Download A Thesis Presented to the faculty of the Department of

PARAMETER DESIGN OF SMALL ELECTROSTATIC FORCE DRIVEN MECHANISMS
A Thesis
Presented to the faculty of the Department of Mechanical Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Mechanical Engineering
by
Adewale George Ogbogho
SUMMER
2014
PARAMETER DESIGN OF SMALL ELECTROSTATIC FORCE DRIVEN MECHANISMS
A Thesis
by
Adewale George Ogbogho
Approved by:
__________________________________, Committee Chair
Akihiko Kumagai
__________________________________, Second Reader
Tien I. Liu
____________________________
Date
ii
Student: Adewale George Ogbogho
I certify that this student has met the requirements for format contained in the University
format manual, and that this thesis is suitable for shelving in the Library and credit is to
be awarded for the thesis.
__________________________, Graduate Coordinator
Akihiko Kumagai
Department of Mechanical Engineering
iii
___________________
Date
Abstract
of
PARAMETER DESIGN OF SMALL ELECTROSTATIC FORCE DRIVEN MECHANISMS
by
Adewale George Ogbogho
The millimeter or insect scale sized electrostatic force driven mechanisms using a parallel
plate actuation with multilayered dielectric considering air gap as a dielectric has not
been delved into so much.
This thesis is directed towards design parameters of small mechanisms taking into
consideration, the dielectric material, which in this study, Teflon was the material of
choice due to its excellent dielectric properties, the size, area and thickness of the
capacitor plates and breakdown voltage and its effects on a small scale mechanism
operating on the principle of electrostatic actuation having a size such as that of an insect.
These parameters were determined using detailed theoretical and practical analyses
showing how these parameters relate to ensure an excellent design taking into
consideration the effect of breakdown voltage and its effect on the maximum force
exerted by the mechanism and on the overall dielectric properties of the dielectrics inbetween the capacitor plates.
iv
The maximum force was determined experimentally by connecting a parallel plate
actuator with Teflon and air as the dielectrics to a triple output DC power supply which
was connected to a power controller and DC high voltage power converter which
amplifies the input voltage by 4310 times. The resulting maximum forces and breakdown
voltage for different Teflon thicknesses and air gap distances were measured to determine
how to optimize the design of these mechanisms considering the effect of dielectric
constant and dielectric strength of the material on the force generated and to aid in
material selection for a mechanism as such.
_______________________, Committee Chair
Akihiko Kumagai
_______________________
Date
v
DEDICATION
This Thesis is dedicated to The Lord God Almighty, Jesus Christ my savior for seeing me
through all the easy and tough times and giving me the grace to pull through with my
masters degree program successfully. I give all thanks and praise to you Lord.
vi
ACKNOWLEDGEMENTS
Firstly, I thank the Lord God Almighty, Jesus Christ my savior for His unending love,
care and protection over me during my entire time at California State University,
Sacramento.
I also want to express my sincere gratitude to my thesis supervisor and graduate
coordinator of the mechanical engineering department, Professor Akihiko Kumagai for
his unconditional assistance in ensuring that I complete this thesis on time and with the
much expected quality by providing his positive criticism, advices and assistance all
through the duration of this thesis. God will continue to bless you sir. I’m also expressing
my gratitude to all the professors who took their time to teach and lecture me in various
courses all through the entire time of my studies.
I also appreciate my wonderful parents, Engr. Mike and Pastor (Barr.) Mrs. Adeola
Ogbogho, for their support financially, morally and spiritually and for their constant
communications to check up on me from a distant land even when it was inconvenient.
To my wonderful gift from God and Fiancée, Shanae K. Theall for her sweet words of
encouragement, time and assistance to ensure that I had the easiest time going through
my thesis and my degree. Also this isn’t complete without expressing my thanks to my
siblings, Mr. Olusola Ogbogho, Mrs. Omotola Bolarinwa, Mrs. Temitayo Akinnola and
Mrs. Omolola Ajasa for their prayers and care all through my entire program. God Bless
you all.
vii
I also want to appreciate the effort of Mr. Mike Newton at the ECS Tech Shop for the
assistance he rendered toward the modification of the base plates. And to the Ogbeide
Family, especially Daddy and Mummy Ogbeide for taking me into their house upon my
arrival from Nigeria. To my friends Anthony Ogbeide, Kio, Anji and Ibim Amachree, Joy
Ndem for being there as brothers and sisters to me, to the pastorate, Pastor Toks and
Toyin Adewunmi and entire members of the Redeemed Christian Church of God (Peace
Assembly), Sacramento and to all who contributed in one way or the other to my life
during this entire time. I can’t thank you all enough but God bless you all richly.
viii
TABLE OF CONTENTS
Page
Dedication ................................................................................................................... vi
Acknowledgements .................................................................................................... vii
List of Tables .............................................................................................................. xi
List of Figures ............................................................................................................ xii
Nomenclature ............................................................................................................. xiii
Chapter
1. INTRODUCTION……………………………………………………….. ...............1
1.1 Electrostatic Field ............................................................................................ 1
1.2 Dielectrics ....................................................................................................... 3
1.3 Capacitors ....................................................................................................... 4
1.4 Aim & Objectives ........................................................................................... 5
2. LITERATURE REVIEW ........................................................................................ 7
2.1 Parallel Plate Capacitors .................................................................................. 8
3. THEORETICAL ANALYSIS
.......................................................................... 12
3.1 Force equation of a parallel plate capacitor with more than one dielectric .. 12
3.2 Dielectric and Material Selection ................................................................. 15
3.3 Expression of dielectric strength of combined materials ................................18
4. EXPERIMENTAL SETUP, DATA AND RESULTS.............................................21
4.1 Setup ..............................................................................................................21
ix
4.2 Data and Test ..................................................................................................24
4.3 Results.............................................................................................................24
5. CHALLENGES, RECOMMENDATIONS AND CONCLUSION .......................27
5.1 Challenges.......................................................................................................27
5.2 Recommendations...........................................................................................27
5.3 Conclusion ......................................................................................................27
Appendix A
MATLAB program showing the force equation 3.8 ............................29
Appendix B
Table showing the experimental data indicating the breakdown
voltage of Teflon for different separation distances and dielectric
thicknesses ...........................................................................................30
References
..............................................................................................................31
x
LIST OF TABLES
Tables
1.
Page
Table 3.1 Table of dielectric materials showing relative permittivity
and Force …………………………… ........................ .……………16
2.
Table 4.1 Table showing the Maximum Forces (F max ) at a range of
F max (Low)< F max (Average)< F max (High) , for each range of air gap
distance d1 at the range of (1mm to 4mm) for each Teflon
thickness d2 of 0.5mm, 1mm, 1.5mm and 2mm…......................... 25
xi
LIST OF FIGURES
Figures
1.
Page
Figure 1.1 Electrostatic Field showing the force of attraction between two
terminals of opposite charges ………………. .……………………2
2.
Figure 1.2 (a) Parallel Plate Capacitor………………………… ....................... 5
3.
Figure 1.2 (b) The Electric field between the parallel plates………………… .. 5
4.
Figure 2.1 Xerographic photocopying process……………… .. ………………7
5.
Figure 2.2 (a) Parallel Plate Capacitor ................................................................ 9
6.
Figure 2.2 (b) An Illustration of a parallel plate actuator ................................... 9
7.
Figure 3.1 Parallel plate capacitor with two dielectrics ................................... 12
8.
Figure 3.2 Graph Showing the relationship between Force and dielectric
constant from table 3.1 showing the location of an ideal
dielectric as well as that of Teflon(dielectric material of choice
for the experiment)......................................................................... 17
9.
Figure 4.1 Schematic diagram of the experimental setup ................................ 22
10.
Figure 4.2 Figures A-D shows pictures of the experimental setup for the
Teflon in its holders and Figures E-G show the pictures of the
connection port labels on the power supply and power controller
as well as the pin numbers and labels on the high voltage
DC converter ................................................................................... 23
xii
NOMENCLATURE
SYMBOLS
MEANING
UNIT
𝐹𝐸 𝑜𝑟 𝐹
Electrostatic Force
N
𝑘
Relative permittivity of air
none
𝑘𝑒
Coulomb’s constant
N·m2/C2 (m/F)
𝑞
Charge
C
𝑟 𝑜𝑟 𝑑
Distance
mm or m
∁
Capacitance
F
ɛ𝒐
Permittivity of free space
F/m
Area
mm2 or m2
𝐸
Energy
J
𝑉
Voltage
V
𝑥𝑜 𝑜𝑟 𝑥
Distance/Displacement
mm or m
𝜀𝑖
Absolute Permittivity
F/m
𝑘𝑖
Relative Permittivity
None
Breakdown Voltage
V
𝑆𝑖
Dielectric Strength
V/m
Maximum Force
N
𝐴
𝑉𝑏
𝐹𝑚𝑎𝑥
xiii
1
CHAPTER 1
INTRODUCTION
Electrostatic forces operate on the principle of an electric charge (q) which can be either
positive or negative depending on the force it generates. There is always a force of
attraction between opposite charges and repulsion between charges of the same polarity.
The relationship between the electric charge and force can be seen in the Coulombs’ law,
which is expressed mathematically as
𝐹𝐸 = 𝑘𝑒
|𝑞1 𝑞2 |
𝑟2
(1.1)
Where F E is the electrostatic force that is exerted when each of the two point charges q 1
and q 2 a distance r apart, acts on each other, and 𝑘𝑒 is the Coulomb’s constant. This leads
to the production of a vector field which is dependent on the properties of the charge,
called the electric field, otherwise known as the electrostatic field, when the charges are
static [1].
1.1 Electrostatic Field
Electrostatic field is said to occur when two materials of different electrical charges are in
close proximity to each other in which the electric charges are either stationary or moving
slowly [1][2]. This is the basis for all small and miniature sized electrostatic
mechanisms, and is being widely used in the design and application of MicroElectroMechanical Systems otherwise known as MEMs [3].
2
Electrostatic fields are used due to the small size of the mechanisms, for example in the
case of small electrostatic suction devices, the device is able to attract electrical charges
on the surface of an object by reorienting the electrical dipoles through polarization while
in small grippers, it causes a force of attraction between the capacitor plates. Electrostatic
fields even though similar to electromagnetic fields should not be mistaken for it.
Electrostatic fields occur as a result of the potential difference due to static electrons.
Unlike electromagnetic fields, electrostatic fields are blocked by metals but can pass
through some other materials [4]. This leads to the question, what materials are good
conductors or carriers of electrostatic fields? Such materials are called dielectric
materials; they are usually poor conductors of electricity but strong conductors of
electrostatic fields [5].
Figure 1.1: Electrostatic Field showing the force of attraction between two terminals of opposite charges
(Electrostatic fields are electric fields which do not change with time, which happens when the charges are
stationary) [17]
3
1.2 Dielectrics
As earlier said, dielectrics can also be referred to as insulators because they allow only
very minimal to no electric charge to pass through them. They allow electrostatic fields to
pass through them while at the same time expending minimal energy in the form of heat.
These materials are used to increase the capacitance of the capacitor plates between them.
Most of these materials include dry air, ceramics, plastics, glass, polymers and oxides of
metals, vacuum, amongst others [5]. They basically have these major functions;
I.
They keep the capacitor plates from coming in contact with each other.
II.
To improve capacitance by reducing the electric field strength
III.
To limit the effect of a dielectric breakdown when high voltage is being
passed through the system [6].
The important factors in knowing the characteristics of a dielectric material are as
follows:
a. Dielectric loss, which is also known as the dielectric loss rate is how much of
energy a material looses when alternating current is applied to it and how
effective the material can prevent energy loss in the form of heat.
b. Relative Permittivity or Dielectric constant is the measure of electrostatic energy
stored within the material and is also how well a material can accommodate or
concentrate the electrostatic lines of flux.
4
c. Dielectric Strength is the ability of a material to withstand continuous increasing
voltage before it breaks down. This voltage is referred to as the breakdown
voltage [7].
1.3 Capacitors
A capacitor is a device that stores electric charge, with one end connected to a positive
electrode and the other end to a negative electrode. The charges on a capacitor are usually
equal and opposite [8]. Capacitors have a wide range of applications and can be
applicable to a lot of engineering applications such as being used to store potential
energy, frequency filtration, delaying voltage change and used as independent voltage
dividers when combined with resistors and in other various applications [8].
The major types of capacitors can be divided into three categories;
1. Parallel Plate Capacitors
2. Cylindrical capacitors
3. Spherical Capacitors [8]
5
Figure 1.2: (a) parallel plate capacitor. (b) The electric field between the parallel plates. [18]
1.4 Aim & Objectives
Electrostatics is a widely used and understood concept but its use has been limited to the
design of big mechanisms due to its consumption of very high voltage such as the Van de
Graff generator which is used mainly for research purposes, Copying machines which use
xerography, laser printers, ink jet printers, robotics applications [9], amongst others and
also limited to the design of MEMs. Not much research has been done in the design of
small millimeter sized mechanisms which operate on the electrostatic force principle for
use in various small applications.
The aim of this thesis is to set design parameters in the form of in-depth analyses in the
design of small (millimeter sized) electrostatic force driven mechanisms taking into
consideration the effects of;
1. Force generated in small mechanisms as well as the maximum force that can be
generated or exerted by these small millimeter sized devices.
6
2. Dielectric materials, taking into consideration factors such as the material type
(that is, the best dielectric material suitable for the design of such small
mechanisms), dielectric constant, the dielectric strength and the effect of
multilayered dielectrics on the overall productivity of this type of mechanism.
3. Size, in terms of the surface area of the capacitor plates, the thickness of the
dielectric material and the separation distance between the two plates.
4. Breakdown voltage and how the design of these small mechanisms can be
optimized by limiting the effect of breakdown voltage.
7
CHAPTER 2
LITERATURE REVIEW
As aforementioned, the use of electrostatics in various applications has been limited to
some use in some big applications such as xerography in the photocopying process and
majorly used in micro-engineering applications.
One of the most common use of electrostatics in the modern age is in xerography also
known as ‘dry writing’ is the combination of electrostatic printing coupled with
photography which is used in photocopy machines in which the copying powder or toner
which is negatively charged, sticks to parts of the surface of the drum where the image is
taken and then attracts the positive charges from the paper to be printed on, thereby
copying the image to the paper [9][10].
Figure 2.1: Xerographic photocopying process. [10]
8
In this era where ‘smaller’ seems to be ‘better’, the demand for smaller devices and
machines is very high. This has led to rapid research in the use of electrostatic forces in
the design of MEMs which has been the foundation for the rapid development of micro
and nanotechnologies as well as micro tools. The need for very small electrostatic
devices has made design of MEMs and nanotechnology applications very complex, using
techniques such as the conversion of metals to liquid or gaseous states in order to use
them, wire explosion laser ablation, pyrolysis and other very complex methods [11][12].
Although there is the use of simple applications such as the use of parallel plate
capacitors in the design of microgrippers fabricated using standard surface
micromachining technology [12], not much research has been done in the area of
electrostatic forces in the design of millimeter-sized mechanisms which have a much
broader range of applications ranging from medicine to robotics.
In this thesis, the design analyses of small electrostatic mechanisms using the parallel
plate capacitor will be discussed.
2.1 Parallel Plate Capacitors
A parallel plate actuator is simply a set of two conductive plates of opposite sides with
each connected to different polarities/electrodes. One end can either be fixed with the
other end moving towards or away from the fixed plate [13]. When the voltage across the
electrode is zero, the electrostatic force between the electrodes is also zero which
produces a distance x o called the rest gap or separation distance. Fig. 2.2 (b) below,
shows the schematics of this kind of actuator, where one end of the spring is connected to
9
the positive end of the voltage source and the other end to the moveable electrode while
the fixed electrode is connected to the negative end. The spring force F s pulls the
moveable electrode towards the spring in opposite direction to that of the electrostatic
force F e. Dean [13] made us understand that as the voltage between the two electrodes is
increased from zero, the resulting electrostatic force between the two electrodes pulls
them together until the electrostatic force equals the spring force. However, taking a
gripper as an example, a good gripper must be able to hold the object within its grasp
firmly with a force that is sufficiently high enough to keep it within grips yet not too high
enough to damage the material, therefore as the input voltage is increased, the parallel
plates get closer until they reach a point where the electrostatic force is too large and the
dielectric between the plates collapse by allowing voltage to pass through it and conduct,
which is referred to as break down voltage. How to limit the effect of break down
voltage is one the focuses of this thesis work. Fig 2.2 (a) and (b) shows the design of a
parallel plate capacitor and a parallel plate actuator.
Figure 2.2: (a) Parallel Plate Capacitor, (b) An Illustration of a parallel plate actuator [13]
10
If the material between the two plates is considered to be uniform, a parallel plate
capacitor such as the model above can be described by this equation:
∁=
ɛ𝒐 𝒌𝑨
(2.1)
𝒅
Where C is the capacitance between the parallel plates, A is the area of the plates, d is the
distance separating both electrodes from each other while ɛ𝑜 and 𝑘 are the permittivities
of free space and air respectively.
Energy stored in a capacitor is given as
1
𝐸 = 2 𝐶𝑉 2 [13]
(2.2)
Where E is the energy, C is the capacitance and V is the applied Voltage
Substituting the capacitance C from equation (2.1) into equation (2.2) will give
𝐸=
ɛ𝑜 𝑘𝐴
2𝑑
𝑉2
(2.3)
In the application of a parallel plate actuator according to Fig 2.1 (b),
𝑑 = 𝑥𝑜 + 𝑥
(2.4)
Where 𝑥𝑜 is the separation distance between the fixed electrode and the movable electrode, and
𝑥 is the displacement of the spring.
Therefore, substituting the value of 𝑑 from equation (2.4) into equation (2.3) will yield
𝐸=
ɛ𝑜 𝑘𝐴
𝑉2
2(𝑥𝑜 +𝑥)
(2.5)
11
Equation (2.5) is the Energy stored in a parallel plate actuator in terms of the separation distance
between the plates, permittivity of the dielectric, area of the parallel plates and applied voltage.
The electrostatic force in a parallel plate capacitor according to Dean [13] is given as
𝐶
𝐹 = 2𝑑 𝑉 2
(2.6)
Where F is the force generated by applying a Voltage to the Capacitor plates having a
separation distance d between the plates.
Substituting the value of Capacitance from equation (2.1) into equation (2.6) will yield
𝐹=
ɛ𝑜 𝑘𝐴
2𝑑2
𝑉2
(2.7)
Equation (2.7) is the general equation showing the amount of force generated in a parallel
plate capacitor having one dielectric material.
Now, considering a parallel plate capacitor with two different dielectrics between it, say
for instance, a gripper having the metal plates attached to the first dielectric which is the
gripper material such as plastic or paper and the second dielectric which is another
dielectric or in the case of this thesis, air. In order to get materials that will work with this
kind of small device, only polymers will be suitable dielectrics. Even though having low
dielectric constants and low energy densities, they are most suitable because of the size of
the device and also because they have excellent electrical and processing/manufacturing
properties which includes they being inexpensive, lightweight and easy to produce
commercially, having high breakdown strength, high dc resistance and low hysteresis
[14].
12
CHAPTER 3
THEORETICAL ANALYSIS
3.1 Force Equation of a parallel plate capacitor with more than one dielectric
In most studies, the electrostatic force generated by a parallel plate capacitor could only
be described with one dielectric in between the plates. This is not always the case as
devices that use electrostatic forces such as grippers and suction cups/plates will most
times have one or more dielectric materials. Air is considered to be a dielectric and most
times can be referred to as vacuum in most electromagnetic applications but is actually
different from a vacuum in some electrostatics applications, especially in small
applications such as MEMs or Insect-sized devices.
Having said these, we will be considering a parallel plate capacitor with two or more
dielectrics in between.
𝜀2
𝑑
𝜀1
𝑑1
1
𝑑2
1
𝑑2
2
2
Figure 3.1: Parallel plate capacitor with two dielectrics, in this case, air is considered a dielectric
The figure above shows two parallel plates embedded in the dielectric material which
also parallel to each other.
13
𝑑 is the total distance between the capacitor plates
Where 𝑑 = 𝑑1 + 𝑑2
(3.1)
𝑑1 and 𝑑2 are the air gap or separation distance between the dielectrics and the thickness
of the dielectric material respectively.
𝜀1 and 𝜀2 are the absolute permittivity of air and the dielectric material respectively.
Where 𝜀1 and 𝜀2 is 𝜀𝑖 = 𝑘𝑖 𝜀0
(3.2)
𝑘𝑖 is the relative permittivity of the ‘ith’ material which in this analysis are the dielectric
material and air while 𝜀0 is vacuum permittivity.
From the previous chapter, the capacitance for a parallel plate capacitor was given as
∁=
𝒌ɛ𝒐 𝑨
𝒅
In a situation where there is more than one dielectric material which in this case
comprises of the first dielectric which houses the capacitor plates, and air as the second
dielectric, both dielectrics are parallel to the capacitor plates and in other words can be
described as a system of three capacitors connected in series on top each other [15].
For capacitors connected in series, the equation is given as
1
𝐶
1
1
=𝐶 +𝐶
1
2
(3.3)
Therefore substituting the value of C from equation (2.1) into (3.3), will yield
14
1
𝐶
=
1
𝑘1 𝜀 0 𝐴
𝑑1
+
1
(3.4)
𝑘2 𝜀 0 𝐴
𝑑2
Therefore, expanding and solving equation (3.4) will yield the value of C to be
𝜀 𝐴
𝐶 = 𝑑1 0 𝑑2
(3.5)
+
𝑘1 𝑘2
From equation (3.2), the absolute permittivity was given as 𝜀𝑖 = 𝑘𝑖 𝜀0
Therefore, substituting the value of 𝜀𝑖 from equation (3.2) into (3.5) and solving, will
yield
𝐶 = 𝑑1
𝜀1
𝐴
𝑑
+ 2
𝜀2
=
𝐴
𝑑1 𝜀2 +𝑑2 𝜀1
𝜀1 𝜀2
=𝜀
𝜀1 𝜀2 𝐴
2 𝑑1 +𝜀1 𝑑2
(3.6)
Equation (3.6) can be said as the capacitance where there is more than one dielectric
parallel to the capacitor plates.
The electrostatic force in this mechanism which is a function of the energy stored in the
capacitor and the distance moved is given as
𝐸
𝐹=𝑑
(3.7)
The energy stored in a capacitor is given in equation (2.2), therefore, substituting this
value of E into the force equation (3.7) will give equation (2.6), now substituting the
value of d from equation (3.1) and C obtained from (3.6) into the force equation from
(2.6) will give a simplified version of the electrostatic force between a parallel plate
capacitor with more than one dielectric shown below.
15
𝑭 = 𝟐(𝜺
𝜺𝟏 𝜺𝟐 𝑨𝑽𝟐
𝟐 𝒅𝟏 +𝜺𝟏 𝒅𝟐 )(𝒅𝟏 +𝒅𝟐 )
(3.8)
The force can be seen as a function of an applied voltage, the area of the capacitor plates,
the absolute permittivities and the separation distance, all of which has an impact on how
much force can be generated depending on the altercation of any of these variables and
what material is used as the dielectric.
3.2 Dielectric and Material Selection
In the design of small mechanisms, selecting the right material is essential as this will
have a strong effect on the overall operation and reliability of the entire mechanism.
Choosing the right material depends on the type of application that mechanism is to
perform, for example, a very small mechanism such as a small gripper or suction cup will
require very light weight material with very high strength having the ability to withstand
high voltage without breaking down easily. Table 3.1 shows a table of dielectric materials
with their dielectric strength, permittivity and calculated force using the force equation in
equation 3.8. A MATLAB program was written to calculate the forces in the table which
can be seen in the appendix.
16
Table 3.1: Table of dielectric materials showing relative permittivity and Force [16]
Dielectric Material
Dielectric
Constant /
Permittivity
Air
Paper
Rubber/Paraffin
Teflon
Waxes/Mineral
Polystyrene
Polyethylene
Polycarbonate
Asbestos Fiber
Epoxy Resin/Nylon
Formica
Porcelain
Graphite
Ammonia
Methanol
Glycerol
Force
(N)
1
3
2
2.1
2.2
2.6
2.3
2.9
3.1
3.4
4.6
5.1
10
20
30
40
50
0.0492
0.0553
0.059
0.0596
0.0601
0.0611
0.0615
0.0629
0.0635
0.0643
0.0648
0.0672
0.0703
0.072
0.0726
0.0729
0.0731
Using this table and the formulas generated earlier on, an analysis showing the general
relationship between each of these parameters in graphical form is discussed below.
The parameters and values used for plotting the above table 3.1 and Fig 3.3 were;
𝜀0 = 8.854 ∗ 10−12 F/m (Permittivity of free space)
𝑘1 = Relative permittivity (Dieletric constant) of dieletric materials
𝑘2 = 1 (Relative permittivity of air)
𝜀1 = 𝑘1 𝜀0 (Absolute permittivity of dielectric materials)
𝜀2 = 𝑘2 𝜀0 = 1 ∗ 𝜀0 (Absolute permittivity of air)
17
𝑑1 = 0.002𝑚 (Separation distance between the dielectric materials)
𝑑2 = 0.001𝑚 (Thickness of dielectric material)
𝑉 = 20000 𝑉 (Applied Voltage)
𝐴 = 25 ∗ 10−5 𝑚2 (Area of the parallel plate capacitors)
This analysis is very important to help with the design of electrostatic force driven
mechanisms especially with small mechanisms to determine the maximum allowable
force and voltage a mechanism can handle and how to optimize the design by selecting
the right materials and parameters in the design process at low cost.
0.075
Ideal Material
0.07
Force (N)
0.065
Teflon
0.06
0.055
0.05
0.045
0.04
0
5
10
15
20
25
30
35
40
45
50
55
Dielectric Constant
Figure 3.2: Graph showing the relationship between Force and Dielectric Constant from table 3.1 above
and the location of an Ideal dielectric as well as that of Teflon (dielectric material of choice for the
experiment)
18
From the graph in Fig 3.3 above, the ideal material will be having a dielectric constant of
10 and a force value of about 0.07N since it lies at the saturation point of the graph but
dielectric materials at this dielectric constant value and higher cannot be used due to
having the chemical and dielectric properties/composition not suitable for an application
such as a small parallel plate actuation. Teflon was chosen as a good choice because not
only does it have excellent dielectric and chemical properties, it can also exert forces
close to that of the ideal material which makes it suitable for this application.
3.3 Expression of dielectric strength of combined materials
This analysis as stated in the first paragraph of this chapter deals with a mechanism with
more than one dielectric in-between the capacitor plates, which will have some effect of
sort on the dielectric strength of dielectrics combined together, which also has an effect
on the overall performance of the mechanism especially with respect to breakdown
voltage.
To check for the dielectric strength of multilayered materials, some factors have to be
taken into consideration:
i.
There is the maximum force at which breakdown voltage occurs.
ii.
In order to get the maximum force, an experiment will be set up using a
variable voltage regulator connected to a high DC voltage power source and
the parallel plate actuator in order to determine the voltage at which this force
occurs. This is explained in the next chapter.
19
To find the maximum force when the dielectric strength of the material is known, the
following analysis shown below;
1
Energy stored in a capacitor is given as 𝐸 = 2 𝐶𝑉 2 from equation (2.2)
𝐸
𝐶
And since 𝐹 = 𝑑 , therefore, 𝐹 = 2𝑑 𝑉 2 .
A formula can be derived for finding the maximum force in terms of the dielectric
strength, and this maximum force and also the breakdown voltage are going to be
achieved experimentally for this analysis.
Breakdown voltage for any given dielectric is given as
𝑉𝑏 = 𝑆𝑖 𝑑
(3.9)
Where 𝑉𝑏 is the breakdown voltage;
𝑆𝑖 is the dielectric strength of the dielectric material in which in this case Teflon
was the dielectric used in this experiment and
𝑑 is the total distance between the two capacitor plates (Including the thickness of
the dielectrics between the plates).
The maximum force which in this analysis is also called the “break-down” force can be
expressed as thus,
𝑉2
𝑏
𝐹 = 𝐶 2𝑑
(3.10)
20
Which can also be expressed by substituting the value of 𝑉𝑏2 into equation (3.9)
𝐹=𝐶
𝑆𝑖2 𝑑
2
(3.11)
Equation 3.11 can be used to find the maximum force in terms of the dielectric strength
of the material.
21
CHAPTER 4
EXPERIMENTAL SETUP, DATA AND RESULTS
4.1 Setup
For the experimental setup as seen in figures 4.1 and 4.2 below, A Hewlett Packard –
E3630A Triple Output DC power supply set to +/- 20 volt and 4 amps setting was
connected from the “com” port to the negative output port of a Kikusui PMC35-1A
power controller and to “pin 1” of an Ultravolt 20A12 – P4 DC high voltage power
converter. The +20V port of the power supply connected to “pin 2” of the power
converter and the positive output port of the PMC35-1A is connected to “pin 6” of the
power converter. This converter amplifies the input voltage by approximately 4310
which is then connected from “pin 8” to the negative; and the red high voltage output
wire to the positive terminals of the Teflon holders respectively. The terminals are made
of thin copper wires connected to the capacitor plates made from Aluminum, the Teflon
holders as well as the base on which they rest, are made of Delrin plastic material.
Teflon was the choice of dielectric material because of its excellent dielectric properties
such as high dielectric strength and very good dielectric constant. And Delrin was used as
the base plate for the setup to ensure proper insulation throughout the process which can
be seen in the experimental setup and figures below.
22
HP-E3630A Power Supply
Kikusui PMC35-1A Power Controller
UV 20A12-PA DC power converter
Teflon
- +
d1
Figure 4.1: Schematic Diagram of the experimental setup
Figure A
Figure B
23
Figure C
Figure E
Figure D
Figure F
Figure G
Figure 4.2: Figures A-D shows pictures of the experimental setup for the Teflon in its holders and
Figures E-G shows the pictures of the connection port labels on the power supply and power
controller as well as the pin numbers and labels on the high voltage DC converter.
24
4.2 Data and Test
In the experimental setup, series of tests were taken with different thicknesses (d2) of
Teflon material and air gaps (d1) ranging from 1mm to 4mm to find and determine the
effect of the factors such as material thickness (Teflon thickness of 0.5mm, 1mm, 1.5mm
and 2mm), and air gap in the breakdown of the material. This is achieved by increasing
the voltage across the system using the Kikusui PMC35-1A power controller until the
Teflon material breaks down. Teflon Dielectric strength table showing the breakdown
voltage from the experiment performed can be seen in the index page.
4.3 Results
Using the Experimental table, the maximum force produced at each value of d1 (mm) and
d2 (mm) can be determined by applying the force equation 3.8 and substituting the value
of Voltage (V) to become the breakdown voltage (V b ) to become
𝐹𝑚𝑎𝑥 = 2(𝜀
𝜀1 𝜀2 𝐴𝑉𝑏 2
2 𝑑1 +𝜀1 𝑑2 )(𝑑1 +𝑑2 )
(4.1)
Table 4.1 below shows the values of the maximum force generated by Teflon at each
value of air gap distance (d1) ranging from 1mm to 4mm and each Teflon thickness
(d2) values of 0.5mm, 1mm, 1.5mm and 2mm.
25
Table 4.1:Table showing the Maximum Forces (F max ) at a range of F max (Low)< F max (Average)< F max (High) , for
each range of air gap distance d1 at the range of (1mm to 4mm) for each Teflon thickness d2 of 0.5mm,
1mm, 1.5mm and 2mm.
d2
d1
F max (Low)
V b (Low)
F max
V b (Average)
F max
V b (High)
(mm)
(mm)
(N)
(V)
(Average) (N)
(V)
(High) (N)
(V)
0.5
1
0.0106
13317.9
0.0113
13796.31
0.0123
14395.4
2
0.0043
14697.1
0.0048
15503.07
0.0053
16291.8
3
0.0029
17153.8
0.0034
18576.1
0.0036
19308.8
4
0.0019
18231.3
0.0022
19623.86
0.0025
20946.6
1
0.0055
12111.1
0.0066
13292.04
0.0085
15085
2
0.0031
14438.5
0.0035
15300.5
0.0043
16938.3
3
0.0020
15990.1
0.0023
16977.09
0.0028
18619.2
4
0.0018
18877.8
0.0021
20636.28
0.0028
23877.4
1
0.0042
12714.5
0.0052
14162.66
0.0060
15214.3
2
0.0021
13404.1
0.0027
15218.61
0.0033
16938.3
3
0.0017
16248.7
0.0021
17839.09
0.0029
20989.7
4
0.0014
18058.9
0.0017
20084.6
0.0024
23920.5
1
0.0026
11723.2
0.0034
13317.9
0.0040
14610.9
2
0.0019
14136.8
0.0021
15089.31
0.0025
16421.1
3
0.0014
15817.7
0.0017
17326.2
0.0022
19696.7
4
0.00099
16334.9
0.0012
18140.79
0.0015
20084.6
1.0
1.5
2.0
26
Where:
F max (Low) , F max (Average) , and F max (High) are the lowest, average and highest maximum
forces that can be exerted by the dielectric material just before is breaks down.
From the data obtained in experimental breakdown voltage table (see Appendix), it
can be inferred that the breakdown voltage of Teflon is not dependent on its thickness
(d2) but the maximum force that can be generated are dependent of factors such as
the material thickness, the separation distance and the voltage.
From the above tables (4.1) and the index table, it can be seen that:
i.
At constant separation distance, the maximum force decreases as the thickness
of the dielectric material is increased. This infers that in the design of small
millimeter sized electrostatic devices, in order to get the required or maximum
force the device can exert, for example, the force a gripper of such size will
exert on the object it is gripping, the thickness of the material will have to be
taken into consideration as it is an important factor.
ii.
At constant dielectric material thickness, the maximum force decreases while
the breakdown voltage increases as the separation distance increases. This
infers that in the case of a small millimeter sized gripper using electrostatic
actuation; it would require a much higher voltage but lesser gripping force to
pick objects as they get bigger.
27
CHAPTER 5
CHALLENGES, RECOMMENDATIONS AND CONCLUSION
5.1 Challenges
Due to the limited voltage range of the Ultravolt power converter, the experiment could
only be conducted for separation distances of 1mm to 4mm, clear and concise readings
could not be determined for distances above 4mm.
5.2 Recommendations
More research can be conducted in finding the direct relationship between the dielectric
constant and dielectric strength as it can be seen from the experiments and research that a
good material for such a design is not only dependent on the dielectric constant of the
material or dielectric strength alone but on a combination of these two parameters.
Also, Teflon is a good material of choice but there are other dielectric materials that may
yield better results than Teflon. Teflon was chosen for this experiment because of its
good dielectric properties such as good dielectric constant, a very high dielectric strength
and having low cost and available.
5.3 Conclusion
The design of small electrostatic force driven devices is a large pool with a lot of
potentials to be researched and determined. It can be seen from this research that the
effect of multilayered dielectrics such as the combination of a dielectric material and air
28
gap (air is considered a dielectric in this kind of situation and its effect is not negligible)
has a large effect on the design of very small devices that work on the principle of
electrostatic actuation especially the amount of force these combinations can exert.
Also in order to either make a fresh design of a small device that operates on this
principle or if it is optimizing an existing design, the parameters to be considered are the
type of material, the separation distance between the plates, the maximum amount of
force that can be generated or exerted by the device, the effect of breakdown voltage and
the thickness of the material, depending on what action the design is to perform.
This design analysis can be applied to all dielectric materials.
29
APPENDIX A
MATLAB program showing the force equation 3.8
30
APPENDIX B
Table showing the experimental data indicating the breakdown voltage of Teflon
for different separation distances and dielectric thicknesses.
Dielectric strength
-PPM experiment plan
31
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