints, the
true cause of the flaw was not found.
The coefficients of thermal expansion were calculated after the average
expansivity of the fused silica over the relevant temperatures were added to the slopes in
Figure 9. These values are determined with an uncertainty of 6%, and are illustrated in
Table 1.
Four other Capacitec probes and amplifiers were tested for resolution, linear
range, and drift throughout this project in the search of the best combination. A
probe/amplifier combination was “good” if it gave high resolution (less than 2 milliinches/volt), wide linear range (at least 20 milli-inches), and low drift (less than 0.1
volt/hour). Although a good probe was found, it broke shortly before the experimental
set-up was fit for taking measurements. A next best probe was subsequently used for
taking data.
14
5.0 Conclusions
In this project an experimental setup was improved and calibrated. The
coefficient of linear thermal expansion for the Printed Wiring Board material was
determined over a temperature range from 36 oC to 161 oC. Since a significant change in
the coefficient of thermal expansion in this temperature range was not observed, the glass
transition temperature is expected to be above 161 oC. Data were not taken at
temperatures higher than 161 oC due to suspected equipment flaw, which caused
unreasonable displacement jumps above 161 oC.
In future work, data at higher temperatures will be obtained until the glass
transition temperature is found. It is expected that above the glass transition temperature,
the coefficient of thermal expansion will rise noticeably. The glass transition temperature
is believed to be around 180 oC, which is also the Tg of many similar materials. More
probes and amplifiers should be tested in search for a better probe/amplifier combination.
15
Tables
Table 1: Printed Wiring Board Coefficient of Thermal Expansion
Temperature (oC)
Average CTE (µm/m)/ oC
36 to 75
14 ± 6%
85 to 161
15 ± 6%
Table 2: Fused Silica Expansivity Values [6]
Temperature (oC)
Expansivity (µm/m)/ oC
25
0.49
47
0.53
67
0.56
87
0.58
107
0.60
127
0.61
147
0.62
167
0.63
16
Table 3: Invar 36 Coefficient of Thermal Expansion [7]
Temperature (oC)
CTE (µm/m)/ oC
25 to 100
1.66
100 to 150
2.16
150 to 200
2.80
17
Figures
∆L
Top View
L=2.998 in.
Wall of the
Invar frame
H=0.045 in.
Front View
Side View
W=0.500 in.
Figure 1: Geometry of the Printed Wiring Board
signal
ground
signal
x
power
signal
Z
Figure 2: Model Printed Wiring Board [1]
18
3
1
2
Figure 3: Geometry of the Epoxy/Glass Weave Adhesive [1]
Front
View
Side View
0.249 in.
3.034 in.
Fused silica
V-notch frame made of
Al 6061 T6511
Figure 4: Geometry of the Fused Silica Specimen Used for Calibration and its V-notch Frame
19
Specimen (either PWB or Fused silica)
Capacitive Displacement
Transducer
Fixed Mount
Gap Distance
Exterior of Oven
Thermocouple
Module
Interior of Oven
Voltmeter
Computer with
Labview Data
Acquisition
Invar Frame
Thermocouple
embedded in
Invar frame
Teflon
Grounded Insulation
Target
Excitation and
Amplification
Figure 5: Schematic of the Experimental Set-up
20
Gap distance = d
Capacitive Displacement Transducer Probe
∆d
Conductive Target
Acts like a parallel plate capacitor with
capacitance = C
C ∝ (1/d)
Calibration curve
Smart Linearization Circuitry
Voltage
Output Voltage ∝ ∆d
∆d
Figure 6: Description of the Capacitive Displacement Transducer
21
12
10
Voltage (V)
8
6
4
y = 0.5233x + 0.1475
2
R = 0.9997
2
0
0
2
4
6
8
10
12
14
16
-3
Displacement (10 inches)
Figure 7: Calibration of the Capacitative Displacement Transducer
22
18
20
∆LPWBcorrected = ∆LPWBuncorrected - ∆LFused silica
8
7
PWB (uncorrected)
6
Fused Silica
PWB (corrected)
Change in Displacement (10
-3
inches)
9
Apparent displacement jump
or bias error
5
4
3
2
1
0
0
20
40
60
80
o
100
120
o
Change in Temperature ( C) (Initial point: 36.2 C)
Figure 8: Change in Displacement versus Change in Temperature
23
140
6
-3
Change in Displacement (10 inches)
y = 0.0419x + 0.6733
2
R = 0.9964
5
4
3
y = 0.0402x - 0.0149
2
2
R = 0.9977
1
0
0
20
40
60
80
o
100
120
o
Change in Temperature ( C) (Initial point: 36.2 C)
Figure 9: Change in Displacement versus Change in Temperature for the
Printed Wiring Board Material
24
140
Expansion measured in the Fused silica
experiment, ∆Lfused silica
Device
expansion, d
Fused Silica
expansion, s
Expansion measured in the Printed Wiring
Board experiment, ∆LPWB(uncalibrated)
Device
expansion, d
Printed Wiring Board
expansion, p
p = (d+p) – (d+s) + s
known
actual
measured
Figure 10: Strategy for Calculation of the Printed Wiring Board Expansion
25
References
1.
Watson, K., “Delamination of Multi-layer Microelectronic Packages,” The
University of Texas at Austin, December 1995.
2.
ASTM E 228, “Standard Test Method for Linear Thermal Expansion of Solid
Materials with a Vitreous Silica Dilatometer,” Annual Book of ASTM Standards,
December 1995.
3.
Touloukian Y. S., et al., Thermal Expansion – Non-Metallic Solids, Purdue
Research Foundation, New York, 1977.
4.
ASTM E 289, “Standard Test Method for Linear Thermal Expansion of Rigid
Solids with Interferometry,” Annual Book of ASTM Standards, December 1995.
5.
ASTM E 831, “Standard Test Method for Linear Thermal Expansion of Solid
Materials by Thermomechanical Analysis,” Annual Book of ASTM Standards,
September 1993.
6.
“Fused-Silica Thermal Expansion,” Standard Reference Material 739 Certificate,
National Institute of Standards and Technology, Gaithersburg, Maryland,
December 1991.
7.
“Carpenter Invar 36 free cut elec alloy unannealed ground,” Certificate of Tests,
Carpenter Technology Corporation, Houston, Texas, March 1998.
26
Appendix
A1 Error Analysis
The following table gives a list of the uncertainties involved. Microsoft Excel was used
to evaluate the statistical uncertainties.
Uncertainty
Percentage
Comments
δLo
1.2
δ(∆V/∆L)
0.9
δ(∆L)
1.4
δ(∆T)
0.3
δ(∆L/∆T)1
5.5
δ(∆L/∆T)2
4.6
δα
0.5
δ(∆L)teflon
2.5
Initial length of fused silica was 36 milli-inches longer than
the Printed Wiring Board, which was 2.998 inches.
This is two times the standard error in the curve fit of the
probe calibration data.
This is the maximum uncertainty encountered when averaging
over 40 data points (~13 minutes) to get each of the several
stabilized Printed Wiring Board displacements. The
uncertainty is two times the standard deviation.
This is the maximum uncertainty encountered when averaging
over 40 data points (~13 minutes) to get each of the several
stabilized temperatures. The uncertainty is two times the
standard deviation.
This is two times the standard error in the curve fit of the
expansion versus temperature curve from 36 oC to 75 oC.
This is two times the standard error in the curve fit of the
expansion versus temperature curve from 85 oC to 161 oC.
The uncertainty of the fused silica expansivity is obtained
from the NIST certificate.
The teflon tape that was used to insulate the Printed Wiring
Board from the ground wrinkled under heat.
The total uncertainty is estimated by the Pythagorean, with all weighting factors being
equal to one.
UCTE/CTE = √(1.22+0.92+1.42+0.32+5.52+0.52+2.52) = 6 % from 36 oC to 75 oC.
UCTE/CTE = √(1.22+0.92+1.42+0.32+4.62+0.52+2.52) = 6 % from 85 oC to 161 oC.
27
220
9.00
200
8.50
180
8.00
160
7.50
140
7.00
120
6.50
Temperature (mV)
Displacement (V)
100
6.00
80
5.50
60
5.00
40
4.50
20
4.00
0
10000
20000
30000
40000
50000
60000
70000
80000
Time (sec)
Voltage Readings from Thermocouple and Displacement Transducer for
the Printed Wiring Board
28
Displacement (V)
Termperature (mV)
A2 Raw Data
220
10.00
200
9.50
180
9.00
8.50
Temperature (mV)
140
8.00
Displacement (V)
120
7.50
100
7.00
80
6.50
60
6.00
40
5.50
20
5.00
0
20000
40000
60000
80000
100000
120000
140000
Time (sec)
Voltage Readings from Thermocouple and Displacement Transducer for
the Fused Silica
29
Displacement (V)
Termperature (mV)
160
A3 Equipment List
Name
Number
Capacitec Series 3000 Amplifier Model 3218-4S, Channel 1A
UT#615928
Capacitec capacitive displacement probe
(labeled with a yellow tape)
National Instruments Labview 3.1.1
(software)
Keithley multimeter 168 autoranging DMM
SN 36394
HiTEC calibration wheel Model CS2
SN138
Standard Environmental Systems Inc. environmental test
chamber model No. RTT/4E, with PRO-SES microprocessorbased controller
Fluke thermocouple with 80TK thermocouple module
SN 86119 (oven)
UT#615734 (controller)
Keithley 175 autoranging multimeter
UT#388280
Invar measuring frame
------
30
------