D T I ELECTROSTATIC FEEDBACK FOR MEMS
Institutionen för fysik, kemi och biologi
ELECTROSTATIC FEEDBACK FOR MEMS
SENSOR –
DEVELOPMENT of
In Situ TEM INSTRUMENTATION
Huai-Ning Chang
Nanofactory Instruments AB
2008 April
Handledare
Alexandra Nafari & Johan Angenete
Examinator
Anita Lloyd Spetz
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© Huai-Ning Chang
Datum
Date
2008-04-22
Avdelning, institution
Division, Department
Physics
Department of Physics, Chemistry and Biology
Linköping University
Språk
Language
Svenska/Swedish
Engelska/English
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Licentiatavhandling
Examensarbete
C-uppsats
D-uppsats
Övrig rapport
ISBN
ISRN:
LITH-IFM-EX--08/1927--SE
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Serietitel och serienummer
Title of series, numbering
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URL för elektronisk version
Titel
Title
ELECTROSTATIC FEEDBACK FOR MEMS SENSOR – DEVELOPMENT of In Situ TEM INSTRUMENTATION
Författare
Author
Huai-Ning Chang
Sammanfattning
Abstract
This thesis work is about further developing an existing capacitive MEMS sensor for in situ TEM nanoindentation developed by Nanofactory
Instrument AB. Today, this sensor uses a parallel plate capacitor suspended by springs to measure the applied force. The forces are in the micro
Newton range. One major issue using with this measurement technique is that the tip mounted on one of the sensor plates can move out of the TEM
image when a force is applied. In order to improve the measurement technique electrostatic feedback has been investigated. The sensor’s
electrostatic properties have been evaluated using Capacitance-Voltage measurements and a white light interferometer has been used to directly
measure the displacement of the sensor with varying voltage. Investigation of the sensor is described with analytical models with detailed treatment
of the capacitive response as function of electrostatic actuation. The model has been tested and refined by using experimental data. The model
showed the existence of a serial capacitor in the sensor. Moreover, a feedback loop was tested, by using small beads as load and by manually
adjusting the voltage. With the success of controlling the feedback loop manually, it is shown that the idea is feasible, but some modifications and
improvements are needed to perform it more smoothly.
Nyckelord
Keyword
Electrostatic force modeling, Force feedback, Force sensor
Abstract
This thesis work is about further developing an existing capacitive MEMS sensor for in situ
TEM nanoindentation developed by Nanofactory Instrument AB. Today, this sensor uses a
parallel plate capacitor suspended by springs to measure the applied force. The forces are in
the micro Newton range. One major issue using with this measurement technique is that the
tip mounted on one of the sensor plates can move out of the TEM image when a force is
applied. In order to improve the measurement technique electrostatic feedback has been
investigated. The sensor’s electrostatic properties have been evaluated using CapacitanceVoltage measurements and a white light interferometer has been used to directly measure the
displacement of the sensor with varying voltage. Investigation of the sensor is described with
analytical models with detailed treatment of the capacitive response as function of
electrostatic actuation. The model has been tested and refined by using experimental data. The
model showed the existence of a serial capacitor in the sensor. Moreover, a feedback loop was
tested, by using small beads as load and by manually adjusting the voltage. With the success
of controlling the feedback loop manually, it is shown that the idea is feasible, but some
modifications and improvements are needed to perform it more smoothly.
Abbreviations
ADC – Analog to Digital Converter
CAPDAC – Computer-Aided Piping Design & Construction
CDC – Capacitance to Digital Converter
DOF – Degree of Freedom
DUT – Device under Test
MEMS – Microelectromechanical Systems
PSU – Power Supply Unit
SOI – Silicon on Insulator
TEM – Transmission Electron Microscopy
Contents
1
Introduction
1
1.1
Microengineering ....................................................................................................... 1
1.2
Electro-Mechanical MEMS sensor ............................................................................. 1
2
The Nanoindenter and Motivation
3
2.1
Nanoindenter Sensor .................................................................................................. 3
2.2
Motivation for the Feedback Loop Control ................................................................. 5
3
System Modeling and Analysis
7
3.1
Introduction................................................................................................................ 7
3.2
Spring Constant.......................................................................................................... 7
3.3
Elastic Force on Electrostatic Actuation ..................................................................... 8
3.4
Electrostatic Micromechanical Actuator with Extended Travel Range ...................... 11
3.5
Extra Capacitors....................................................................................................... 12
3.6
Bent Plate................................................................................................................. 14
3.7
Analytical Model for Nanoindenter Sensor............................................................... 17
4
Experimental Methods
20
4.1
Introduction.............................................................................................................. 20
4.2
C-V Measurements................................................................................................... 20
4.3
Optical Surface Profile ............................................................................................. 22
4.4
Tilting Effect and Surface Roughness....................................................................... 24
5
Experimental Results
26
5.1
Capacitive behavior.................................................................................................. 26
5.2
Optical profiler measurement ................................................................................... 27
5.3
Nanoindenter Feedback Loop ................................................................................... 30
6
Conclusion
33
6.1
Summary.................................................................................................................. 33
6.2
Recommended Future Work..................................................................................... 33
Acknowledgement ............................................................................................................... 35
Bibliography ........................................................................................................................ 36
Appendix ............................................................................................................................. 38
Appendix A -- Analytical Spring Constants ......................................................................... 38
Appendix B -- AD7746 & LCR Meter ................................................................................. 41
Appendix C -- Spring Design ............................................................................................... 45
Chapter 1
Introduction
1.1 Microengineering
Mankind has always been curious of the mini-world. The desire to go beyond to nano world
devices could trace back to the second half of the 20th century, envisioned by Feynman:
“What are the possibilities of small but movable machines? They may or may not be useful,
but they surely would be fun to make” [1]. It seemed to be quite obvious that it is not only fun
to miniaturize machines but would also be useful in many fields like industries and medical
applications. The small miniaturized world revealed exciting possibilities in engineering, or a
new paradigm: Microengineering.
Though small in size, the background disciplines and the potential applications developed
by microengineering techniques are enormous. The involved knowledge comes in many fields
where multiple physical domains (electrical, mechanical, thermal, optical, chemical, and
magnetic) meet in the micro scale range. A collection of technological capabilities are
involved in this engineering field. One of the branches that has emerged within
microengineering was MEMS. Microelectromechanical systems (MEMS) refer to mechanical
devices that have a characteristic size smaller than millimeter but more than microns. In
MEMS electrical and mechanical components are fabricated using integrated circuit batchprocessing technologies. Feynman’s vision has become a reality and this multidisciplinary
field has witnessed explosive growth during the last decade.
MEMS possess big potential for innovation. However, the actual implementation is
hampered by the multidisciplinary complexity, which results in an approach that is best
described by leap forward rather than being able to comprehend theories and disciplines
behind. MEMS techniques have not yet matured which might result in risks and even failures
in some application; however, some successful and encouraging achievements have opened
up a prosperous market for MEMS techniques [2], e.g. micromachined ultrasonic transducer,
optical MEMS devices and RF MEMS technology in wireless application.
1.2 Electro-Mechanical MEMS sensor
The strong coupling of various domains involved in MEMS devices is a distinct characteristic.
When the characteristic dimensions of the device element decreases from the macro scale
level to the micro scale, some effects might become minor or negligible such as gravity
whereas some might be comparatively dominant like adhesive and friction effects. This
implies that the experiences and disciplines based on the macro scale level are no longer
-1-
Chapter 1. Introduction
always valid. The coupling between domains can therefore be much stronger than at macro
scale and can be incentive to new innovative applications [2].
The coupling between the electrical and mechanical domain is not exclusive of the MEMS
field, but there are characteristics that are unique when it is applied in MEMS. Besides, the
electrostatic force has stronger effect in miniaturized devices than in the devices in macro
scale, which enables the electrostatic force to be used in actuation. Some examples of
electrical and mechanical coupling applications are electrical motors, air compressors as well
as true RMS-to-DC converters. The last one is a very good example of the coupling between
the electrical and mechanical domains at the macro level [2].
-2-
Chapter 2
The Nanoindenter and Motivation
2.1 Nanoindenter Sensor
Indentation is a straight forward method to test the mechanical properties of materials. The
basic principle which lies behind indentation is to apply a load to deform the testing materials
and intrinsic material properties such as hardness and elastic modulus can be obtained [3].
In the indentation process, the displacement and the applied load are required information
in order to perform further investigation. For example, the hardness and the effective elastic
modulus can be calculated from:
H=
E eff =
Pmax
A
1 π dP
2 A dh
Figure 1.1: Load-displacement curve for indentation.
-3-
Chapter 2. The Nanoindenter and Motivation
H is the hardness, Pmax is the maximum applied load, A is the contact area. Eeff is the effective
dP
elastic modulus and dh is the derivative of the unloading curve.
In the original design of the nanoindenter sensor developed by Nanofactory Instrument AB
[4], the load applied could be obtained from the capacitance readout, calibration was done
before the testing, and precise value of the applied load could be obtained. The displacement
of the experiment is measured by using a piezoelectric positioning system on which the
sample is mounted.
For small volume test, in situ TEM nanoindentation was often applied. Since TEM gives a
good real time imaging, the indentation depth and the contact area could be precisely obtained.
The TEM holder and the positioning setup designed by Nanofactory Instruments AB are
shown in Fig. 1.2.
Figure 1.2: Specimen holder and a detailed sketch for sample and sensor relative positions.
In the indenting process, an indenter tip is mounted on the sensor holder to indent sample.
The tip is designed as a cylindrical base with various kinds of tip. Different types of tips are
chosen for different purposes.
Figure 1.3: Sketch of the nanoindenter force sensor. (a) The sensor design seen from above.
-4-
2.1. Nanoindenter Sensor
(b) The sensor in cross section. The arrow ↨ indicates that the motion of the plate is restricted
to up and down.
This thesis work is focused on electrical and mechanical coupling of an electrostatic
actuated parallel-plate capacitive sensor. Moreover, there were some work of testing preloaded sensors and use parallel plate capacitor models for analysis. Besides, different masses
of small beads were placed on the sensor holders to simulate the load which applies when the
diamond tip indents a sample.
The sensors used in the experiment were developed by Nanofactory Instruments AB [3],
Fig. 1.4. The sensors functions as a capacitive sensor. The parallel plate capacitor consists of
two plates, one upper plate connected through the springs to a Si membrane and a lower Al
electrode deposited on glass.
The fabrication methods include photolithography, deep reactive ion etching and anodic
bonding [3]. The upper plate of the capacitive sensor is made of p+ doped silicon, and the
lower electrode is made of aluminum, Fig. 1.4.
Figure 1.4: Materials of the sensor. The upper electrode is p+ doped Si and the lower
electrode is Al. The springs are also made of p+ doped Si.
2.2 Motivation for the Feedback Loop Control
For the indenting process, the load and the indentation depth are fundamental information, Fig.
1.1. The applied load is obtained from the capacitance readout. The indentation depth is
supposed to be the same as the extension or contraction of the piezo-tube. However, the actual
indentation depth is more complicated. When the indenter tip starts indenting the sample, the
tip bounces back due to the elastic force of the connected springs, which leads to errors in the
indentation depth.
The concept for the feedback loop is to apply voltages on the plates and then produce an
electrostatic force [5]. Since the main displacement errors were due to the spring’s elastic
force, the idea of using electrostatic force to balance elastic force was developed and tested.
The basic feedback loop is shown in fig. 1.5; the piezo-tube movement h is the distance of
the piezo-tube extension, which is used to push the tip; δ is the compression of the sensor
plates and ∆ is the indentation depth. In the indentation process, the tip is moving toward the
sample, Fig. 1.5 (a).When the tip encounters sample, the tip deforms the sample with
indentation depth ∆ and the sensor plates are compressed with δ at the same time. The
extension of piezo-tube h is used for both indentation depth and the compression of the sensor.
Thus h = ∆ + δ . In order to make the feedback loop, the applied voltage V1 is turned down to
a certain value V2, which makes the compression δ zero, Fig. 1.5 (d). The electrostatic force at
-5-
Chapter 2. The Nanoindenter and Motivation
V2 is less than the force at V1, and this will release the compression of the two plates and
decrease the capacitance. Once the capacitance drops back to the original value, there is no
change of the gap, i.e. δ = 0 , and one loop is completed.
If the capacitance feedback and the voltage control respond fast enough, there would be
almost no change of the plates’ distance at any time, and thus the goal of recording the
indentation depth of the tip is accurately reached.
Figure 1.5: Indentation process diagram. (a) The indenter tip is moving toward the sample,
the actuation voltage is V1. (b) The tip is pushed forward by piezo-tube of extension h. (c) Tip
is in contact with sample surface and starts deforming the sample with indentation depth ∆.
The capacitor’s electrodes were also pushed together. (d) Use voltage V2 to control the
feedback loop (V1>V2). V2 was chosen to keep the electrode distance constant, i.e. δ=0; and
force due to piezo-tube will be used to indent sample instead of pushing electrodes.
-6-
Chapter 3
System Modeling and Analysis
3.1 Introduction
In this section, an overview of some parameter models for the surface micro systems will be
presented. Micromechanical equations of motion, spring constants, and capacitive position
sensing will be discussed.
3.2 Spring Constant
In the original sensor spring design, there were 3 types of serpentines structures [3]; in this
thesis work, two types of springs were chosen for testing, Fig. 2.1, Appendix C
Figure 2.1: Sketch of the sensor spring design. (a) Type II spring, designed with two turns. (b)
Type I spring, designed with one turn.
For the spring constants of the serpentine structure, the conditions will be different in the
three directions of motion. The nanoindenter sensor is designed to be very stiff in both x- and
y- direction, thus only the motion in z-direction needs to be considered. The model for
analyzing spring constant in z-direction is shown in Fig. 2.2 [6].
-7-
Chapter 3. System Modeling and Analysis
Figure 2.2: Two turns serpentine spring schematic.
By using free-body diagrams for calculation of the z-directed spring constant, Fig. 2.3,
the results show that the spring constants are 1000 N/m for the type I, 1 turn spring design
and 430 N/m for type II, 2 turns spring design. Detailed analysis and calculation are in
Appendix A.
Figure 2.3: Free-body diagram of a serpentine spring.
3.3 Elastic Force on Electrostatic Actuation
Consider a device which is a two parallel plate capacitor with one plate connected through the
springs to the upper plate, Fig. 2.4. The plate’s movement is designed to be in the normal
direction. Once a voltage is applied on the electrodes, the electrostatic force pulls the two
plates closer and connects the plate to the spring is affected by the elastic force. If the plate
displacement is noted as δ, thus the force exerted on the plates could be expressed as below
[5]:
F = Fe + Fk
-8-
( 2.1 )
3.3. Elastic Force on Elastic Actuator
Figure 2.4: Parallel plate capacitor with displacement by electrostatic force.
where Fe is the electrostatic force and Fk is the elastic force. When the system is at
equilibrium, the total force is zero:
F = Fe + Fk = 0
( 2.2 )
Therefore, the equation becomes:
F=
Aεε 0V 2
− kδ = 0
2( d 0 − δ ) 2
( 2.3 )
A is the electrode area, ε 0 = 8.854 × 10 −12 N −1m −2C 2 is the dielectric constant and ε = 1 since
the dielectric permittivity of air is very close to vacuum at room temperature, V is the applied
voltage, d 0 is the original distance between two plates without applying any voltage or force
and k is the spring constant.
The plot of Fe and Fk versus the displacement, δ, are shown below, Fig. 2.5. It is worth
noting that the displacement range is limited to be less than d0; as it is physically unfeasible if
the displacement goes beyond this range, solutions beyond this range are out of consideration.
The equilibrium position for the system can be obtained from solutions of Eq. (2.3). Fig.
2.5 shows the plot of the two forces with the displacement. The intersection points are the
solutions to the system [2, 5].
Figure 2.5: Dependence of electrostatic force and elastic force on displacement. The
intersections indicate equilibriums. Intersection c is an invalid solution since it is not a
physical solution.
-9-
Chapter 3. System Modeling and Analysis
As we can see from the graph, two solutions are of the equilibrium positions in the range
of [0, d0]. The first one is a stable solution; the second one is an unstable solution. If a small
perturbation of displacement is applied on the system around equilibrium positions, the elastic
force restoring to the equilibrium position is larger than the pushing–away electrostatic force
at position a, therefore, the system intends to remain at the equilibrium position. On the other
hand, it is the opposite case at the equilibrium position b, a small displacement at b will push
the displacement even further until the movable plate failing into contact with the fixed
electrode.
To describe the equilibrium position mathematically,
∂F
<0
∂δ
( 2.4 )
That is:
Aεε 0V 2
−k <0
(d 0 − δ ) 3
(2.5 )
Combine with Eq. (2.3), we then have
1
δ < d0
3
( 2.6 )
This means that the displacement for stable conditions is when the displacement is less than
one third of its original distance from the fixed electrode.
The pull-in (or snap-in) voltage V po is the minimal applied voltage to reach the critical
1
d 0 . For V > V po , Fe is larger than Fk . The movable plate will therefore fall
3
into contact with the fixed electrode.
The pull-in effect of the microstructure sets the limit for the device operation at a micro
scale. This special property is due to Paschen’s law [2, 7]. The value of the pull-in voltage
V po can be calculated from Eq. (2.3). Following notations are used:
displacement,
Aεε 0V 2
F
δ
~
and p = eo
, Fkd = − kd 0 , Feo =
y=
2
2d 0
Fkd
d0
Then we have
F
~
y (1 − ~
y ) 2 = eo = p
Fkd
( 2.7 )
1
For the critical displacement δ = d 0 , ~
y = 1 / 3 , and ~y (1 − ~y ) 2 is 4/27 . Therefore, the
3
condition for a stable solution is:
p=
Feo
4
≤
Fkd 27
which leads to the pull-in voltage [5]
- 10 -
( 2.8 )
3.3. Elastic Force on Elastic Actuator
V po =
8kd 03
27 Aεε 0
( 2.9 )
3.4 Electrostatic Micromechanical Actuator with Extended Travel Range
According to the above discussion, the travel range of the movable plate in the normal
direction is limited to one third of the initial gap. The movable electrode will be pulled into
contact with fixed electrode when the voltage is higher than the pull-in voltage. However, the
first test, which is presented in Section 4.2, show that the model discussed above does not
describe our MEMS system good enough, and some experimental results, e.g. higher snap-in
voltage and lower capacitance readouts can not be explained by the simple model. Therefore a
more complex model which includes serial and parallel capacitors is assumed.
Several methods have been suggested to extend the travel range for the electrostatic
actuators. The manufacturing solution to increase the entire initial gap is preferable for optical
applications. Other methods include closed-loop voltage control, series feedback capacitance
and leveraged bending [8]. The following is a detailed discussion of the MEMS system with
serial capacitor [5].
Figure 2.6: Extended travel range by a serial capacitor.
Use the notation C M for the capacitance of the mechanical structure of the sensor and C S
1
1
1
for the capacitance due to the serial capacitor. Since =
+
, where C is the total
C C M CS
C C
capacitance being measured, the resulting capacitance is C = M S . Moreover, denoting
C M + CS
δ
Aεε 0
~
y=
and C0 =
, gives that the mechanical structure capacitance between the plates A
d0
d0
Aεε 0
C0
≡
and B is C M =
. For capacitors in series, the amount of charge Q is the
(d 0 − δ ) (1 − ~
y)
same over each plate, Q = C SVS = C M V M , where VS is the voltage drop on the serial capacitor
and VM is the voltage drop on C M . Thus,
- 11 -
Chapter 3. System Modeling and Analysis
VM =
CsV
C s + C0
1
y
1− ~
=
C s (1 − ~
y )V
~
Cs (1 − y ) + C0
( 2.10 )
When the elastic force of springs and the electrostatic force are in equilibrium, it requires
Aεε 0V M2
= kδ
2( d 0 − δ ) 2
Suppose that C s = bC0 and yˆ ≡
have:
( 2.11 )
b ~
y , b is a factor to simplify the equations below, we then
1+ b
yˆ (1 − yˆ ) 2 =
b 3 Aεε 0V 2
2(1 + b) 3 kd 03
( 2.12 )
From Eq. (2.12), the maximum stable region for yˆ ranges from 0 to 1/3 before the pull-in
effect occurs.
However, the original mechanical sensor and series capacitor form a voltage divider circuit;
thus, higher voltage needs to be applied in order to compensate the bias drop over the serial
capacitor C S . Therefore, a higher driving voltage is needed to achieve the same displacement
in the MEMS capacitor CM. According to Eq. (2.12), the voltage VM for the maximum
displacement (i.e., ~
y = 1 ) is:
3
VM =
2kd 0
Aεε 0 b 2
( 2.13 )
Compare the pull-in voltage V po and the bias on the sensor VM , from Eq. (2.9), the relation
between V po and VM is [6]:
Vm =
27
V po
4b 2
( 2.14 )
3.5 Extra Capacitors
From the comparison of theoretical fitting and experimental result, it is suspected that besides
the sensor capacitance existed, there are also serial and parallel capacitances. When analyzing
the capacitive sensor in detail, a serial capacitor can be expected at the connection point from
the Al electrode to the upper plate in Si, Fig. 2.7.
- 12 -
3.5. Extra Capacitors
Upper electrode
connection
Press contact beneath Si
plate
Lower electrode
connection
Figure 2.7: Schematic plot for sensor electrodes connection. (a) Press contact is used to
connect one electrode to the Si plate. (b) The lower electrode is an integrated design; no extra
connection between the electrode and the Al plate is needed.
Fig. 2.7 shows the electrode connection. The serial capacitor is believed to exist at the
press contact. Press contact was used to connect one electrode to the upper plate, it consists of
three layers, p+ doped silicon, aluminum and glass. It is well-known that aluminum oxidizes
easily and quickly once it is exposed to oxygen. Aluminum oxide Al2O3 is a dielectric
material with dielectric constant 9.1ε 0 at 25ºC [9]. The press contact could therefore form a
capacitor in series with the sensor capacitor, Fig. 2.8.
Another factor which influences the capacitance readout is the parallel capacitor, Cp. The
parallel capacitor could be due to several sources, such as electronic devices, measurement
setups and wires for circuit connection. With a capacitor Cp in parallel with the mechanical
capacitor CM, the capacitance shifts by a constant, C = C M + C p , and the voltage applied on
the sensor is the same with additional parallel capacitor. Comparing to the serial capacitor, the
parallel capacitor is of minor importance since serial capacitor changes the shape of the curve
of the C-V measurement whereas the parallel capacitance affects the measurement by adding
an offset.
- 13 -
Chapter 3. System Modeling and Analysis
Figure 2.8: Side view of sensor capacitor and position of the suspected serial capacitor.
3.6 Bent Plate
Besides extra capacitors in the system, it is found out that the mechanical sensor CM is not of
purely parallel plates. It is discovered that the upper plate is bent at the four corners from the
interferometer measurement, Fig. 2.9.
Figure 2.9: White light interferometer measurement shows the bent degree of the upper plate.
The central part is higher than the positions at the four corners.
The possible reasons for the bending of the plate might be stress and strain from type of
the Si wafer type used. The plate was made of SOI (silicon on insulator) wafer, where the
different lattice constants can give deformation when two materials are bonded. Moreover, the
holder placed on the upper plate can also lead to some asymmetrical factors for the sensor’ s
plate.
To analyze the effect of a bent-plate on the capacitance, a modified model instead of
parallel plate capacitor was used. From Wyko measurement, the bent profile along one edge is
shown in Fig. 2.9. Instead of using the arc shape bending curve, an asymmetric trapezoid
curve without the bottom line was used for the estimation, Fig. 2.10.
- 14 -
3.6. Bending Plate
Figure 2.10: Modified curve for bent plate. The light color line is the real profile for the
bending shape; the dark line is the trapezoid curve for the bending.
It is worth emphasizing that most of the bending is at the four corners of the plate, the
central circle and the middle part of plate edge is flat. With the simplified bending profile
shown in Fig. 2.10, the upper plate of a bent-plate capacitor model is shown in Fig. 2.11. To
calculate the capacitance for this model, the flat part is considered as a parallel plate capacitor
whereas integral was used to calculate the capacitance for the four bent parts of the plate [10].
Thus, the total capacitance for the model is:
Ctotal = C Parallel _ plate + C Bent _ plate
C Parallel _ plate =
εA'
d0 − δ
C Bent _ plate = ∑ ε ∫∫
4
i =1
dxi dy i
z i ( x, y )
( 2.15 )
( 2.16 )
( 2.17 )
where A'is the flat area, z is the distance at each point to the lower plate and the bent plate
capacitance is the sum of the four corners. Since z is different at every point on the plane, z is
a function of x and y.
- 15 -
Chapter 3. System Modeling and Analysis
Figure 2.11: Schematic 3D plot of the modified bent-plate model. (a) Vertical view from
above. (b) Side view.
- 16 -
3.6. Bending Plate
The calculation of bent plate is shown in Fig. 2.12. Voltages at 10V, 13V, 16V and 19V were
chosen for the analysis. Since the voltage is the readout from the whole system, serial and
parallel capacitors were also taken into consideration for the bent plate model with Cs 15pF
and Cp 1.1 pF. The deformation degree of the bending does not change as the voltage varies.
Also, it is discovered that the central part of the plate is arched up as the corners are bent
down. Thus, the resulting capacitance is not as high as expected. Compensation mechanism
exists and does not result in prominent values for the capacitance. Besides the plate bending,
the springs are also bent along the longitudinal direction. However, due to limited time, there
was no further investigation of how the bent springs influence the capacitance performance.
Figure 2.12: Calculation of bent plate model in comparison with experimental data and
theoretical modeling.
3.7 Analytical Model for Nanoindenter Sensor
In this section, a model of the electrostatic behavior of the sensor with all factors taken into
consideration will be presented. Establishing a model for the sensor is like investigating a
black box, Fig. 2.13. The box cannot be opened to give any clues of the components inside.
The models are the capacitance and voltage readouts, which are the result of all the
components inside the black box. Only the displacement, measured by Wyko NI1000, could
be specified to the mechanical sensor CM.
- 17 -
Chapter 3. System Modeling and Analysis
Figure 2.13: Schematic sketch of the sensor model. C and V are the measurements done on
the system.
From section 2.4 and section 2.5, the capacitance of a systems with both serial and parallel
CM CS
capacitors can be expressed as C =
+ C P , where C M indicates the mechanical
CM + C S
capacitance – sensor’ s capacitance; C S is the serial capacitance and C P is the parallel
capacitance. As for the voltage, since parallel capacitor does not divide the applied voltage as
serial capacitor does, from Eq. 2.10, the total voltage for the system can still be expressed as
δ
C0 + C S (1 − )
d0
× VM , where C0 is the initial capacitance without any applied voltage.
V =
δ
C S (1 − )
d0
2k
Aεε 0
and VM =
δ ⋅ (d 0 − δ ) 2 in the above expressions, C and V can be
εA
(d 0 − δ )
rewritten as:
Use C M =
Aεε 0
× CS
(d 0 − δ )
C=
+ CP
Aεε 0
+ CS
(d 0 − δ )
V =
C0 + C S (1 −
C S (1 −
δ
)
d0
δ
)
d0
×
2k
δ ⋅ (d 0 − δ ) 2
εA
( 1.18 )
( 2.19 )
Higher total voltage is required with the serial capacitor in the system. Since the voltage is
distributed on both the CS and CM, V is defined as the total applied voltage and VM is the
voltage drop on the mechanical capacitor.
To plot the C-V relation, the capacitance and the voltage can be view as functions of δ,
C (δ ) and V (δ ) . By eliminating δ, the C-V relation is presented below.
- 18 -
3.7. Analytical Model for Nanoindenter Sensor
Figure 2.14: Analytical modeling prediction for the C-V relation of the sensor
- 19 -
Chapter 4
Experimental Methods
4.1 Introduction
For the reliability and accuracy of the measurement, two different methods were utilized to
measure the relations between capacitance and voltage. One way was to use a commercial
chip, AD7746 [11, 12] to measure the capacitance while an external DC bias was applied over
the sensor from a power supply unit. The other measurement method was to use a LCR meter,
HP 4284A [13], to apply voltage on sensor and simultaneously measure the capacitance.
Moreover, white light interferometer, Wyko NT1100 [14], was used to directly measure the
displacement while applying a voltage and measuring the capacitance.
4.2 C-V Measurements
To use IC chip AD7746 in combination with an external applied DC bias, the circuit used in a
setup show in Fig. 3.1. AD 7746 applies an AC square wave voltage and a DC constant
voltage 2.5V to the sensor [12], Fig. 3.2. This preloaded DC voltage gives an offset voltage
− 2.5 V for the voltage supplied by PSU. The frequency of the square wave is 32 kHz and the
excitation voltage 2.5V. The sensor capacitor, marked as CS, was placed in series with a
capacitor CB. CB should have capacitance far exceeds CS in order to give low impedance in the
circuit. According to the formula of impedance [15]:
1
Z=
( 3.1 )
2πfC S
The sensor capacitor gives resistance around 1MΩ whereas CB, which is 100nF here, gives
around 50Ω. One end of capacitor electrode is powered by the PSU with a resistor in parallel
to the ground; the other electrode is connected to the USB socket of computer. A big
resistance with respect to the sensor capacitor, R, should also be used in the circuit to avoid
leakage current. Moreover, according to the datasheet for AD7746 [12], parasitic resistance
more than 30MΩ should be used to get leakage current small enough (i.e. I < 150nA). The
supplied voltage was read from the PSU and the capacitance was read out from an evaluation
software program for the AD7746.
Different voltages were applied from the power supply unit in the range of 0~25V. At the
same time, the capacitances of the sensors were measured with the AD7746. 10 sensors were
chosen to repeat this test to obtain general characteristics of the sensor performances. Among
the 10 sensors, 4 of them are of spring type I, one turn design, and the rest sensors are type II,
two turns design.
- 20 -
4.2. C-V Measurement
50M
Figure 3.1: Circuit design to test AD7746.
Figure 3.1: Voltage supply from AD7746. A square wave with frequency 32 kHz in
combination with a constant voltage 2.5V were supplied together.
- 21 -
Chapter 4. Micromechanical Test
Figure 3.2: Photograph of the experimental setup. (a) Experimental set up for testing with IC
chip AD7746. (b) Photograph of IC chip AD7746 evaluation board.
The second way of testing the C-V relation was to use a LCR meter, HP 4284A. No
additional circuit was needed to be constructed for the testing. The DC voltage from the LCR
meter is applied to the sensor directly by two probes in contact with two electrodes on the
sensor. The capacitance output was read out directly from the screen on the LCR meter.
The HP 4284A efficiently gives the C-V curve compared to AD7746. In the circuit design
for AD7746, a long RC recovery time around 20 seconds prolongs the measurement period.
Also, HP’ s 4284A LCR meter provides more choices for testing e.g. the dependence of the
amplitude and the frequency of the excitation voltage on the capacitance. On the other hand,
AD7746 has a much smaller volume as compared to the LCR meter, which is the most
dominant advantage and the reason to be chosen as one device in the feedback loop design.
Figure 3.3: Experimental set up for testing with LCR meter HP 4284A.
4.3 Optical Surface Profile
The capacitive sensor is actuated by the electrostatic force. For 1-DOF model of the parallel
plate capacitor, the gap distance and electrode displacement is of substantial importance for
the resulting capacitance. Optical surface profiler Wyko NT1100 is a convenient way to
profile the sensor surface at different applied voltages [16]. From surface profile
measurements, the sensor plate movements ∆d can be extracted. The focus of lens should be
adjusted to get circular fringes on the plate; and as for data analysis, sample tilting should be
taken into consideration.
- 22 -
4.3. Optical Surface Profile
Figure 3.4: Optical image for the upper electrode of sensor capacitor. Corner 1 is fixed and
independent of applied voltage. Oppositely, corner 2 is sustained by the connected spring and
drops down as the applied voltage increases.
By the white light interferometer, there is no direct way to determine the distance between
two electrodes under different voltages. One alternative is to measure the upper electrode’ s
displacements instead. A specified corner was chosen out from the four corners, the white
light interferometer was used to make a scan of the corner at different voltages. A point on a
fixed corner 1 was chosen and compared to another point on the movable electrode 2, and the
two points were used to measure the difference in distance differences. The d 0 was obtained
by breaking a sensor, measuring the spring thickness and subtracting this from the profile
meter measurements from the spring down to the Al electrode. The displacements, ∆d , due to
different voltages come from subtraction of the difference in displacement with applied
voltage minus distance difference without voltage.
The method to extract d 0 is demonstrated here. The optical profile Fig. 3.5 gives the total
distance from upper electrode down to the bottom. This distance includes gap distance and
thickness of spring. Then from photos taken by microscopy, the thickness of the springs was
also known. As a result, the original gap separation could easily be obtained.
Figure 3.5: Wyko 1100 optical image to measure d 0 . (a) The arrow indicates one ditch for
depth profile. (b) The depth profile for the ditch. It indicates that the distance from electrode
surface down to the bottom of aluminum electrode is 16.4 µm.
- 23 -
Chapter 4. Micromechanical Test
Figure 3.6: Microscope photo for the side view of the spring. These springs were picked up
from broken sensors. Estimation of spring side thickness is around 14.4µm.
In order to measure the spring constant, k, tests were done to measure the displacement
with a known mass applied. The spring constant could then be calculated using Hooke’ s law.
However, due to large bead cross-section area and no suitable mass alternatives found, the
idea of using optical profiler to get spring constant was discarded.
4.4 Tilting Effect and Surface Roughness
The reason which supports the method to take only one corner for displacement measurement
is that assuming the plate does not tilt and that, the whole plate moves simultaneously and
uniformly, meaning that displacement at one corner could be generalized to the whole plate.
2
1
Figure 3.7: Two corners at diagonal relative positions were selected for tilting test.
To verify this, one sensor with one turn of spring as seen in Fig. 3.7 was chosen for the
measurement, the two corners are at relatively diagonal positions. Comparing the capacitance
calculated from the displacement at each corner, it is shown that the capacitance correlated to
each other. Therefore, this demonstrates that it is feasible to choose only one corner for
analysis. Moreover, this definitely saves time in data analysis.
- 24 -
4.4. Tilting Effect and Surface Roughness
Figure 3.8: Experimental result for tilting effect. The dotted line is the measurement from IC
chip AD7746. ∆ comes from calculation of displacement at corner 1 and □ comes from
calculation of displacement at corner 2. Wyko measurements at two corners were
photographed while measuring the capacitance with chip AD7746.
- 25 -
Chapter 5
Experimental Results
5.1 Capacitive Behavior
To ensure the capacitive characteristics of the MEMS sensor, the impedance Z of the sensor
was measured using the LCR meter. Impedance Z is the total opposition a circuit or device
offers when an alternating current of certain frequency flows through the circuit. Z contains a
real and an imaginary part, which could be expressed by impedance and reactance in polar
form [17], Z = z ∠θ , Fig 4.1 Using LCR meter to measure the impedance and the angle of
the sensor, Fig. 4.2, and the phase measurement show a clear capacitive behavior for the
frequencies above 10 kHz. [15]. However, 100 kHz had an unexpected affect on the
measurement; thus, the measurement was done using 1 MHz.
Figure 4.1: Definition of impedance. (a) Voltage, current and impedance. (b) Vector
representation of impedance.
- 26 -
5.1. Capacitive Behavior
Figure 4.2: The impedance Z and phase shift θ measured by LCR meter under different
frequencies and excitation voltage amplitude. (a) Plot of impedance Z with applied voltages
(b) Plot of degree θ with applied voltage. .
5.2 Optical Profiler Measurement
Displacement-capacitance and displacement-voltage measurements were done by using white
light interferometer Wyko NT1100. The experimental results together with theoretical
modeling are shown in Fig. 4.3 and Fig. 4.4.
- 27 -
Chapter 5. Experimental Results
Figure 4.3: Displacement-Capacitance relation for type I (one turn) spring and type II (twoturns spring). The measured data is from LCR meter.
Figure 4.4: Displacement-Voltage relation for type I (one turn) spring and type II (two-turns
spring). The measured data is from LCR meter.
- 28 -
5.2. Optical Profiler Measurement
Neglecting the fringe effect, the theoretical fitting for the displacement-capacitance
measurement shown in Fig. 4.3 is
εAI
8.854 × 10 −12 × 769600 × 10 −12
×C SI
× C SI
d −δ
1.8 × 10 −6 − δ
( 4.5 )
+ C PI for type I
C= 0
+ C PI =
εAI
8.854 × 10 −12 × 769600 × 10 −12
+ C SI
+ C SI
d0 − δ
1.8 × 10 −6 − δ
εAII
8.854 × 10 −12 × 699200 × 10 −12
×C SII
× C SII
−6
d0 − δ
×
−
1
.
8
10
δ
+ C PII =
+ C PII for type II ( 4.6 )
C=
εAII
8.854 × 10 −12 × 699200 × 10 −12
+ C SII
+ C SII
d0 − δ
1.8 × 10 −6 − δ
where C SI is 15 pF, C PI is 0.8 pF, C SII is 9.3 pF and C PII 1.1 pF. The modeling fitting of
displacement-voltage relations shown in Fig. 4.4 is
V =
δ
)
2 × 999.49
1.8 × 10 −6 ×
δ ⋅ (1.8 × 10 −6 − δ ) 2 for type I
−12
−12
δ
8.854 × 10 × 769600 × 10
C SI (1 −
)
−6
1.8 × 10
C0 I + C SI (1 −
( 4.8 )
V =
δ
)
2 × 428.1
1.8 × 10 −6 ×
δ ⋅ (1.8 × 10 −6 − δ ) 2 for type II
−12
−12
δ
8.854 × 10 × 699200 × 10
C SII (1 −
)
−6
1.8 × 10
C0 II + C SII (1 −
( 4.9 )
where C0I and C0II are the initial capacitances without any applied voltages for type I and type
II sensor.
To make a comparison, capacitance-voltage measurement with modeling of the parallelplate capacitor excluding serial and parallel capacitors are shown in Fig. 4.5.
Figure 4.5: Fitting without serial and parallel capacitors. The left one is type I sensor, the
right one is type II sensor.
In Fig. 4.5, there were obvious discrepancies between the theoretical fitting and the
measurement data for both AD7746 chip and LCR meter. On the contrast, if combining Fig.
- 29 -
Chapter 5. Experimental Results
4.3 and Fig. 4.4, the modeling fits well to the C-V measurement if serial and parallel
capacitors are taken into consideration, Fig. 4.6.
Figure 4.6: C-V characteristics for type I (one turn, stiff) spring and type II (two turns spring,
soft). The measured data is from LCR meter.
5.3 Nanoindenter Feedback Loop
Different masses were applied with a pre-loading voltage. The original purpose was to
construct a feedback loop of indentation. In our design of the nanoindenter, one problem is
that the probe is pushed back during indentation due to the pressed springs recovering force.
At larger forces the probe can move out of the imaged area, which makes the analysis
incomplete. The idea behind the feedback loop was that during indentation, the capacitance
increases because the two plates become closer; after a decrease of the voltage, the plates
were released and the capacitance went back, which could make elastic force small and keep
the nanoindenter always under the microscopy focus area.
In the preliminary test, small beads with a known mass were mounted on the sensor holder
to simulate applied load during indentation [3], Fig. 4.7. From this preliminary test, maximum
workload and the corresponding tuning voltage range could be extracted.
- 30 -
5.3. Nanoindenter Feedback Loop
Selected mass
Load force
of beads (mg)
(µN)
16.35
160.23
12.62
123.68
7.05
69.09
4.06
39.79
2.02
19.79
0.85
8.33
Figure 4.7: (a) Schematic plot of the nanoindenter force-sensor with an added mass. (b)
Table of selected mass of beads and the corresponding load force.
Fig. 4.8 shows the C-V measurement of a type I, stiff spring sensor with small bead 16.29
mg and an initial actuated voltage 15V. The capacitance is measured by AD7746 with
conversion time 62.0 ms /16.1 Hz and excitation voltage at VDD/2 level, VDD = 5V. When
comparing the capacitance readout with mass applied and without mass, it shows that the
capacitance increases with 0.2 pF when a mass of 16.29 mg is applied. The increased
capacitance implies that the two electrodes are pressed closer together due to the mass. Now
with applied voltage and extra load due to the mass, the force exerting on this system could be
expressed as:
kδ =
εAV 2
+ mg
2( d 0 − δ ) 2
- 31 -
( 4.10 )
Chapter 5. Experimental Results
Figure 4.8: C-V measurement with and without applied mass. The curve shape with applied
mass is quite similar to the other one without mass.
By measuring C-V with mass applied to the system, it demonstrates the possibility to
construct feedback loop mechanism for the nanoindenter sensor. Based on the measurement
presented in Fig. 4.8, a manual loop was executed and satisfying result was obtained. The
schematic demonstration of manual feedback loop is shown in Fig. 4.9.
Figure 4.9: Manual feedback loop based on measurement in Fig. 4.8. (a) Initial voltage is
15V on the sensor and the capacitance is 4.15pF. (b) Add a mass with 16.29mg, the
capacitance increased to 4.38pF. (c) Turn down voltage from 15V to 10V, capacitance drop
back to the initial value.
The bead mass put on the sensor holder is used to simulate applied load on capacitive
sensor when the tip is indenting sample, Fig. 1.5. Fig. 4.9(b) characterizes compressed plates
as in Fig. 1.5(c). And Fig. 4.9(c) gives the ideas for the voltage needs to be turned down to
keep electrodes distance constancy as in Fig. 1.5(d).
Repeating this measurement for different initial voltages, the voltage is needed to
compensate for the load is shown in Fig. 4.10. It shows that higher initial voltage gives larger
forces range whereas lower initial voltage decreases the forces range.
Figure 4.10: Manual feedback loop for voltage turned down at different applied load.
- 32 -
Chapter 6
Conclusion
6.1 Summary
In this thesis work, the performance of an in situ nanoindentation sensor combined with IC
chip AD7746 has been investigated. To test the possibility and feasibility of the feedback loop
for the sensor, C-V measurement was done as preliminary test. Several unexpected results
were measured, like offset voltage, higher snap-in voltage and higher capacitance readout. To
explain these unexpected phenomena by a parallel-plate capacitor model, the simple model
has been improved and modified. Serial capacitor, parallel capacitor and bending plate were
parameters that were added to the model. With those parameters the model could be
completed. To establish the model additional displacement measurements were done by using
white light interferometer.
The feedback loop which was the final goal of the project has been tested using manual
feedback. Small beads with known mass were used to replace the role of real load, and the
manual feedback loop was successfully established. With current sensor design, the feedback
loop works well if the applied load is less than 200 µN. The original plan of developing an
automatic feedback loop for the sensor was postponed for several reasons, like the big RC
constant results in a long response time, the uncertainty about the serial capacitance and
designing new software program interface. Hence, the construction of feedback loop will wait
until a new design of nanoindenter sensor is performed.
6.2 Recommended Future Work
According to the basic investigation of nanoindenter capacitive sensor, some work remains
before producing the new generation of nanoindentation sensor. Unavoidably, some
mysterious phenomena were found and will remain as new tasks for further studies. These
include different offset voltages for different spring type sensor, control over the serial
capacitance, different curve shape of the measurements of LCR and AD7746, the bending
degree of plates and the precise value of spring constant.
For the next generation nanoindenter sensor, the following items should be considered:
• First, during the design and manufacturing process, the existence of serial capacitor
should be taken into account. Some alternative such as discarding the press contact
and using other ways to connect electrodes might be possible.
• Second, a modified design of the sensor electrode might be needed to achieve an
efficient, reactive feedback loop. The existing circuit design gives the reaction time
around 20~30 seconds, which are too long for an efficient feedback loop.
- 33 -
Chapter 6. Conclusion
•
•
•
Third, the feedback loop which has been tested so far can achieve maximum load
around 200 µN. However, for the actual application it is preferred to have forces up to
1000 µN. Therefore, a new investigation of the spring design is recommended.
Moreover, it is recommended to investigate the bending plate further. Several related
factors like wafer manufacturing, thickness of sensor plate design and plate shape
could be further considered before designing. Besides, it is recommended to do
simulation for the bending plate model. It will be more valid and convinced if there
are simulation results in comparison with the analytical model presented in this work.
Finally, it is required to develop a program and new interface for the controlling of
nanoindenter sensor based on AD7746.
- 34 -
Acknowledgement
Acknowledgement
First and foremost, my thanks go to my two supervisors, Alexandra Nafari and Johan
Angenete. I would not make anything without their generous help. Moreover, I would like to
appreciate them for giving me the chance to do this project in Nanofactory Instruments AB.
And thanks for reading and revising my report patiently, carefully and giving me excellent
and practical suggestions. Deep and sincere appreciation to Alexandra, for her patience and
willingness to teach me, take me to try many new things and give me many opportunities. By
working with her, I leaned many useful and interesting new skills.
Secondly, I am grateful to all my colleagues in Nanofactory Instruments AB (alphabetical
order: Andrey Danilov, Ann-Jeanet Jörgensen, Björn Ahlgren, Christelè Grimaud, Dan
Olofsson, Jens Dahlström, Klas Nordström, Ludvig de Knoop, Mikael Johansson, Mikael von
Dorrien, Oleg Lourie, Paul Bengtsson). They are friendly and willing to give me a hand when
I need help. Klas is especially thanked for his help with building up the electronic circuits for
AD7746 testing, and I sincerely appreciate his patience with teaching and discussing with me
the fundamental knowledge of electronics. I would like to mention Gittan here, too. She
always brings happiness and laughs wherever she is present, and this makes the working
environment more joyful and enjoyable.
Two other people who can not be missed here are Professor Peter Enoksson at the
BioNanoSystem Laboratory, Chalmers and Sjoerd Haasl at Imego. Thanks for providing
valuable ideas for result discussion and pointing out direction for further investigation. And
these help us to make this project a satisfying work.
Finally, I am obliged to my family, the teachers and friends I have met in Linköping and
Göteborg. Thanks for encouraging and supporting me to do my thesis work in Nanofactory
Instrument AB and to pursue my studies in Sweden.
- 35 -
Bibliography
Bibliography
[1] R.P. Feynman, “There’ s Plenty of Room at the Bottom” in Miniaturization, pp. 282-296,
Reinhold Publishing, New York, 1961
[2] Dynamics and Nonlinearities of the Electro-Mechanical Coupling in Inertial MEMS, Luis
Alexandre Rocha, PhD thesis, Delft University
[3] MEMS Sensor for In Situ TEM Nanoindentation, Alexandra Nafari, Licentiate thesis,
Chalmers University of Technology, 2007
[4] Nanofactory Instruments AB, www.nanofactory.com
[5] Handbook of Sensors and Actuators, Micro Mechanical Transducers, Pressure Sensors,
Accelerometers and Gyroscopes, M,-H. Bao, Elsevier, 2004
[6] Simulation of Microelectromechanical Systems, K. Fedder, PhD thesis, University of
California at Berkeley, (1994)
[7] Electrical Breakdown Phenomena for Devices with Micron Separations, Ching-Heng
Chen, J Andrew Yeh and Pei-Jen Wang, Journal of Micromechanics and
Microengineering, 2006
[8] Electrostatic Micromechanical Actuator with Extended Range of Travel, Edward K. Chan,
and Robert W. Dutton, Fellow, IEEE
[9] http://www.accuratus.com/alumox.html
[10] Fundamentals of Engineering Electromagnetics, David K. Cheng, Addison Wesley (1993)
[11] Analog Devices, www.analog.com/en/
[12] Datasheet for Analog Devices AD7746
[13] http://www.testwall.com/datasheets/HP-4284A.pdf
[14] Veeco, www.veeco.com
[15] Introduction to Electric Circuits, Richard C. Dorf, James A. Svoboda, Wiley
[16] Surface micromachined RF MEMS variable capacitor, Dong-Ming Fang, Shi Fu, Ying
Cao, Yong Zhou, Xiao-Lin Zhao, Microelectronics Journal 38, 2007
- 36 -
Bibliography
[17] http://www.murata.com/cap/measure.pdf
[18] Datasheet for LCR meter HP 4284A
[19] Mastering MATLAB 7, Duane Hanselman, Bruce Littlefield, PEARSON Prentice Hall
(2005)
- 37 -
Appendix A
Appendix
Appendix A -- Analytical Spring Constants
Serpentine flexure was adopted for the sensor spring design. The snake-like meandering of
spring segments gives the name serpentine. Meanders are of length a and width b. For each
spring, one end is a guided-end and the other one is a fixed-end, Fig. A1. This boundary
conditions lead to the fact that the spring motion is limited to the preferred direction.
Fig. A.2 shows the free-body diagram for serpentine spring with n meanders. The
longitudinal segments are called span beams whereas the horizontal ones are defined as the
connector beams, Fig. A.1. Besides, the connector beams are indexed from i=1 to n and the
span beams are indexed from j=1 to n-1.
Figure A.1: Serpentine springs schematic.
Necessary terms are defined first and later will be used to calculate the spring constant.
Consider one guided-end beam having length L, width w, and thickness t. If force is applied
on the beam, the beam is bending with bending moment of inertia. The bending moment of
inertia about the z-axis is defined as I z . For the rectangular beam cross-section [5],
Iz = ∫
t/2
Ix = ∫
w/ 2
−t / 2
Similarly,
∫
x 2 dxdz =
tw3
12
( A.2 )
∫
z 2 dzdx =
t 3w
12
( A.3 )
w/ 2
−w/ 2
−w / 2
t/2
−t / 2
With torsion, the torsion modulus G is related to Young’ s modulus E and Poisson’ s ratio υ
G=
E
2(1 + ν )
Since the material of springs is silicon, having E=165 GPa, υ=0.3. And we have torsion
modulus G = 6.346 × 1010 .
- 38 -
( A.4 )
Appendix A
Also, the torsion constant for a beam of rectangular cross-section is
 192 t ∞ 1
1
 iπw  
J = t 3 w1 − 5
tanh 

∑
5
3
π w i =1,iodd i
 2t  

The z-direction spring constant for the flexure is
k z ,beam =
Fz
δ z . For n even,
12 S ea S eb S ga S gb
S eb S ga a 2 (S gb a + S ea b)n 3 − 3S ea S eb S ga a 2bn 2 +


2
2
3
S ea b(2 S eb S ga a + 3S eb S gb ab + S ga S gb b )n − S ea S ga S gb b 
where
k z ,beam =
kz =
( A.5 )
( A.6 )
S ea ≡ EI x,a S eb ≡ EI x ,b S ga ≡ GJ a
S ≡ GJ b
,
,
, and gb
. For n odd,
12 S ea S eb S ga S gb ( S gab(n − 1) + S eb an)
S eb S ga a 2 ( S eb S gb a 2 + (S ea S eb + S ga S gb )ab + S ea S gab 2 )n 4 −



2
3
S eb S ga a b((3S ea S eb + S ga S gb )a + 4 S ea S gab)n +



2
3
2
2
2
2
2
3
2
S eab(2 S eb S ga a + (5S eb S ga + 3S eb S gb )a b + 4S eb S ga S gb ab + S ga S gbb )n − 

2
2
2
2
2 2
2 2 
2S eb S ga b (S eb S ga a + 2 S eb S gb ab + S ga S gb b )n + S ea S gab ( S gab − 3S eb a ) 
( A.7 )
For type I spring, n=3, a=40µm, b=340µm, w=30µm, and t=14µm. Therefore, we
have I xa = 6.86 ×10 -21 , I xb = 6.86 ×10 -21 , Sea = 1.13 ×10 -9 , Seb = 1.13 ×10 -9 , Sga = 1.19 ×10 -9 ,
Sgb = 1.19 ×10 -9 , and J a = J b = 1.94 ×10 -20 . Similarly, for type II spring, all parameters are the
same except n=4, b=330µm.
Since there are 8 springs connected to the upper plate, it is equivalent to 8 springs connected
in parallel, which results in the spring constant k z = 8 × k z ,beam . Finally, the spring constant for
type I is 999.5 N/m and it is 428.1 N/m for type II.
- 39 -
Appendix A
Figure A.2: Free-body diagram of a serpentine spring.
- 40 -
Appendix B
Appendix B -- AD7746 & LCR Meter
AD7746 is a 24-bit, capacitance-to-digital converter with temperature sensor manufactured by
company Analog Devices [10]. The capacitance input range is ±4 pF and by using the on-chip
program, up to 17 pF capacitance input could be adopted by the device.
The core structure of AD7746 is a converter made by a second order modulator and a third
order digital filter. For capacitive input, it functions as a CDC whereas for voltage input or for
the voltage from a temperature sensor, it works as an ADC.
Fig. B.1 shows the simplified CDC diagram. The input of the measured capacitance is
connected to one Σ-∆ modulator and a square wave excitation source. During the conversion,
the modulator samples the charge flaw through Cx continuously. Afterwards, the signals from
the modulator were processed by the digital filter to streams of 0s and 1s. Finally, after
scaling and calibration, final results come out at the serial interface.
It is worth noticing that since AD7746 is designed for floating capacitive sensors, none of
the measured capacitor Cx plates could be grounded.
Figure B.1: CDC Simplified block diagram.
Two possible capacitance inputs could be used for AD7746, one is single-ended capacitive
input, and the other one is the differential capacitive input. In this thesis work, single-ended
capacitive input was applied for all the measurements.
For single-ended mode, the allowed input range for CDC is from 0 pF to 4 pF, Fig. B.2.
Further, with the programmable CAPDAC, input range could shift up to 21 pF. Fig. B.2
shows how to span a ±4 pF range CDC to measure capacitance between 0 pF to 8 pF and fig.
B.3 shows capacitance between 13 pF to 21 pF measured with a programmable CAPDAC.
Figure B.2: Full ±4 pF CDC span to measure capacitance between 0 pF to 8 pF.
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Appendix B
Figure B.3: Shift the input range up to 21 pF.
The CDC architecture used in the AD7746 measures the capacitance Cx connected
between the EXC pin and the CIN pin.
Figure B.4: Σ-∆ CDC architecture.
The basic loop for measuring an unknown capacitor is shown in Fig. B.5. A reference
voltage is applied on CREF. The charge signals are transmitted to the comparator which
compares to the signals of charge from CSENSOR. The feedback loop is supposed to work as the
following descriptions: the excitation voltage for the CSENSOR generates a signal and sends it
to the integrator; meanwhile, VREF also generate a signal and send it to the integrator. At the
comparator, these two signals are compared and the feedback loop will execute some
procedures like charging (by VREF (+)) or discharging (by VREF (-)) CREF so that at the end
CREF and CSENSOR would have the same amount of charge.
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Appendix B
Figure B.5: Measurement circuit.
HP 4284 A LCR meter is one of the most common used LCR meter, the widest application
for LCR meter is to measure capacitance and dissipation factor of capacitor.
The basic principle lies behind LCR meter is the “automatic balancing bridge method” [16],
Fig. B.6. The name LCR meter implies it can be used to measure the impedance of inductors,
capacitors and resistors and other components.
Figure B.6: Principle diagram of Automatic Balancing Bridge.
To measure the capacitance, two modes can be chosen from which depends on if the
circuit is serial equivalent or parallel equivalent [16].
According to Eq. 3.1, small capacitance gives high impedance, which therefore gives more
prominent parallel resistance compare to the serial capacitance, fig. B.7 (a). This effect leads
to the possibility to ignore the effect of serial capacitance in the circuit, thus small capacitor
gives parallel capacitor equivalent circuit. Conversely, if the capacitance is large (low
impedance), then Rs has relatively more significance than Rp, so the series circuit mode
should be used [16, 17], fig. B.7 (b).
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Appendix B
Figure B.7: Capacitance measurement circuit mode. (a) Model to use for small capacitance.
(b) Model to use for large capacitance.
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Appendix C
Appendix C -- Spring Design
The detailed design of sensors spring and the size parameters are presented here.
Type I, stiff spring:
- 45 -
Appendix C
Type II, soft spring:
- 46 -
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