Dynamic Analysis of an Electrostatic Energy
Harvesting System
by
Feifei Niu
Submitted to the Department of Civil and Environmental Engineering
in partial fulfillment of the requirements for the degree of
ARCHNE
Master of Science in Civil and Environmental Engineering
MASSACHUSETTS iNSTI-lit
OF TECHNOLOGY
at the
at
theJUL
0 8 2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
I
LIBRARIES
June 2013
© Massachusetts Institute of Technology 2013. All rights reserved.
.................
A uth o r ..........................................
Department of Civil and Environmental Engineering
May 23, 2013
....
....
Konstantin Turitsyn
Esther and Harold E. Edgerton Assistant Professor
Thesis Supervisor
Certified by ....................................
Certified by.....................
.
-
Pedro Miguel Reis
Esther and Harold E. Edgerton Assistant Professor
Thesis Reader
Accepted by.....................
Heidi 1\. Nepf
Chair, Departmental Committee for Graduate Students
2
Dynamic Analysis of an Electrostatic Energy Harvesting
System
by
Feifei Niu
Submitted to the Department of Civil and Environmental Engineering
on May 23, 2013, in partial fulfillment of the
requirements for the degree of
Master of Science in Civil and Environmental Engineering
Abstract
Traditional small-scale vibration energy harvesters have typically low efficiency of energy harvesting from low frequency vibrations. Several recent studies have indicated
that introduction of nonlinearity can significantly improve the efficiency of such systems. Motivated by these observations we have studied the nonlinear electrostatic
energy harvester using a combination of analytical and numerical approaches. The
analytical approach was based on the normal vibration mode analysis around an
equilibrium point. The numerical model was implemented and tested using Modelica
language. It was found that the efficiency of energy transfer strongly depends on three
parameters: the ratio between the maximal electrical and mechanical energies in the
system and ratio of natural frequencies of electric and mechanical modes, and finally
the dimensionless degree of nonlinearity in the system. The dependence of the transfer factor on these three parameters was studied and characterized both theoretically
and numerically. It was found that the transfer factor Tr has a sharply pronounced
peak as a function of e providing a possibility of efficient energy conversion between
modes with highly different normal frequencies.
Thesis Supervisor: Konstantin Turitsyn
Title: Esther and Harold E. Edgerton Assistant Professor
Thesis Reader: Pedro Miguel Reis
Title: Esther and Harold E. Edgerton Assistant Professor
3
4
Acknowledgments
I would like to extend my gratitude to the many people who helped to bring this research project to fruition. First, I would like to thank Professor Kostya Turitsyn for
providing me the opportunity of taking part in this research. I am so deeply grateful
for his help, professionalism, valuable guidance and financial support throughout this
project and through my entire program of study that I do not have enough words to
express my deep and sincere appreciation.
I would also like to acknowledge Professor Pedro M. Reis as my thesis reader, and I
am gratefully indebted to him for his valuable comments for this thesis.
I would also like to thank Mr. Petr Vorobev and my roommate Daniela Miao, who
have willingly proof read my thesis.
Finally, I must express my very profound gratitude to my parents and my friends
Sha Miao and Xin Xu for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching
and writing this thesis. This accomplishment would not have been possible without
them. Thank you.
5
6
Contents
15
1 Introduction
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.2
Review of technologies
. . . . . . . . . . . . . . . . . . . . . . . . . .
16
. . . . . . . . . . . . . . . . . . .
16
1.2.1
Electromagnetic Generators
1.2.2
Piezoelectric Generators
. . . . . . . . . . . . . . . . . . . . .
22
1.2.3
Electrostatic Generators
. . . . . . . . . . . . . . . . . . . . .
27
1.3
Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
1.4
Nonlinear harvesters
. . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2 Physical model and linearized analysis
2.1
2.2
Physical model
41
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.1.1
Convertor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.1.2
Design of mechanical system . . . . . . . . . . . . . . . . . . .
43
2.1.3
Design of electric circuit
. . . . . . . . . . . . . . . . . . . . .
44
2.1.4
Configuration of the energy harvesting system . . . . . . . . .
45
Linearized analytical studies . . . . . . . . . . . . . . . . . . . . . . .
46
2.2.1
Equation of motion . . . . . . . . . . . . . . . . . . . . . . . .
47
2.2.2
Equilibrium point . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.2.3
Linearization
. . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.2.4
Mode shapes
. . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.2.5
Transformation matrix . . . . . . . . . . . . . . . . . . . . . .
55
2.2.6
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
7
3 Numerical analysis
3.1
Modelica language for modeling
3.2
67
. . . .
67
Modelica model . . . . . . . . . .
. . . . . . . . . .
68
3.2.1
The constant capacitor
. . . . . . . . . .
68
3.2.2
The variable capacitor
. . . . . . . . . .
69
. . . . . . . . . .
70
3.3 Test the Modelica model.....
3.4 Numerical studies of different cases without external input and resistor
3.5
4
71
3.4.1
Base . . . . . . . . . . . ..
. . . . . . . . . .
73
3.4.2
Case 1 . . . . . . . . . . .
. . . . . . . . . .
74
3.4.3
Case 2 . . . . . . . . . . ..
. . . . . . . . . .
76
3.4.4
Case 3 . . . . . . . . . . . .
. . . . . . . . . .
77
3.4.5
Case 4 . . . . . . . . . . . .
. . . . . . . . . .
78
3.4.6
Case 5 . . . . . . . . . . . .
. . . . . . . . . .
79
Forced vibration simulation
81
Conclusions
85
A Modelica script
89
A.1 Constant capacitor . . . . . . . . . . .
89
A.2 Variable capacitor . . . . . . . . . . . .
89
A.3 Model used to test energy conservation
90
A.4 Model Base . . . . . . . . . . . . . . .
91
A.5 Case 1 . . . . . . . . . . . . . . . . . .
91
A.6 Case 2 . . . . . . . . . . . . . . . . . .
92
A.7 Case 3 . . . . . . . . . . . . . . . . . .
93
A.8 Case 4 . . . . . . . . . . . . . . . . . .
94
A.9 Case 5 . . . . . . . . . . . . . . . . . .
94
8
List of Figures
1-1
Typical schematics of electromagnetic generators [4] . . . . . . . . . .
1-2
(Upper left) isometric, (upper right) side, and (lower) schematic views
[17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
1-3
Schematic design by Rajeevan and Anantha [1]
1-4
Schematic of Prez-Rodrguez, Serre, Fondevilla, Cereceda, and Morante's
design [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-5
17
18
19
Generator by Williams, Shearwood, Harradine, Mellor, Birch and Yates
[2 3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-6
17
19
Simplified energy harvesting system by Williams, Shearwood, Harra-
dine, Mellor, Birch and Yates [23] . . . . . . . . . . . . . . . . . . . .
20
1-7
Electromagnetic generator designed by Mizuno and Chetwynd [12] . .
21
1-8
Micromachined Silicon Generator [5]
. . . . . . . . . . . . . . . . . .
22
1-9
a) 3-3 mode (left); b) 3-1 mode (right)
. . . . . . . . . . . . . . . . .
22
1-10 Schematic design by Umeda, Nakamura and Ueha [21] . . . . . . . . .
23
1-11 Schematic drawing of experiment by Xu, Akiyama, Nonaka and Watan-
. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .
24
. . . . .
25
. . . . . . . . . . . .
26
. . . . . . . . . . . . . . . . . . . . . . .
26
1-15 Three types of electrostatic generators [4] . . . . . . . . . . . . . . . .
28
. . . . . . . . . . . . . . . . . . . . . . . .
28
1-17 Schematic drawing of an electrostatic transducer [9] . . . . . . . . . .
29
. . . . . . . . . . . . . . . . . . .
30
abe [24]
1-12 Two approaches to harvest piezoelectric energy in shoes [18]
1-13 Curved PZT unimorph excited in 3-1 mode [25]
1-14 Schematic of PZT model [7]
1-16 Controller architecture [9]
1-18 Design by Peano and Tambosso [4]
9
1-19 Schematic design of electrostatic generator with electret [2] . . . . . .
31
1-20 Schematic design of an electret based in-plane overlap electrostatic
generator [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-21 Design by Miyazaki, Tanaka and Ono [11]
32
. . . . . . . . . . . . . . .
33
1-22 Design by Tashiro and Kabei [20] . . . . . . . . . . . . . . . . . . . .
33
1-23 Design by Meninger, Mur-Miranda and Amirtharajah [10]
. . . . . .
34
1-24 Schematic design by Cottone, Vocca and Gammaitoni [6]
. . . . . .
36
1-25 Envelope modulation 6(t) [22] . . . . . . . . .
. . . . . .
37
1-26 Three types of nonlinear energy sinks [16]
. .
. . . . . .
38
2-1 Configuration of the variable capacitor . . . .
. . . . . . . . . .
42
2-2
Charge trapping configuration . . . . . . . . .
. . . . . . . . . .
43
2-3
Configuration of mechanical system . . . . . .
. . . . . . . . . .
44
2-4
Configuration of electrical subsystem . . . . .
. . . . . . . . . .
45
2-5
Configuration of the energy harvesting system
. . . . . . . . . .
46
2-6
Electrical energy generation . . . . . . . . . .
. . . . . . . . . .
47
2-7
Analytical analysis model
. . . . . . . . . . .
. . . . . . . . . .
47
2-8
Plots of 1
. . . . . . . . . . . . . . . .
. . . . . . . . . .
64
2-9
Plots of (1 - a2)-E . . . . . . . . . . . . . . . .
. . . . . . . . . .
64
3-1
Energy Plots
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3-2
Numerical analysis model
. . . . . . . . . . . . . . . . . . . . . . . .
71
3-3
Numerical results of model Base . . . . . . . . . . . . . . . . . . . . .
74
3-4 Numerical results of model Case 1 . . . . . . . . . . . . . . . . . . . .
75
3-5 Numerical results of model Case 2 . . . . . . . . . . . . . . . . . . . .
76
3-6 Numerical results of model Case 3 . . . . . . . . . . . . . . . . . . . .
78
3-7 Numerical results of model Case 4 . . . . . . . . . . . . . . . . . . . .
79
3-8 Numerical results of model Case 5 . . . . . . . . . . . . . . . . . . . .
80
3-9 Model with external input and resistor . . . . . . . . . . . . . . . . .
81
3-10 Numerical results of model with external vibration input
82
-
a2-
10
4-1
Transfer factor (1 - a2) plots
. . . . . . . . . . . . . . . . . . . . . .
11
86
12
List of Tables
3.1
Parameters of different cases . . . . . . . . . . . . . . . . . . . . . . .
13
72
14
Chapter 1
Introduction
1.1
Motivation
Mechanical energy is the most common form of energy observed in our daily life.
Common processes like people walking, vibration of a running car, tiny oscillation of
a tall building under wind load all involve dissipation of mechanical energy. Eventually most of the mechanical energy is dissipated and transferred to heat. Modern
technology offers us an opportunity to harvest this ambient mechanical energy from
the environment, and convert it to other forms. This harvesting process would enable
and drastically reduce the cost of a number of sensing technologies that are currently
reliant on expensive batteries. Conversion of the mechanical energy to electrical is
the most attractive approach as the electrical energy can be directly used by all kinds
of low power electronics. Although the output of electrical energy is limited, it is
enough to power some wireless sensors, as shown in different experiments.
Vibrations occur naturally in many Civil Engineering problems. They are one of
the major causes of material fatigue and are vital for the serviceability criteria. Vibrations may induce problems like structural damage caused by the wall moving in
and out of its at-rest position. They can affect people in a high-rise office boardroom
or disturb sensitive medical and industrial equipment. Harvesting these vibrations
to power wireless electronic devices, like structure health monitor devices, can help
15
lower the cost.
This thesis is structured as follow. In Chapter 1, different types of energy generators is introduced and current technologies is reviewed. In Chapter 2, we present
the energy harvesting system used in this thesis and develop analytical studies of the
system. In Chapter 3, the system is analyzed in a numerical approach. Chapter 4 is
the conclusions.
1.2
Review of technologies
Kinetic energy harvesters are composed of several parts. A converter, based on electromagnetic, piezoelectric, or electrostatic technologies that converts mechanical energy to electrical one. Electric circuit that transfers or makes use of electrical energy. Mechanical system that enables coupling to external vibrations. Most of the
vibration-based micro-generators can be modeled as simple spring-mass systems.
There are several standard mechanisms of conversion that are reviewed below.
1.2.1
Electromagnetic Generators
Electromagnetic induction, first discovered by Faraday in 1831, is the generation of
electric current in a conductor located within a magnetic field. [4]A typical schematic
as shown in Figure 1-1, is made up of a coil and a magnet. Electric current is generated in the coil if there is relative movement between the magnet and coil, or the
magnetic field changes with time.
Figure 1-2 depicts a schematic of an electromagnetic approach via a system, which
was designed by Sari, Balkan and Kulah [17]. In this work, a microelectromechanical system based electromagnetic vibration-to-electrical power generator has been
16
B
B
Din
Motgon
41)
CON
Wo
N "
00
a
usMain
-0
Figure 1-1: Typical schematics of electromagnetic generators [4]
Mal
Rwg
MIU
~r
4 NIu Id
I WSbago,
Dw"wqm
Amaent
e
capi
wswsen my~
Parwsuedtwe Re
Woontamu byli nuif
ein
14b
PwineCus
cy
fthq
upcvie
Figure 1-2: (Upper left) isometric, (upper right) side, and (lower) schematic views
[17]
designed to harvest energy from low-frequency external vibrations. It has been found
that the efficiency of an energy harvester is proportional to the frequency of its external excitation. The authors used the frequency upconversion technique to design
the generator in order to transfer the low-frequency vibrations in its environment to
a higher frequency. A generator of the size 8.5 x 7 x 2.5mm 3 , consisting of 20 cantilevers has been fabricated to increase the generated voltage and power. It has been
shown that by upconverting the input frequency of 95 - 2kHz to 70 - 150Hz, the
generator can effectively harvest energy and generates 0.57mV voltage with 0.25nW
power from a single cantilever.
17
--- sprng,k
VCMloomutrp
SOOT
swo
mass, M
sw
* ow
i re c
,t I
--- -..... ..----.-------------------
Criica Pah
prmanent
magneto B
(a) Generator Mechanical Schematics
C-
S
(b) Detailed Block Diagram of Self-powered
DSP System
Figure 1-3: Schematic design by Rajeevan and Anantha [1]
Rajeevan and Anantha [1] designed and tested a chip to operate a digital system powered by vibrations in its environment using similar approach. The power generator,
as shown in Figure 1-3a is a moving coil electromagnetic transducer. The tested chip
includes an ultra-low power controller and a low power sub-band filter DSP circuit.
The controller has been used to control the voltage of the generator by delay feedback
control. It has been found that theoretically power on the order of 40OpW can be
generated. In their tests, 500 kHz self-powered operation of the sub-band filter has
been used. The experiments have shown that the entire system, including the DSP
load, dissipates 18pW of power and that 23 ms of valid DSP operation is generated
by a single generator excitation at a 500 kHz clock frequency. Based on the above,
the authors concluded that it is possible to create a portable digital system that will
no longer depend on a battery and will be powered entirely by vibrations in its environment.
Figure 1-4 shows a schematic of Prez-Rodrguez, Serre, Fondevilla, Cereceda, and
Morante's [14] design of an electromagnetic inertial micro-generator. The device is
based on a fixed coil of 29 turns with track width 30pm, and separation 20[pm. The
coil surface is about 1cm 2 , and the cross-section shape is nearly circular.
18
Planar coil
h
Membrane
Figure 1-4: Schematic of Prez-Rodrguez, Serre, Fondevilla, Cereceda, and Morante's
design [14]
A resonant structure was manufactured, using a thin polyimide film, whose Young's
modulus is significantly lower than that of Si related materials. This choice of material can broaden the bandwidth of input vibration, ranging from some Hz to several
kHz by varying membranes thicknesses between 25 and 127pm. The design has been
optimized, with respect to the values of series resistance and parasitic damping. It
has been found that the generated power ranges from a few pW up to 0.1 - 1W.
0 A3os
pinr
Figure 1-5: Generator by Williams, Shearwood, Harradine, Mellor, Birch and Yates
[23]
Williams, Shearwood, Harradine, Mellor, Birch and Yates' [23] design of the prototype micro-electromagnetic inertial generator is shown in Figure 1-5. The generator
19
is composed of two parts: an upper mass-spring on a substrate attached to a lower
planar coil and substrate. A vertically polarized permanent magnet forms the inertial
mass and is attached under a polyimide membrane. A planar gold coil is attached to
the spring-mass wafer. Therefore, the magnet forms the inertial mass, the membrane
forms the spring, and the magnet and coil together form the electromechanical generator.
Figure 1-6: Simplified energy harvesting system by Williams, Shearwood, Harradine,
Mellor, Birch and Yates [23]
The device has been modeled as a general linear inertial electric generator, as shown
in Figure 1-6. A mass, m, is suspended on a spring with spring constant k, equivalent
elasticity of the membrane, and a damper, d, includes all mechanical and electrical
damping losses of the system. Both finite element model and mathematical symbolic
analysis has been developed to optimize the configuration in terms of the separation between the magnet and planar coil, the coil radius and magnet volume. In
their experiment, it has been shown that a millimeter scale device is able to gener20
ate power of 0.3pW at an input frequency of 4MHz. It has been found that this
result is in agreement with their model predictions. They concluded that the power
produced by such kind of devices was proportional to the cube of the input frequency.
ico-cantilevers
Permanent man
S N
Base, Back plate
Ci
Figure 1-7: Electromagnetic generator designed by Mizuno and Chetwynd [12]
Another typical schematic of an electromagnetic generator takes use of a cantilever
vibration as an input vibration. In Mizuno and Chetwynd's [12] design, a cantilever
serves as a resonant element. The generator consists of a coil patterned on the cantilever surface with a fixed permanent magnet close to its end, as shown in Figure
1-7. When the cantilever moves, the coil cuts the magnetic flux and induces an emf at
both ends of the coil. Therefore, current can be generated in the coil. In this design,
an external vibration source is coupled to the fixed end of the cantilever. Power can
be generated as the cantilever's free end moves at larger amplitude.
Figure 1-8 shows Beeby and Tudor's [5] design schematic. The design has a fourmagnet arrangement. The coil is designed to move laterally relative to the magnets.
Two magnets are located within etched recesses in the Pyrex wafers and two Pyrex
wafers are bonded to each surface of the silicon wafer. The coil is placed on a silicon
cantilever. Both mechanical and electrical models have been developed by authors to
analyze the mechanical and electromagnetic behavior of the silicon vibration powered
generators of 3 models with different beam types.
21
Figure 1-8: Micromachined Silicon Generator [5]
1.2.2
Piezoelectric Generators
In 1880, J and P Curie found that crystals became electrically polarized if they were
subject to mechanical press and that the degree of polarization was proportional to
the applied strain. This effect is called the piezoelectric effect. This kind of material
deforms when exposed to an electric field. Piezoelectric materials are in many forms,
such as single crystal, screen printable thick-films and polymeric materials.
F
-
F
Figure 1-9: a) 3-3 mode (left); b) 3-1 mode (right)
Orthogonal axes 1, 2 and 3 are used to show the directions in a piezoelectric element.
The 3-axis is conventionally set to be parallel to the direction of polarization of the
22
material, established during manufacture. Piezoelectric generators typically work in
either 3-3 mode (Figure 1-9 a) or 3-1 mode (Figure 1-9 b). In the 3-3 mode, a force
is applied to the same surfaces that charge is collected on, such as the compression of
a piezoelectric block that has electrodes on its top and bottom surfaces. In the 3-1
mode, a lateral force is applied to the perpendicular surfaces that charge is collected,
an example of which is a bending beam that has electrodes on its top and bottom
surfaces. Generally, the 3-1 mode has been the most commonly used coupling mode
although it has a lower coupling coefficient than that of the 3-3 mode. Common
energy harvesting structures such as cantilevers or double-clamped beams typically
work in the 3-1 mode because the lateral stress on the beam surface is easily coupled
to piezoelectric materials deposited onto the beams.
Steel Ball
(Ms6)
Free
FallI
Heightlh
11A
Holder
f,
To
Load
RL
Piezoelectric
Vibrator
Figure 1-10: Schematic design by Umeda, Nakamura and Ueha [21]
A schematic design of Umeda, Nakamura and Ueha [21], is shown in Figure 1-10.
The piezoelectric ceramic and the vibrator are fixed to the holder on one edge. The
vibrator consists of a bronze disk of 27mm in diameter and 0.25mm thick, and the
ceramic is 19mm in diameter and 0.25mm thick. The vibrator is connected to an
electrical circuit, consisting of a load impedance RL. The experiment used to test
the harvester was performed as follow. A steel ball with mass 5.5g was placed at the
height h from the vibrator. After the ball fell on the vibrator, the vibrator started to
23
steel periment
pipe
storage oscilloscope
Insulator
R
PC
Pt
Si3 N4R
Figue 111:Schmatc
drwin ofexprimnt
b Xu Akyam, Nnak
Steel
PC
Al203
Free
Waan-
(a) Schematic Drawing of constant rate load- (b) Schematic Drawing of impact loading experiment
ing experiment
Figure 1-11: Schematic drawing of experiment by Xu, Akiyama, Nonaka and Watanabe [24]
oscillate due to mechanical energy input. The ceramic deformed, since it is attached
to the vibrator. Due to piezoelectric effect, voltage was generated in the circuit. The
efficiency of the system was then analyzed both analytically and experimentally. It
has been concluded that if the ball oscillates with the vibrator, the output energy
increases, and that the load resistance affects the waveform of the voltage.
Xu, Akiyama, Nonaka and Watanabe [24] studied the electrical response of PZT ceramics under slowly applied stress and impact stress by doing experiments. Schematic
drawings of constant rate loading experiment and impact loading experiment are
shown in Figure 1-11a and Figure 1-11b. Five cylindrical specimens with different
diameters and thicknesses for each experiment has been fabricated. The experiments
have shown that two electrical output currents with opposite directions but same value
are generated, when specimens are pressed either in increasing stress or decreasing
stress. There is no electro-mechanical coupling factor for impact stress condition, and
such kind of relation only exists with slowly applied stress. The voltage observed in
both experiments are of the same order, even though the impact stress experiments
results in less electrical energy.
24
PZT
dimorph
PZT unimorph
Alo
MWtI
PVDF
stave
PZT unimnoip
Figure 1-12: Two approaches to harvest piezoelectric energy in shoes [18]
A group of researchers in MIT Media Laboratory [18] designed a piezoelectric generator in shoe sole. Two methods have been used, explained in Figure 1-12, of piezoelectrically converting shoe power in bending 3-1-mode operation. One way is to harvest
the energy dissipated in bending the ball of the foot, using a PVDF stave under the
sole. The other way is to harvest foot strike energy by putting PZT dimorph under
the heel. This device, called a dimorph, consists of two back- to-back, single-sided
unimorphs. Although this application is very novel, its efficiency is relatively low. It
can generate high voltage on the order of hundred V, but very low current on the order
of 10~ 7 A. After trying different methods, they finally developed an offline, forwardswitching converter, consisting of a small number of inexpensive, readily available
components and materials.
Yoon, Washington and Danak [25] from the Ohio State University have studied an
initially curved PZT unimorph structure, shown in Figure 1-13, to find more efficient 3-1-mode generators, by using linear piezoelectric theory, composite laminate
theory and shell theory.
An equation, relating the dimensional parameters of the
PZT unimorph beam to the charge generation, has been developed, in order to find
25
Figure 1-13: Curved PZT unimorph excited in 3-1 mode [25]
the optimal design parameters. Nine samples have been fabricated to test their performance under mechanical loads. The experimental results were compared with
numerical simulations based on the equation derived in the paper. Although there
are some differences between the experimental and analytical prediction, they are
strongly correlated. A circuit has been designed to prove the feasibility of using PZT
unimorph as generators, and its performance has been studied under mechanical vibrations. It was shown that this kind of PZT device could be used to harvest kinetic
energy, such as human walking.
Fin
mechanical
port
b
X1
M
n
Fr
Fr nt
t
e
R
electrica
port
PZT stack dynamice
Figure 1-14: Schematic of PZT model
[71
Goldfarb and Jones [7] have studied a piezoelectrical system, as shown in Figure 1-14
26
with a commercial PZT stack model. An analytical model has been developed and
the efficiency of the system has been found to be a function of external force input frequency and load resistance. Their analysis shows that the main problem of an energy
harvesting system based on PZT is that most of the energy is stored in PZT and is
then transferred to the mechanical system. The maximum efficiency occurs when the
external input frequency is several orders of magnitude below the structural natural
frequency of the stack. To better understand the system, several experiments have
been analyzed. Both analytical and experimental results indicate that efficiency is
highly dependent on the input frequency and weakly sensitive to the load resistance.
The amplitude of the input external force also affects the system's efficiency.
Mateu and Moll [8] have studied the maximum deflection of bending beams with
different geometries and boundary conditions, under different loadings. Based on the
study, suitability of each beam for shoe inserts has been analyzed. These bending
beams have been divided into two groups, according to their properties. One property is the vertical structure (homogeneous bimorph, and symmetric or asymmetric
heterogeneous bimorph). The other property is the support: cantilever with a triangular horizontal shape for maximum efficiency, and simple support at both ends,
either with a point load, or with a distributed load. The power generated by different structures has been calculated analytically and analyzed. The results indicate
that the force applied on the insert is strong enough to create a deflection limited
by the shoe cavity dimensions. Therefore, the cavity dimension should be taken into
account. It was concluded that the deeper the cavity the greater the energy generated.
1.2.3
Electrostatic Generators
The fundamental principle of an electrostatic generator is the variable capacitor. The
variable capacitance structure is driven by mechanical vibrations. If the charge on the
capacitor is constrained, it will escape from the capacitor as the capacitance decreases.
Thus, mechanical energy is converted to electrical one. Electrostatic generators can
27
be classified into three types: in-plane overlap (Figure 1-15a) varying the overlap
area between two electrode plates, in-plane gap closing (Figure 1-15b) varying the
gap between electrode plates and out-of-plane gap closing (Figure 1-15c) varying the
gap between two large electrode plates.
(a) In-plane overlap
(b) In-plane gap closing
(c) Out-of-plane gap closing
Figure 1-15: Three types of electrostatic generators [4]
These three types can be operated either in charge-constrained or voltage- constrained
cycles depending on the electric circuit used. In general, harvesters working in voltageconstrained cycles provide more energy than those in charge-constrained cycles.
aw*e.ts.
Figure 1-16: Controller architecture [9]
Meninger [9] designed a low power open loop controller, shown in Figure 1-16, based
on the electrostatic transducer, as shown in Figure 1-17, to convert ambient mechan28
7pmmetd
Anchor
11 1jU
121"
Devioe
Wsa
HowdlWahr
SUlcon
SteadayComb
C~] Oxdde
X
Aluminum
(b) Transducer Cross-sectional View
(a) Transducer Plan View
Figure 1-17: Schematic drawing of an electrostatic transducer [9]
ical vibration energy from its environment into electrical energy. The controller is
very novel because the input filter (L and Cres) also serves as the output filter, and
the DC output voltage and input voltage are equal to the voltage across Cres. Energy
dissipation of the digital core has been measured and appeared to be within 20% of
the analytical predicted values. A closed loop controller has been designed to reduce
the energy dissipation. By employing discontinuous feedback, the controller's performance is so satisfactory that the energy dissipation of the controller is only a few LW.
Peano and Tambosso [4] have developed a method to optimize an electret-based
capacitive converter, using the nonlinear dynamical model.
As the procedure re-
quires a numerical solution of the governing equations for each combination of free
parameters, a series of constraints (technology driven) on the design of the device
have been adopted in order to reduce the number of free parameters, and, consequently, the required computational time. An example of application was reported
showing that the nonlinear behavior of the converter is crucial in the optimization
process and has to be taken into account to get correct results. The 911Hz vibration
source with an oscillation amplitude of 5pim can generate a maximum output power
of 50pW. These values can be achieved using the optimal combination of dimensional
parameters calculated with the nonlinear model. Such a operating point could not be
found using the techniques of the small signal, linear theory. Indeed, a much lower
power (i.e., 5.8pW ) could be extracted from a device whose design parameters are
29
Figure 1-18: Design by Peano and Tambosso [4]
optimized with the linear model.
Figure 1-19 depicts the electrostatic generator schematic design by Arakawa, Suzuki
and Kasagi [2]. It has been found that electret materials should meet three requirements: be compatible with MEMS fabrication technique, be easy to be formed into
thick film, and have high dielectric strength. CYTOP has been studied as electret materials in order to develop a micro seismic electret power generator. It has been found
that the CYTOP could be powered to a charge density at a maximum 0.68mC/m 2 .
The experimental results show that 6mW power could be generated when the generator is excited by external vibration with 10Hz frequency and 1mm amplitude.
The prediction of the model is similar with the experimental results. Furthermore,
it is possible that 0.5W power could be generated by external oscillation with 2kHz
frequency and 0.3mm amplitude.
30
V
Figure 1-19: Schematic design of electrostatic generator with electret [2]
Roundy, Wright and Pister [15] have compared three different types of electrostatic
generators, in-plane gap closing, in-plane overlap, and out-of-plane gap closing, taking
use of the environment vibration as a power source for wireless sensor nodes. Silicon
MEMS technique has been used to fabricate those generators, since it can closely integrate generators with silicon microelectronics. It has been stated that experimental
results have shown that in-plane gap closing electrostatic generator is the preferred
design compared with those two both quantitatively and qualitatively. The experiment has shown that a vibration source of 2.25m/s 2 with 120Hz frequency could
generate an output power density of 116pW/cm3 . This vibration source very closely
matches the casing of a microwave oven.
Figure 1-20 represents an electret based in-plane overlap electrostatic generator
schematic design by Sterken, Fiorini and Baert [19].
This generator has several
advantages. First, relative high values of capacitance can be achieved using this
gemoetry. Second, the capacitor presented in their paper is not sensitive to rotation,
which improves the reliability. Finally, the maximum capacitance is linked to the
resonance frequency, avoiding the combination of a large capacitor on a small mass
at low resonance frequencies.
31
MMEU
Adhbsive
bondina
5
Movable
Electrode
E
de
Glass
Figure 1-20: Schematic design of an electret based in-plane overlap electrostatic generator [19]
In-plane gap closing type
Figure 1-21 describes an in-plane gap closing electrostatic generator design by Miyazaki,
Tanaka and Ono [11]. The system consists of a variable capacitor, shown as Figure
1-21b, an externally powered timing-capture controller, shown as Figure 1-21c, and
a charge transporting LC tank circuit. The efficiency of the system has been studied by estimating the mechanical energy loss, the charge transportation loss, and the
timing-capture loss. It has been found that the parasitic elements in the charge transporter and the timing management of the capture scheme are the main factors for
the system's efficiency, based on that the system has been optimized to maximize the
efficiency. Comparing the experimental data with the theoretical results, the system's
conversion efficiency is 21%, resulting from a 43% mechanical-energy loss and a 63%
charge-transportation loss.
Tashiro and Kabei [20] have designed a variable-capacitance-type electrostatic
(VCES) generator, shown in Figure 1-22, harvesting ventricular vibration, in order
to drive a cardiac pacemaker. Since the generator is handmade, it is too large to be
placed in the thoracic cavity of a laboratory animal. The left ventricular wall motion
has been measured and reproduced using a vibration mode simulator. The simulator
provides external vibration to the generator and the produced power is supplied to
the cardiac pacemaker. Animal experimental results show that this generator could
32
PMOS
a[J=z IFL
Pr
C.
(a) Energy harvesting system
(b) Variable capacitor
(c)
Externally
powered
timing-capture controller
Figure 1-21: Design by Miyazaki, Tanaka and Ono [11]
successfully power the cardiac pacemaker for more than two hours.
In-plane overlap type
Peronl oeie
fix Dat Rcrdt
VC Resomaw
Figure 1-22: Design by Tashiro and Kabei [20]
Figure 1-23 shows the energy harvesting system designed by Meninger, Mur-Miranda
and Amirtharajah [10]. Cpar is used to provide maximum energy conversion. Controller IC, power switch size and Cpar have been optimized. Experiments have shown
that the maximum usable power is 8.6pW, allowing for a self-powered electronic system. It has been found that the output energy could be easily increased by designing
the system for higher voltage operation.
33
wim &V*
CM*
51 PS
Am%
DeviceWafer
Hanle WSlW
=silicon
..1Oxide
=mliriw
(b) MEMS device side view
(a) MEMS device plan view
(c) Energy harvesting system
Figure 1-23: Design by Meninger, Mur-Miranda and Amirtharajah [10]
1.3
Challenges
Mechanical vibrations with frequency as low as a few Hz are very common in our
daily life. According to Pachi's study [13], the frequency of people walking ranges
from 1Hz to 3Hz.
High-rise buildings vibrate in horizontal or vertical directions under seismic load or
wind load. The higher the building, the stronger is effect of wind load. Typically,
the first three mode shapes, which are critical during the conceptual structure design
stage, are X translation, Y translation, and Z torsion. The first natural frequency of
a high-rise building with N stories can be estimated using the equation
fi=
10
N Hz.
(1.1)
Thus the first natural frequency of a high-rise building with 20 stories, approximately
75m, assuming the typical floor height is 3.5m, is about 0.5Hz.
34
Both wind load and seismic load can excite vibrations of a bridge. There are several types of bridges commonly used nowadays: beam bridge, suspension bridge, and
cable-stayed bridge. The first natural frequency of a cable-stayed bridge can be calculated using equation
C
L
fi=-Hz
(1.2)
where C is 105 for reinforced concrete cable-stayed bridge, 110 for steel cable-stayed
bridge, L is the bridge's span. Thus the first natural frequency of a reinforced concrete cable-stayed bridge with span 100m is 1.05Hz.
In general, the first natural frequency of either a high-rise building or a long-span
bridge are pretty low. Such kinds of low frequency vibrations are very common in our
daily life, but were ignored as an energy source in the past. Vibrations of high-rise
buildings and bridges could be harvested to power wireless electronic devices, such
as wireless sensors used for system monitoring. Harvesting low frequency mechanical
vibrations to power wireless electronic devices is one of the most important topic of
research in the development of energy harvesting systems.
Unfortunately, the current energy harvesting systems' performance is not satisfactory under low frequency external vibration source. Most of the current harvesters
are based on high quality factor linear systems that are very efficient only when they
are excited at their resonance frequency. Therefore, a little change in the ambient
vibration frequency leads to a significant drop of the output power, and the system
does not perform well in the case of a broadband excitation. Therefore, we need to
improve the current generators to better harvest low frequency mechanical vibrations.
35
1.4
Nonlinear harvesters
As linear harvesters are only efficient when they are excited near their resonance, to
harvest energy from low frequencies we need to design a mechanical system with a
low resonance frequency, which means, with mass fixed because of size limitations,
a system should have small stiffness. This would consequently imply large oscillations, which is contrary to the assumption of a small device. However, interaction
can provide a better way to improve energy harvesting from ambient low frequency
mechanical vibrations.
Some experiments and simulations have been done proving that nonlinearity can
increase the efficiency of an energy harvesting systems, since it can broaden the
bandwidth of ambient vibrations that can be harvested.
micetric
xyzstag
I
X
1
Figure 1-24: Schematic design by Cottone, Vocca and Gammaitoni [6]
Cottone, Vocca and Gammaitoni [6] proposed a new method, exploiting the dynamical features of stochastic nonlinear oscillators. Figure 1-24 shows the schematic design
36
of an energy harvesting system, consisting of a piezoelectric inverted pendulum, designed by them. They reproduced the ground vibration by attaching two magnets
whose magnetic excitation was properly designed, near the base of the pendulum.
Under the excitation, the pendulum oscillates, alternatively bending the piezoelectric
beam and thus generating a measurable voltage signal. The dynamics of the inverted
pendulum tip can be controlled with the introduction of an external magnet conveniently placed at a certain distance A and with polarities opposed to those of the tip
magnet.
When the external magnet is far away from the rest position, that is to say A is
large, the inverted pendulum behaves like a linear oscillator whose dynamics is resonant with a resonance frequency determined by the system parameters. On the other
hand, when A is small enough, two new equilibrium positions appear. The random
vibration makes the pendulum swing in a more complex way with small oscillations
around each of the two equilibrium positions and large excursions from one to the
other. In this case, the potential is bi-stable with a very pronounced barrier between
the two wells. In between there is a range of distances A where Vm, reaches a maximum value. In this condition the pendulum dynamics is highly nonlinear and the
swing reaches its largest amplitude with noise assisted jumps between the two wells.
0(c),OPP-amadm
F0.2ioulappn6t)["
-0.41
Figure 1-25: Envelope modulation 6(t) [22]
37
Vakakis [22] has analyzed energy pumping in impulsively loaded vibrating systems
with strongly nonlinear attachments.
It has been found that in a two degree-of-
freedom (DOF) system, nonlinear energy pumping coincides with the zero crossing of
a frequency of envelope modulation 6(t), as shown in Figure 1-25. This finding led to
the formulation of a criterion for inducing energy pumping in a two DOF system. An
impulsively loaded multi-DOF chain with a nonlinear attachment has been analyzed,
showing that after some initial transients the response of the nonlinear attachment
settles to a motion dominated by a single "fast" frequency. This frequency coincides
with the lower bound of the propagation zone of the linear chain, and corresponds
to in-phase standing wave oscillations of all particles of the chain. This property
enables the reduction of the problem of energy pumping in the chain compared with
the simpler problem of energy pumping in a two DOF system. Thus, the previous
results derived for a two DOF system are directly extended to the chain problem.
Authors of [22] stated that nonlinear attachments, if appropriately designed, could
act as passive sinks of unwanted disturbances. This result can be extended to one- or
two-dimensional elastic continuous applications in the area of electromagnetic fault
arrest in extended power networks.
Linear
Limearspring
(a)
aper
Linear spring
Lin
(b)
Linearsprng
Iear dampers---
~*inear
damper
(c)
Figure 1-26: Three types of nonlinear energy sinks [16]
Sapsis, Quinn, Vakakis and Bergman [16] have studied the stiffening and damping
effects that local essentially nonlinear attachments can have on the dynamics of a
primary linear structure. These local attachments can be designed to act as nonlinear energy sinks (NESs). Three types of NESs, shown in Figure 1-26, have been
38
designed and their effects on the stiffness and damping properties of the linear structure have been studied via (local) instantaneous and (global) weighted-averaged effective stiffness and damping measures. It has been found that these attachments
could dramatically increase the effective damping of a two-degrees-of-freedom system
and, to a smaller degree, the stiffening properties as well. The essentially nonlinear
attachments could introduce significant nonlinear coupling between distinct structural modes, redistributing nonlinear energy between structural modes. This feature,
coupled with the well-established capacity of NESs to passively absorb and locally
dissipate shock energy, can be used to create effective passive mitigation designs of
structures under impulsive loads.
39
40
Chapter 2
Physical model and linearized
analysis
In this chapter, we describe the design and mathematical model of an electrostatic
energy harvesting system analyzed in this thesis. We discuss in details about the
physical structure of individual system components and governing equations that describe their dynamics. We describe the analytical studies aimed to understand how
the system parameters affect the response of the system.
2.1
2.1.1
Physical model
Convertor
This work is focused on an in-plane overlap electrostatic generator that can be used to
harvest ambient environmental vibrations. Electrostatic design was chosen because
of its simplicity, potential for high values of energy density and direct conversion between mechanical and electric energies.
An in-plane overlap varying capacitor, shown in Figure 2-1, with one electrode fixed
to the ground and the other connected to an external mechanical vibration source,
41
Id
(a) Variable capacitor
(b) Variable capacitor mov- (c) Variable capacitor moving in the right direction ing in the left direction
Figure 2-1: Configuration of the variable capacitor
is the key element used for conversion the mechanical energy into the electrical one.
It consists of two electrodes connected to electric circuit.
The upper electrode of
the capacitor is constrained to move in the horizontal direction only. The horizontal
movements of the mobile electrode decrease the overlap area of the capacitor, resulting in the redistribution of the charge on each electrode. Thus charge will flow in or
out the electrodes and the current will be generated in the circuit.
This phenomenon can be also explained theoretically. The capacitance of plane capacitor is given by
C
_o(Lo - |xi)w
d
(2.1)
where co is the dielectric constant(8.854 x 10- 1 2 F/m), LO is the length of the fixed
electrode, x is the relative displacement between those two electrodes of the capacitor, w is the width of the electrodes, d is the gap between two electrodes. If the
movable plate of the variable capacitor is connected to an external vibration source,
time dependent relative motion will incur between two electrodes, that is to say x as
well as C1 will change with time. At any moment of time the charge on the capacitor
plates
Q and
the voltage between the plates V will be related via
Q=CV
(2.2)
The charge of the variable capacitor will change with time as well. The escaped
charge will generate the current in the circuit.
42
C11
P1
C,
N1
P2
N2 C2
Figure 2-2: Charge trapping configuration
To prevent the complete discharge of the system, a serial interconnection of capacitors
is used to trap the charge between inner plates, as shown in Figure 2-2.
We denote the charge of variable capacitor C1 to be qi =
QP1
= -QN1.
The
charge of constant capacitor C2 is q2 = QP2 = -QN2. If some charge dq escapes from
electrode P1, the same amount of charge will also escape from electrode N1. Since
electrodes NI and P2 are connected, and there is no circuit to allow the charge to
escape from them, the total charge QN1 + QP2 is constant. Therefore,
-qi + q2 = Qo(constant).
(2.3)
Since the charge of the variable capacitor changes with time, if its mobile electrode
is connected to an external vibration source, qi will change with time. Hence, q2 will
change with time, but the total charge stored in capacitors C1 and C2 will remain
constant. This way, the charge is transferred between the variable capacitor and constant one and cannot escape from them, or in other words the charge is trapped in
the capacitors.
2.1.2
Design of mechanical system
The mechanical subsystem, shown in Figure 2-3, contains one moving mass, with
mass m and one spring, with spring constant k, that can be used to tune the natural
43
frequency of the system to match that of the external vibration source. The mobile
electrode forms the moving mass. One end of the spring is connected to the mobile
electrode and the other end is fixed on the ground. To better analyze the system, we
separate the mass from the capacitor and assume the capacitor has no mass. Thus,
the mechanical part of the system purely consists of a moving mass and a spring.
spring1
massi1
Electrical System
fixed1
Figure 2-3: Configuration of mechanical system
A few external vibration sources are connected to the moving mass, driving it move
horizontally. The external energy input can be modeled in the form of external forces,
applied to mass m
f
(2.4)
Ficos(Qit+ s3 )
=
i=1
where F is the amplitude of the vibration with frequency Qj and phase
#.
The existence of the spring will change the response of the system to external forcing
and affect energy harvesting and conversion rates.
2.1.3
Design of electric circuit
The electric circuit consists of the generator, described in Section 1, an inductor L
and a resistor R, shown in Figure 2-4.
44
sytem
ground1
Figure 2-4: Configuration of electrical subsystem
By adjusting the value of inductance, the natural frequency of the circuit can be tuned
to match that of the mechanical system, in order to increase the energy conversion
efficiency of the system. The frequency of the circuit can be expressed as
f =
(2.5)
27rvTU-
The value of the resistance is determined by the electrical device, which is powered
by this electromechanical system.
2.1.4
Configuration of the energy harvesting system
The complete system, shown in Figure 2-5, is an interconnection of the electric circuit and the mass-spring system, where the moving mass is attahced to the mobile
electrode.
If the moving mass is forced by an external vibration source, the mobile electrode
will move horizontally, incurring decrease of the capacitance. Thus the charge stored
45
fbsdl
'
grodi
Figure 2-5: Configuration of the energy harvesting system
in the variable capacitor C1 will decrease as well. In other words, some charge q will
escape from the variable capacitor and flow to the constant one C2. After some time,
the mass will move in the opposite direction, leading to the movement of the plate in
the same direction as that of the mass. Thus the overlap area of the capacitor's two
electrodes will increase, resulting in the increasing of the capacitance. Therefore, the
charge will be transferred from the constant capacitor to the variable one. Current
will be generated in the opposite direction. This way, the charge can oscillate between
those two capacitors, generating current. In general, if the external vibration source
keeps forcing the mass, electrical energy will be generated continuously, keeping powering the resistor. This process is shown in Figure 2-6.
2.2
Linearized analytical studies
The model used for analytical studies is a simplified version of the model, as shown
in Figure 2-7.
46
-v v
External
vibration
source
*--I
~WIt
Figure 2-6: Electrical energy generation
Nod
I
Figure 2-7: Analytical analysis model
2.2.1
Equation of motion
There are two main methods to derive the equations of motion for a lumped-parameter
electromechanical system. One is the direct approach, and the other is variational
approach, known as Lagrange's approach. The variational approach can be used to
avoid lots of complicated internal forces between the electrical and mechanical subsystem.
This electromechanical system is a two degree-of-freedom system, and x and q are
used as the generalized coordinates.
47
The magnetic coenergy W*, accounting for all the inductance in the system, is expressed as
Lidi = Li2
W*, =
(2.6)
= L412
The electrical energy We, accounting for all the capacitance in the system, is expressed as
2
where C (x)
=
d
-o(Lo-')
and q2
2
q2(2.7)
C(x) + 2Co
We =
q + Qo.
=
The Lagrangian is
L(x,tjq 1 ,j1 ) = T-V+W
=m c2
2
-We
kx+
_
2
(q + QO)
- L1
2
C (x)
2Co
2
(2.8)
where T and V denote the kinetic energy and potential energy in the mechanical
system respectively.
Lagrangian's equations are expressed as
d OL
aL
=
(t ) -q
d ((L
DL
dt 82
x
48
-
qi
(2.9)
(2.10)
where E
and -E, are generalized external forces.
HL
--(
)L= Ld
j -(
)H
dt 892
dt 841
z
d
1
+ -) qi
Co
eow(Lo - xi)
Qo
=: --(
9q1
2
aL = -kx -
_q
C.
d signix)
1
2cow (Lo - xIl)
ax
2
(2.11)
(2.12)
(2.13)
Therefore, substitution of Equation 2.8 into Equations 2.9 and 2.10 gives equations
of motion as below
1
+ -)qi
+(
Li
mz5 + kx +
The existence of nonlinear terms
sign(x)
cowd (Lo-IXI)
2q
2
and
d1
eow(Lo-IxI)
+
-
= 0
(2.14)
2
motion
in equations offmto
i qain
-0
-
q
si_"
2
2cow (Lo-IXI)
indicates that this electromechanical system is a highly coupled nonlinear system.
2.2.2
Equilibrium point
The equilibrium positions can be found by setting
0, di
z
-
0,
=
0
(2.15)
0
(2.16)
Therefore,
_d +
{(
kx +
q
2
1)qi +
oscillate around qi =
,
(2.17)
=n(x
0
*d
2
eow (Lo-IxI)
x
Thus, the equilibrium position is qi =-
=0
-
0. In other words, the system will
x = 0. Any initial conditions differing from this equilib-
rium position will initiate the system's vibration. If there is no external energy input
49
into the system during the oscillation, the total energy of the system is conserved and
energy may redistribute due to the oscillation around the equilibrium position. In
order to better understand the system's behavior, a further analysis of the system's
dynamic response should be carried out.
2.2.3
Linearization
The existence of nonlinera x and qi coupling terms in the equations of motion, Equation 2.14 implies that conversion between electric and mechanic modes is possible.
In Section 2.2.2, we found that qi
-2,
x = 0 is one of the equilibrium positions.
This position is the rest position of the system. Introduction of a small deviations
from equilibrium can excite the system vibrations. We are interested in is the small
oscillations around the equilibrium position and the corresponding energy redistribution between its mode shapes. We can study the behavior of this nonlinear system
near equilibrium position qi
Qo,
x = 0 using linearization. Linearization in this
context means construction an approximation to the nonlinear terms in the equations of motion around the equilibrium point. As long as the motion stays close to
the equilibrium point, the performance of the linearized system is a good predictor of
that of the original nonlinear system. Therefore, we can use the linearized equations
of motion to describe small vibrations about the equilibrium point.
In order to simplify the equations of motion, setting q = qi + 2,
denoting the
deviation from equilibrium, gives
_ + -)(q-
L4 +(
+
mz+kx
Assuming that |xi < Lo and q <
(q-)2
d sign(x)
2
eow
(Lo-\xi) 2
Qo,
)+G
=0
(2.18)
0
the nonlinear terms can be linearized as shown
below.
d
Eow( Lo -|jx||
1
Co
~(Ld
- +
50
|x|
LoCo
(2.19)
9
(q -
)2 d sign(x)
(Lo - |x\) 2
2eow
Q 20 d sign(x)jxI
Qod
4eowL 20
2eowgL20
Q0 sign(x)
2Co
Lo
Q20
Lo
4
(2.20)
C0L2(
Thus, substitution of Equations 2.19 and 2.20 into Equation 2.18 gives linearized
equations of motion:
Lij+
g-
2
C0Lox =0
m + kx -
+
q
(2.21)
Q
x =0
Note that this linearization is only acceptable when the system oscillates close to
equilibrium position q = -
, x = 0. It must be noted that the resulting equations
are still nonlinear because of lxj and sign(x) terms. However, they can be represented
as linear equations in two regions x > 0 and x < 0.
2.2.4
Mode shapes
Equations of motion can be written in the equivalent matrix form
M<I>+ K-k = 0
where <D is the response column vector
<P
=
4(2.23)
X
51
(2.22)
M is the mass matrix
L
0
0
m
(2.24)
J
K is the stiffness matrix
2
Q0
T 2CoLO
o
K -
L
Q0
T 2CoLo
k
+ 42
4CoL' 2o
-
mw2
m
I
(2.25)
where -F is resulted from different regions x > 0 and x < 0.
Normal modes analysis, also called eigenvalue analysis or eigenvalue extraction, is
a technique used to calculate the mode shapes and associated frequencies that a
structure will exhibit.
K-MA
=0
(2.26)
where A = w2. The solution of this eigenvalue problem yields two eigenvalues.
Substituting mass and stiffness matrices into it, the eigenvalue problem can then
be written as
2 -
Lw
2
Qo
T2C0 L0
T 2COLO
2C~o k
+
= 0
(2.27)
mw2
4 202o - m
4CoL
which leads to the quartic equation
2
CO
- L2)(k +
0
2
4C 0 L2 0
- mw 2 )
Q2
0
4CoL 20
0
(2.28)
with two solutions wi and w2 .
Setting We
=
,
Wm =
k
and e
= 4c
2.28 give
52
2
and substituting into Equation
2
w2
(1- We2
e 4
02
6-4CoLo k
~Wm 2
Q0
can
rertenaL as E
be rewritten
can be
+ )
2
(2.29)
=0
2
4C 0
We can easily tell from this equation
2.
that E is a fraction of the electrical energy over twice of the spring potential energy
of the system.
2
To simplify Equation 2.29, defining q = (g)2, and ( = (''a) and substituting them
into Equation 2.29 gives the governing equation
(1 -
)(1 + 6 - n)
-
=
(2.30)
0
The solutions of this equation are
2
(2.31)
2
According to the matrix form of equations of motion,
(2.32)
(K - MW2 )<=O0,
substitution of mass and stiffness matrices gives
[
Q0
T- LCo
T2cOLO
WociLo
+
4COL 2
- MW2
I[q
=0,
(2.33)
To obtain the eigenvectors, or mode shapes, we substitute the frequencies into the
eigenvalue equation. Since the stiffness matrix is different when x changes sign, mode
shapes can be separated into two cases.
Case 1: when x > 0,
53
Mode 1:
Q0
q,
(2
x1
LCow12)Lo
Q0
(2.34)
2~1
2L(1 -- 1)
Mode 2:
=2
_Qo
2Lo(1-
1
2
(2.35)
0
)
Although mode shapes are not unique, they are chosen to keep the same units as q
and xi. This method simplifies future calculation and analysis.
Thus the response can be expressed as a combination of the two mode shapes.
where c+, c+,
#1,
<b+
=
<k+Ctcos(wit
+
#2)
and
#2
depend on initial conditions. c+, ci,
#1,
+ #1 ) +
<I2c2jcos(w 2 t
(2.36)
and
#2
are related
to energy distribution associated with the initial conditions between the two modes
of vibration. In the absence of nonlinear interactions, the energy stored in each mode
remains in that mode forever.
That is to say there is no energy transfer between
normal modes.
Case 2: when x < 0,
Mode 1:
qQoQ
-zi
2
-2Lo(1 - 771)
54
1
(2.37)
Mode 2:
Qo
<b2
=
(2.38)
2
q2
L-2Lo(1
L-X2 -J
- r/2)
The response can be expressed as
<b
where c-, ci,
2.2.5
= (i-c-cos(wit +
#1, and #2 depend on
#1)
+
<D2jc2cos(w2 t
+
#2)
(2.39)
initial conditions.
Transformation matrix
Section 2.2.4 shows that mode shapes are different in two regions of x space. The
system can be treated as two different systems, one is called positive system, where x
is always positive and the other is called negative system, where x is always negative.
Those two systems are not isolated, instead they are related to each other. From last
section, we can tell that once the initial conditions are given, c+, c,
#,
and
#2 can
be easily calculated. In positive system, when x decays to zero, it will cross the zero
axis, that is to say x becomes negative. Then at this instant of time, the values of q
and x become the initial conditions of the negative system. Thus, from the negative
initial conditions, c7 and c2 can be found.
The sign of x is determined by the initial movement direction of the mobile plate
of the variable capacitor. Positive means that at this instant of time the movement
direction is the same as the initial movement direction. Negative means that at this
instant of time the direction of movement is opposite with the initial moving direction.
The relationship between c+, C and c-, c- can be found by matching initial conditions at the moment of crossing x
=
0. If we use x+(t) and q+(t) to denote the value
of x and q just before x's crossing respectively, and x-(t) and q-(t) to denote the
value of x and q just after x's crossing respectively, the following relations should be
satisfied.
55
{
x+(t) = x-(t)
(2.40)
q+(t) = q-(t)
Substitution of the negative and positive systems' responses gives
Cj+X 1 Cos(Wet +
#1) +
C2+X
C1+ql cos(wet +
#1) +
C2+q2 Cos(Wmt
2 Cos(Wmt
+
+
#2 )
c1-(-x 1 ) cos(Wet
+ #1) + c2 (-x
2)
Cos(Wmt
c 1-gi cos(wet + 4 1 ) + c 2 q2 Cos(Wmt + 0 2 )
(2.41)
02) =
These two equations can be written as matrix equation
[ [
X1
X2
C1+
-X1
q2
c 2+
Lq,
C1
-X2
LC2
q2
I
(2.42)
In a much simpler way, a transforming matrix T can be introduced. T is defined by
[
ci+±
Cj
T
Ci
(2.43)
LC2+
Therefore, substituting Equation 2.42 into Equation 2.43, we can get
T
-x1
-X2
x1
X2
q1
q2
q1
q2
1
--x 1 q2
+ x2q11
1
-xiq2 + x2q1i
2
X2
-q 1
-x
[-
xiq2 + x 2 q1
-2x
1
56
qi
J
x1
X2
q1
q2_
2x 2 q2
-(xiq
2
+ x 2q1 )
I
(2.44)
+ #2)
Substitution of mode shapes into Equation 2.44 gives
T -
71 -
2.2.6
L_
2 - 71-
1
772
2(1-q2)
72
(2.45)
-2 +71 +121
-2(1 - 71)
Energy
An understanding of how energy is distributed in each mode is very important for
improving the energy transfer efficiency of the system in the future. Although, as referred before, the energy in each mode remains constant during linear dynamics and
the modes do not interact with each other, when x crosses zero axis, mode shapes
change, so do the initial condition parameters ci and c2, that is to say, the energy
distribution among those two modes will change. This will lead to energy redistribution in the system.
In this section, the amount of redistributed energy will be calculated and the relationship of the redistributed energy and the system parameters will be developed.
Energy stored in the system can be expressed as
E =
2
i>TM4
+
2
(2.46)
kTM(P,
where <D is a general form of response <P+ and <b-. According to the orthogonality
of the mode shapes,
<bM4b2 = 0,
<bTM~pi = 0, <bT K(b2 = 0,1
57
<b K(bi = 0,
(2.47)
energy E can be expressed as
E
1j 2 sin(wt
=
+
41) 2 w12<JiTMI1 +
c2sn(w
2t
)22T
+ c 22 [cos(w2 t + #)2222
+ c12 [cos(wit +
(2.48)
where ci is the general form of ct and c-, and c2 is the general form of cj and c-.
Since
w 12 <JTM
w2 2<D2TMI)2
)1 = <biTK-bi,
= <P2TK4b2,
(2.49)
Equation 2.48 can be simplified as
1
2
-C 12 [sin(wit +
E
+
=
2
C22[sin(w2 t +
c 21
W
2
1
2
#1)
W2
#2) 2
TM~pi +
2
iTML)
1
+ cos(wit + 41)2W
2(D2TM(b2 + cos(w 2t
c2 2w2 2b
2<]17M(b1]
+ #2 )2 w2 2<P 2 TM4b 2]
TMb2
2
(2.50)
x2
When x > 0,
=
= q1 2 L
=
X1
[qi
+x
0]
0
m
]
L
[
q]
X1
2
1 m
2
L+4Lo
0 (1
58
n)2m
(2.51)
[q2
Qo
X21
2
L+
4
(
Lo2
7
[
01
]L0
q2
m
X2
(2.52)
m
Thus,
E+= 1±W2l24QL
+2 2 [Qo L
E-21
c2 0 2
=
2[
+ 4L 0 2 (l _ ql) 2 m] +
1 +2LQo2 2e2
r/1[11
4
= ci
=
2
c=
1+2 QO2
2C 02
C
2
2
+ 4L
2L
r
1c2W22[
2
_ 1) 2 m] +
1 +22
CWe
2 Lo
L +
2LL+4
L + 4L
2
2 1 +2LQo2We
r2[
4
- c2 +
16mnLo2
+LQ 0 2 (1
7 1
4L
2
2
2
(1 -
16mL 02
12)2m]
+ LQ 0 2 (1 -
2
7/2 - 1_
7/2 - (1 + e)(
1 +2Q02
2CO2
+ (1 + 6)(] + cf2 eLo2k 72 2272 - [1 + (1 + )()
r/1
- (1 + e)
72 - (1 + )
c2Lo2kr/ 1 2r1 - [1
(2.53)
When x < 0,
1
M)-
=
-
[ qi
4
L
01
0
m
-Xi]
L + 4Lo2 (1
59
-
2m
[
q1
-zi
(2.54)
0]
<D TM 4b-
x21[L
I
=Iq2
0X
Q0 2
q2
m
-X2
,2L2
= 4 -L +
0m
r2)2
(1-
(2.55)
Thus,
2
E- = c,_2 eLo 2 kr/1 r/1 - [1 + (1 + E)( )
77
2r/2 - [1 + (1 + e)(]
_22EL0 2
72 -
(2.56)
(1 + e)(
Two scenarios should be considered when analyzing the energy redistribution of the
system.
Scenario 1:
Assuming there is only initial electrical energy input into the system, which means
C+ =C
and c+ = 0, substitution into Equation 2.45 gives
[-I I
Cl
c1
0
27/1
=ct
-
72
a
-2(1
]
- r72
-ql
I
(2.57)
where
(2 -
/1 -'r2)
71 -
(2.58)
72
and
-r/)
0-2(1
r/1 - r/2
60
(2.59)
Thus, energy stored in mode 1 when x > 0 is
Elc=
2771 - [1 + (1 + e(
+>l(2.60)
c Lo2k??2[
_C22
Ei
-
(1 + E)2
and there is no energy stored in mode 2 when x > 0, since c2 = 0. Energy stored in
mode 1 when x < 0 is
2771 - [1 + (1 + e)() a
2
E- = c2 eLo2k 1
2
(2.61)
- [1 + (1 + e)(]32
(2.62)
11 - (1 + c)(
and energy stored in mode 2 when x < 0 is
kl
C02
2 71
11 - (1 + E)
Since there is no external energy input into the system, according to energy conservation,
(2.63)
E+ = E-- + EEj can be written in a simpler way
E
=c
+22
~'-2
C1
2,q1 - [1 + (1 +
Lo2[(1
e)(
- (1±+
-
C)(]
2)
(2.64)
= E+(1 - a 2 )
When x > 0 or x < 0, energy stored in each mode is constant once ci and c2 are given,
and the total energy is conserved. Equation 2.64 implies that, when x crosses zero
axis, or in other words when x changes signs, energy stored in mode 1 is transferred to
mode 2, which means energy is redistributed between two modes. This redistribution
phenomenon can be well utilized to convert mechanical energy into electrical one,
if the parameters are designed properly. We find that the amount of redistributed
energy is E2 , which is (1 - a2) times the total energy. An in-depth understanding
of how the system properties, such as we, wm and e, affect the energy redistribution
61
should be done by analyzing the relationship of (1 - a2) and those parameters.
(2 -
2
1
1i -
2) 2(2.6
5)
a[
Substitution of Equation 2.31 into Equation 2.65 gives
[1I (1 + 6)(]2
[~1±)j
1-a 2a2 =1-_
[1 - (1 + e)(]2 + 2c
[1 12
(1 + E)] 2 + 2c6
(2.6 6)
Scenario 2:
Assuming there is only initial mechanical energy input into the system, so c=
and cj
=
0
c+, substitution into Equation 2.45 gives
[I I
0
C2
c
C2
2(1 - 772)
c2
771-172
- 2
+
m1
=cj
where a and
#
+ 772
(2.67)
are the same as those in Scenario 1.
Thus there is no energy stored in mode 1 when x > 0 and energy stored in mode 2
when x > 0 is the total inital energy of the system, denoted as E0 . When x < 0,
energy stored in mode 1 is the total energy transferred from mode 2 to mode 1.
Ei = Eo(1 -a
62
2
)
(2.68)
The transfer factor
2
~
2 +
m1++
2 12
[ 1 - ]2
[1-(1 +
6) ]2
+ 26
(2.69)
is the same as the one in Scenario 1.
For typical systems described in section 2, the value of e can be set to be 0.1, 0.2,
or 0.3. In addition, extreme conditions, such as e = 0.01 and e = 1, should also be
considered. Then substituting these values into Equation 2.66, respectively, gives
0.024
[1-1.01 ]2+0.02
0.24
[1 -1.1g]2+0.26
a2
0.46
[1-1.2 ]2+0.4
0.64
[1-1.3(]2+0.66
[1-2 ]2+2
where (
=
0.01,
0.1,
0.2,
(2.70)
0.3,
1
(,-)2
It can be concluded from Figure 2-8 that all the transfer factors can reach the maximum value 1, no matter what value e is. And the peak gradually moves toward left
as c increases.
The value of ( depends on the inductance and is generally very small for microscale energy harvesters. In our work we consider the following values of ( deferring
the question on how they can be achieved in practice.
(= 0.01, 0.03, 0.05, 0.07, 0.3
63
(2.71)
1
a2
1.0
(=0.01
03
0.6
-
e 0.1
e(0.2
= 0.3
--
02
Figure 2-8: Plots of 1 - a2_
Then substituting these values into Equation 2.66 respectively, gives
1
-
2
2
[1-0.01(1+e)] +0.02E
0.06e
[1-O.03(j±e)] 2 +O.06e
M~E
[1-0.05(j+E)]2+0.1e
0.01,
=
0.03,
=
0.05,
0. 14E
[1-O.07(1+E)] 2 +0.14e
OM6E
2
[1-o.3(1+e)] +O.6E
(2.72)
S0.07,
=
0.3
1 - a2
OE
Figure 2-9: Plots of (1 - a2)_
It can be concluded from Figure 2-9 that all the transfer factors can reach the maxi64
mum value 1, no matter what value ( is. And the peak of the transfer factor moves
to the left slowly with the increasing of .
Those phenomena have simple mathematical explanation.
Equation 2.66 can be
rewritten as
1 2)2
2 -
This implies that (1 -a
2
(2.73)
) increases with ( before ( =
1, reaches its limit at (
=
and decays to zero after that. In addition, the transfer factor (1 - a 2 ) increases with
e before c
=
-
1, reaches its peak at e
=
-
1, and decays to zero after that.
The maximum value of (1 - a 2 ) is 1. That means all the energy stored in mode 1 is
transferred to mode 2 when x crosses zero axis.
65
66
Chapter 3
Numerical analysis
In this chapter, we describe the numerical studies of the system to test the validity
of the theory in Chapter 2 and to better understand result of nonlinear mode interactions. We also study how the system parameters affect the energy transfer efficiency.
3.1
Modelica language for modeling
We use Modelica language to numerically model our electrostatic energy harvesting
system. Modelica is a programming language that allows specification of mathematical models of complex natural or man-made systems. It is also an object-oriented
equation-based programming language. It allows high performance simulations when
applied to computational models with high complexity. The four outstanding features
of Modelica are:
It is mainly based on equations instead of assignment statements. This allows users
to better reuse classes since equations do not specifically have a certain data flow direction. Therefore a Modelica class can be applied to more than one data flow context.
It has multidomain modeling capability, that is to say model components of physical
objects from several different domains such as, electrical, mechanical, thermodynamic,
67
hydraulic, biological, and control applications can be described and connected.
It is an object-oriented language with a general class concept that unifies classes,
generics-known as templates in C++and general subtyping into a single language
construct. This facilitates reuse of components and evolution of models.
It has a strong software component model, which constructs for creating and connecting components. Thus the language is ideally suited as an architectural description
language for complex physical systems and to some extent for software systems.
These are the primary properties that make Modelica both powerful and easy to
use, especially for modeling and simulation.
Since the generator consists of both electrical and mechanical systems, a programming language capable of modeling multidomain components is necessary. Based on
this, Modelica is a good choice.
3.2
Modelica model
in the following sections, we describe the implementation of individual component
models used in our study.
3.2.1
The constant capacitor
A constant capacitor has two electrical pins to communicate with other electrical
elements. Typically, an electrical pin has two variables, current i, and voltage V. A
constant capacitor has two more variables, capacitance C and charge
Q.
Thus, in
total, there are four variables. These four variables are not independent. They have
the following relationship.
68
Q=CV
(3.1)
=dQ
dt
(3.2)
These two relations form the governing equations for the modelica model of a constant capacitor. The modelica script of the constant capacitor used in here can be
found in Appendix A.
3.2.2
The variable capacitor
Different from the standard capacitor, the variable capacitor should have both the
electrical and mechanical pins to interconnect with the electrical and mechanical subsystems. It is described by the following three governing equations
C =eow(Lo - xi)
d
(3.3)
V = -C
(3.4)
i
(3.5)
dQ
dt
An additional equation is needed to describe the mechanical property of the capacitor's mechanical pins. It is the constitutive relation for the horizontal electrical force
between two plates, generated when the overlap area changes. This force can be derived by calculating the partial derivative of the electrical energy with respect to the
relative displacement between two electrodes.
d(2)
dE,
dx
dx
sign(x)Q2 d
2eow(Lo - Ix|)2
69
d(
Q
dx
)
(3.6)
The Modelcia models described above can be referred to Appendix A. Other components of the system are part of the standard Modelica library [3] and are not described
here.
3.3
Test the Modelica model
In order to test the validity of the Modelica model, a numerical simulation was run
without damping and external energy input. The main goal of this test was to check
the energy conservation in this model. The plots of electrical energy, mechanical energy and total energy are shown in Figure 3-1.
IMSX19
-1
x
to-2.167x
-
IS--
LE. 102.105x 1L
IJO'S,
six 10e
(a) Electrical energy
(b) Mechanical energy
Lids
LISS10-
.
.o .
(c) Total energy
Figure 3-1: Energy Plots
As we can see from the plot of total energy, the variations of the total energy with
respect to time are tiny compared with the value of energy, and this phenomenon is
due to numerical calculation errors, which are negligible compared with the value of
the total energy. Thus the total energy of the system is conserved.
70
3.4
Numerical studies of different cases without
external input and resistor
In this section, the theory developed in Chapter 2 will be tested using numerical
analysis method not relying on any approximations like in analytical part. The model
used in this section is the same as that in Chapter 2 and is described in Figure 3-2
again.
Wrigi
Minal
fixed1
Figure 3-2: Numerical analysis model
It was found in Chapter 2 that there are mainly two factors that affect the transfer
factor of the system, e = 4c ok2
=
o If an impulse is applied to the mass, that
is to say the mass has some initial velocity vo, the energy will oscillate between the
mechanical and the electrical systems. Thus, another parameter -Y,defined below,
will affect the transfer factor of the whole system.
1 nvo 22
(3.7)
" kLo
where vo is the initial velocity of the mass, -yis the ratio of the initial mechanical
energy over the largest available spring potential energy.
In general, mainly three factors play an important role in the performance of this
71
electro-static energy harvesting system, e,
, and -y. A few cases with different c,
,
or -y will be analyzed in order to check the validity of the theory found in Chapter 2.
The goal of this work was to understand the efficiency of nonlinear energy conversion
achieved with the nonlinearity discussed in the previous chapter. In order to study
the energy conversion via Modelica language, we have chosen the physical parameters
such that the values of E, , -y are given by Table 3.1.
Table 3.1: Parameters of different cases
E
,
Y
Base
0.10
0.09
0.10
Case 1
0.50
0.09
0.1
Case 2
0.10
0.30
0.1
Case 3
0.10
0.09
0.8
Case 4
0.01
0.09
0.1
Case 5
0.10
0.05
0.1
The parameters can be calculated using the following equations. The actual values
and specific geometry has to be decided depending on the application.
Qo = -/4Cok Lo2E
2m
kCo
VOm
kL
(3.8)
(3.9)
(3.10)
It might be difficult to match the frequencies of electric and mechanical circuits in
micro-scale, but high values of ( can be achieved with the help of large multi-plate
capacitors.
72
In order to check results easily, the following items are defined by making the corresponding terms dimensionless during the post processing stage.
x*=
(3.11)
Lo
q* q=
(3.12)
o
For the sake of testing the validity of Chapter 2's theory, more attention should be
paid to how much energy is transferred if there is only electrical or mechanical energy
input. Thus, assuming that the initial energy is purely kinetic, and the electrical energy should be compared to its equilibrium value. The following modifications should
be made during the post processing stage.
Ee
1
12
2C1
L
2
_
2C 2 2 2
k Lo
2
kmi22
2
E* = mkLo
2Qo
2
4Co
(3.13)
2
(3.14)
2
In order to verify the transferring efficiency of the system, in the following sections,
T,
3.4.1
EE* is defined as the transfer factor.
Base
Figure 3-3a shows that the response of x* appears harmonic and primarily one mode
shape is excited. As one can see from Figure 3-3b, the dynamics are non-harmonic
because the system is nonlinear. This implies that both mode shapes play a role in
the response of q*. In addition it should be noted that each time x* crosses zero, there
is a transition from one mode to another in the response of q*. This phenomenon
proves the validity of the theory found in Chapter 3.
As it can be seen from Figure 3-3c, the electrical and the mechanical energy appears
73
I*
J~~A
A1A AIA
042
r,
0."
0.11
-
t oAor
-04-
-0
0
.
I
(b) Base-q*
(a) Base-x*
EU=V*
0.(y
0.15
040
0
5
10
15
20
t
11
1
20
$
(d) Base-Tr
(c) Base-Energy*
Figure 3-3: Numerical results of model Base
sinusoidal. This also indicates that this system's nonlinearity is very weak and the
coupling is not strong. The plot of transfer factor T, Figure 3-3d, shows that although the almost sinusoidal plot implies the linear property of the system, the small
fluctuations of the peak indicates the weak nonlinearity of the system.
In general, this model is weakly nonlinear and only one mode is excited. The transfer
factor is not satisfactory due to the small coupling of mechanical system and electrical
system.
3.4.2
Case 1
In comparison to the Base Case, we increased the value of e in Case 1 by doubling
the total charge stored in two capacitors, keeping ( and -y the same as in Base. As
74
q.
1*
~1
A
AA
AA
A
A A
I
-OA
-
-0.-0
-0.16
-
-
(b) Case 1-q*
(a) Case 1-x*
Tr
0.0Em
0.10
Ee
OME
OMW
0N0(
0
5
10
i5s
0A
C-
15
2
I
20
(c) Case 1-Energy*
(d) Case 1-Tr
Figure 3-4: Numerical results of model Case 1
we can see from Figure 3-4a and Figure 3-4b apparently, there are two frequencycomponents in the plots, indicating that both modes are excited for both x* and
q*. This implies that increasing e can help excite both modes when ( is small. This
can be explained by analyzing the relationship of mode shapes with e from Chapter 2.
As one can see from Figure 3-4c, the system has a strong coupling between mechanical system and electrical system. In addition, the large overlap area indicates
that the energy transfer between these two systems is very active, suggesting there is
good coupling between the mechanical and electrical susbsytems. Figure 3-4d shows
that there is a significant increase of the transfer factor from a maximum of 0.31 to a
maximum of 0.82, compared with that in Base Case. It implies that the increase of
e helps dramatically improve the performance of the system. According to Equation
3.64, the growth of transfer factor with the increase of e happens only when e <
before the total transfer happens.
75
1,
Case 2
3.4.3
A
0
A
I
I
0.20
A'
A
0.15
0.1o
OA&
005s
I
-0.1-005V
-0.101-
-02
-
-
--
-
-
(b) Case 2-q*
(a) Case 2-x*
Tr
OAA
VM
A
012
0
5
10
15
20
t
(
1d
15
20
I
(d) Case 2-Tr
(c) Case 2-Energy*
Figure 3-5: Numerical results of model Case 2
In Case 2, compared with Base Case, ( is increased from 0.09 to 0.3 by increasing the
inductance. As we can see from Figure 3-5a, there is one dominant mode and another
mode is very weak, but compared with Base x* plot, Figure 3-3a, the second mode
is more pronounced. The second mode is clearly visible in q* plot, Figure 3-5b, and
there exists an obvious transition from one mode to another. Besides, the increase of
6 helps enlarge the amplitude of q*from peak to bottom.
Figure 3-5c shows that there is a small coupling between the mechanical system
and the electrical system, and that the energy profiles are no longer sinusoidal. This
implies that nonlinear effects are more pronounced. But compared with the energy
plot in Case 1, Figure 3-4c, the coupling is much weaker. Figure 3-5d shows that
76
the peak of transfer factor increases to some degree compared with that in Base. In
addition, the moderate fluctuation of the peak indicates that the existence of both
modes are more obvious.
However, compared with the results of Case 2, these improvements for Base are
not satisfactory. This can be explained by comparing Plots of (1 - a2)-e and Plots
of (1 - a2)-( in Chapter 2. As we can see from Plot (1 - a2)-, when is small,
increasing e from a small value, say 0.1 to 0.5, can dramatically increase the transfer
factor. However, in Plots of (1 - a2)-E, when c is small, the gradual increase of ( from
a tiny value, say 0.09, to 0.3, can only help increase the transfer factor slightly. Thus,
increasing the value of Eis a better way to improve the system's performance.
3.4.4
Case 3
In Case 3, the value of -yis raised by increasing the initial velocity of the mass, compared with that in Base. As we can see from Figure 3-6a, similar to x* plot in Case 2,
this plot has small fluctuations, with one dominant mode and one secondary mode.
Figure 3-6b shows evidently that two modes coexist in the response of q*. In addition, the perturbation in q* plot indicates the system's weak nonlinearity. Besides,
the amplitude of q* increases significantly due to the increase of initial energy input.
Figure 3-6c shows that the envelop of either energy is non-sinusoidal, since the nonlinear effects are more pronounced and these two plots tend to overlap. This indicates
that the coupling of the system is not strong enough. Figure 3-6d shows that despite
additional nonlinearity, the transfer factor is still very tiny.
In general, the increase of -y introduces moderate nonlinearity and small coupling
to the system. However, the transfer factor does not increase much. This method is
not recommended.
77
a.
0.
q*
A1A A
A
A
i
LIiXiAUILXI7IULL
I
I'I I I II
OA
I
'I
I
'1
v V
IV
-0.-
(a) Case 3-x*
-
(b) Case 3-q*
r
OA:
OJI
0.I
0
5
10
15
I
20
2151
(c) Case 3-Energy*
(d) Case 3-Tr
Figure 3-6: Numerical results
3.4.5
of model Case 3
Case 4
In Case 4, the value of e is reduced by decreasing the total charge stored in those
capacitors, compared with Base Case. As we can see from Figure 3-7a, the response
of x* is almost harmonic. However, a highly nonlinear behavior is observed as seen
from Figure 3-7b. The profile of q*experiences highly non-harmonic oscillations due
to nonlinear effects.
As can be seen from Figure 3-7c, the two energy plots are far away from each other.
This indicates that the system is highly decoupled and energy transfer is very small.
Figure 3-7d shows that the transfer factor is far less than that in Base Case, although
due to nonlinearity, it has an increasing trend of the transfer factor.
78
q*
03
0.2
02
0.1
I
-0.
-0.3-
0.
-03
(b) Case 4-q*
(a) Case 4-x*
Ewugye
I
I
Mr
0.07
OM&
0.02
OA+
-
OA2
0.0
0
5
10
15
I1L!.ALIi.
20
1)
I
is
____
I
23
(d) Case 4-Tr
(c) Case 4-Energy*
Figure 3-7: Numerical results of model Case 4
In general, the decrease of e introduces high nonlinearity to the response of q*and decoupling to the system. It dramatically decreases the transfer factor at first. We have
observed rapid accumulation of numerical errors in this case leading to significant
non-conservation of energy on long time-scales. Hence, the results after 20 seconds,
which are not shown here, are not satisfactory. But as long as the first 20 seconds
results are considered, this is not a good way to improve the system's performance.
3.4.6
Case 5
In Case 5, the value of ( is reduced from 0.09 to 0.05, by decreasing the inductance,
compared with Base Case. As one can see from Figure 3-8a, the amplitude of x* is
sinusoidal. This indicates that nonlinearity affects the response of x*. Figure 3-8b
79
q*
OA I
-----------
I
(b) Case 5-q*
(a) Case 5-x*
Tr
EnWge
(d) Case 5-Tr
(c) Case 5-Energy*
Figure 3-8: Numerical results of model Case 5
shows clearly the mode transits every time when x* crosses zero. The plot of q* also
indicates that the system is highly nonlinear.
In figure 3-8c, the envelop of both plots are non-sinusoidal, which indicates the high
nonlinearity of the system. In addition, at certain moments of time, two plots overlap.
This implies that the system is moderately coupled. As it is shown in Figure 3-8d,
at some point of time, the peak of transfer factor increases significantly.
In general, the decrease of ( introduces high nonlinearity and moderate coupling
to the system, and its improvement for the system's performance of transferring energy is attractive.
80
3.5
Forced vibration simulation
Numerical studies in Section 4.5 imply that the model in Case 5 is highly nonlinear
and has high transfer factor without the cost of large initial charge. Thus, in this section, Case 5's model is tested with external vibration input and resistor in a numerical
approach. The model is shown in Figure 3-9. The external vibration is applied to the
moving mass in a form of sinusoidal time dependent position. The system is excited
via moving the mobile electrode described by the following relation
(3.15)
x = Asin(Qt)
where A is the amplitude of the vibration source, Q is the angular frequency of the
vibration source. The frequency of the vibration source is 0.5wm and the amplitude
is
jLo.
The resistance is set such that the damping factor of the electric circuit is
0.1. Other parameters are the same as those in Case 5.
4
I
g~d1
Figure 3-9: Model with external input and resistor
The output energy, that is the energy dissipated in the resistor, is defined dimensionless as
81
Ri
2
(3.16)
EUt ==
kLo02W
We
The numerical results are shown in Figure 3-10.
q*
0
-
--
-
----
- --
I
(b) Plot of q*
(a) Plot of x*
0A0
Tr
0A0
OM7O
004
01
0A010.00
&
0
5
10
"A
0
15
02
20
t
(c) Plot of energy dissipated in the resistor
(d) Plot of transfer factor T,
Figure 3-10: Numerical results of model with external vibration input
The plot of q*, Figure 3-10b, shows that sudden application of external vibration
source to the system, similar as an impulse effect, excites a high charge flow in the
circuit. After a small period of time, the profile of q* stabilizes at a combination of
two mode shapes.
The impulse effect is more obvious in the plot of output energy East, Figure 3-10c.
In this figure, the peak of the output energy in the first 10 seconds is about four
times higher in comparison to that of the steady state. As shown in Figure 3-10d,
the transfer factor appears harmonic despite its high value due to the impulse effect.
82
In general, although the energy transfer is not very high, the system still have a
reasonable rate of continuous energy conversion. More studies should be done in the
future, in order to optimize this forced energy harvesting system's design.
83
84
Chapter 4
Conclusions
In this thesis an electro-static energy harvesting system was proposed and studied.
The system was analyzed using a combination of analytical and numerical approach.
As shown in Figure 2-5, the system consists of the electrical and the mechanical subsystems. In the electrical subsystem, the primary energy conversion element is the
in-plane overlap varying capacitor. In addition, a constant capacitor is attached in
series with the variable one in order to trap the charge. An inductor is connected with
the capacitors to induce the vibrations at the electrical frequency. The mechanical
subsystem is a mass-spring system, where the moving mass is the variable capacitor's
moving plate.
In the analytical studies, the system was "linearized" around the equilibrium point
x = 0, q= - Q. After linearization, the system is still nonlinear because of the nonanalytic term Ix| in the potential energy. Therefore, this system can be decomposed
into two fully linear systems in regions x > 0 and x < 0. Both systems can be analyzed using normal mode analysis approach. It was found that systems in regions
x > 0 and x < 0 have same frequencies and similar but different mode shapes differing only in the sign of x. When x crosses zero, the energy is transferred between the
normal modes. The transfer matrix T was defined via a relation between ct, C and
c--, c2 - amplitudes of both of the mode shapes in positive and negative x regions.
By analyzing the energy distribution in each mode, we found that although energy
85
stored in each mode remains constant while the system is away from x
ergy is redistributed among two modes at the x
=
-
0, the en-
0 crossing event. This effect was
called energy redistribution. The energy transferred during this process was found
to be dependent on two parameters, e = 4cQo0k, and
(1
-
a2)
Etransfer
Etotat
can reach its peak 1, when il+E
=
(
(
. The transfer factor
(. Plots of transfer factor with
respect to ( and E are shown in Figure 4-la and Figure 4-1b, respectively.
1-a2
-2
(a) Transfer factor (1 - a 2 ) as a plot
of
(b) Transfer factor (1- a 2 ) as a plot
ofe~
Figure 4-1: Transfer factor (1 - a 2 ) plots
In the numerical analysis, the model was created using Modelica language. Besides
those two factors discussed in the analytical studies, another factor
=
v0
was
studied to understand how the system performs away from the quasi-linear regime.
Analysis of six different cases was carried out by changing one factor each time to
find an optimal design. By comparing energy plots and transfer factor T, plots of
different cases, we found that increasing the factor e by introducing more charge in
capacitors could lead to a significant increase of the transfer factor. What's more,
decreasing the factor Eby reducing the total charge stored in capacitors can lead to a
high increase of the transfer factor. The second method is the most promising, since
it allows operation with small electric charges, but also can increase energy transfer
to a satisfactory level.
Future effort should be put on the analysis of this electrostatic energy harvesting
system with external vibration source and resistors. Naturally the results of this the86
sis should be extended with thorough analysis of the energy transfer in the presence
of external forcing and dissipative elements in electronic subsystem. This could be
accomplished by introducing resistors to the circuit and external vibration or forcing
to the moving mass.
87
88
Appendix A
Modelica script
A. 1
Constant capacitor
model capac
extends Modelica.Electrical.Analog.Interfaces.OnePort;
constant Real w=0.0015 "Width of the plate";
constant Real 1=0.02 "Length of the plate";
constant Real d=0.00004 "Distance between two plates";
constant Real epsilon=0.00000000000885 "Permittivity";
Modelica.SIunits.Charge Q(start=0.000000001);
Modelica.SIunits.Capacitance C(start=l);
equation
C=epsilon*w*l/d;
Q=v*C;
i=der(Q);
end capac;
A.2
Variable capacitor
model FlexCapac-parallel
parameter Real w(start=0.0015)
"Width of the plate";
89
parameter Real d(start=0.00004)
parameter Real L_0(start=0.02)
"Distance between two plates";
"Initial length of the plate";
extends Modelica.Electrical.Analog.Interfaces.OnePort;
extends Modelica.Mechanics.Translational.Interfaces.PartialCompliant;
Modelica.SIunits.Charge Q(start=-0.000000001);
Modelica.SIunits.Capacitance C(start=1);
equation
C=Modelica.Constants.epsilonO*w*(L_0 - abs(s-rel))/d;
v=Q/C;
i=der(Q);
f=sign(s-rel)*Q*Q*d/(2*Modelica.Constants.epsilonO*w*(L_0 - abs(s-rel))^2);
end FlexCapac-parallel;
A.3
Model used to test energy conservation
model testConservation
Modelica.Electrical.Analog.Basic.Inductor inductorl(L=7700000000.0);
Modelica.Mechanics.Translational.Components.Spring springl(s-relO=O, c=0.0036);
Modelica.Mechanics.Translational.Components.Mass
massl(m=0.001,
Modelica.Mechanics.Translational.Components.Fixed fixed1;
Modelica.Mechanics.Translational.Components.Fixed fixed2;
Modelica.Electrical.Analog.Basic.Ground ground1;
FlexCapac-parallel flexCapac-parallell;
capac capac1;
equation
connect(flexCapac-parallell.flange-a,massl.flange-a);
connect(flexCapac-parallell.flange-b,fixed2.flange);
connect(flexCapac-parallell.n,groundl.p);
connect(flexCapac-parallell.p,capacl.n);
connect(capac1.p,inductor1.p);
90
s.start=0.001);
connect(inductorl.n,groundl.p);
connect(mass1.flangeb,spring1.flange-a);
connect(spring.flangeb,fixedl.flange);
end testConservation;
A.4
Model Base
model Base
Modelica.Electrical.Analog.Basic.Inductor;
Modelica.Mechanics.Translational.Components.Spring spring1(s-re10=0, c=0.0036);
Modelica.Mechanics.Translational.Components.Mass massl(m=0.001, v.start=0.012);
Modelica.Mechanics.Translational.Components.Fixed fixed1;
Modelica.Mechanics.Translational.Components.Fixed fixed2;
Modelica.Electrical.Analog.Basic.Ground groundi;
FlexCapac-parallel flexCapac-parallell(Q.start=-0.00000000098);
capac capac1(Q.start=0.00000000098);
equation
connect(flexCapac-parallell.flange-a,massl.flange-a);
connect(flexCapac-parallel.flangeb,fixed2.flange);
connect(flexCapac-parallell.n,groundl.p);
connect(flexCapac-parallell.p,capaci.n);
connect(capac1.p,inductor1.p);
connect(inductor1.n,ground1.p);
connect(mass1.flangeb,spring1.flange-a);
connect(spring1.flangeb,fixed1.flange);
end Base;
A.5
Case 1
model Casel
91
Modelica.Electrical.Analog.Basic.Inductor inductor1(L=7500000000.0);
Modelica.Mechanics.Translational.Components.Spring springl(s-relO=0,
c=0.0036);
Modelica.Mechanics.Translational.Components.Mass massl(m=0.001, v.start=0.012);
Modelica.Mechanics.Translational.Components.Fixed fixed1;
Modelica.Mechanics.Translational.Components.Fixed fixed2;
Modelica.Electrical.Analog.Basic.Ground ground1;
FlexCapac-parallel flexCapac-parallell(Q.start=-0.0000000022);
capac capac1(Q.start=0.0000000022);
equation
connect(flexCapac-parallell.flange-a,massl.flange-a);
connect(flexCapac-parallell.flange-b,fixed2.flange);
connect(flexCapac-parallell.n,groundl.p);
connect(flexCapac-parallell.p,capac1.n);
connect(capac1.p,inductor1.p);
connect(inductorl.n,groundl.p);
connect(mass1.flange-b,spring1.flange-a);
connect(springl.flangeb,fixedl.flange);
end Casel;
A.6
Case 2
model Case2
Modelica.Electrical.Analog.Basic.Inductor inductorl(L=25000000000.0);
Modelica.Mechanics.Translational.Components.Spring springl(s-relO=0, c=0.0036);
Modelica.Mechanics.Translational.Components.Mass massl(m=0.001, v.start=0.012);
Modelica.Mechanics.Translational.Components.Fixed fixed1;
Modelica.Mechanics.Translational.Components.Fixed fixed2;
Modelica.Electrical.Analog.Basic.Ground ground1;
FlexCapac-parallel flexCapac-parallell(Q.start=-0.00000000098);
capac capac1(Q.start=0.00000000098);
92
equation
connect(flexCapac-parallell.flange-a,massl.flange-a);
connect(flexCapac-parallell.flange-b,fixed2.flange);
connect(flexCapac-parallell.n,groundl.p);
connect(flexCapac-parallell.p,capacl.n);
connect(capacl.p,inductorl.p);
connect(inductor1.n,groundl.p);
connect(mass1.flangeb,spring1.flange-a);
connect(springl.flange-b,fixedl.flange);
end Case2;
A.7
Case 3
model Case3
Modelica.Electrical.Analog.Basic.Inductor inductorl(L=7500000000.0);
Modelica.Mechanics.Translational.Components.Spring springl(s-re10=0,
c=0.0036);
Modelica.Mechanics.Translational.Components.Mass massl(m=0.001, v.start=0.034);
Modelica.Mechanics.Translational.Components.Fixed fixedi;
Modelica.Mechanics.Translational.Components.Fixed fixed2;
Modelica.Electrical.Analog.Basic.Ground ground1;
FlexCapac-parallel flexCapac-parallell(Q.start=-0.00000000098);
capac capacl(Q.start=0.00000000098);
equation
connect(flexCapac.parallell.flange-a,massl.flange-a);
connect(flexCapac-parallell.flange-b,fixed2.flange);
connect(flexCapac-parallell.n,ground1.p);
connect(flexCapac-parallell.p,capac1.n);
connect(capacl.p,inductorl.p);
connect(inductorl.n,groundl.p);
connect(mass1.flange-b,spring1.flange-a);
93
connect(spring1.flangeb,fixed1.flange);
end Case3;
A.8
Case 4
model Case4
Modelica.Electrical.Analog.Basic.Inductor inductorl(L=7500000000.0);
Modelica.Mechanics.Translational.Components.Spring springl(s-relO=0, c=0.0036);
Modelica.Mechanics.Translational.Components.Mass mass1(m=0.001, v.start=0.012);
Modelica.Mechanics.Translational.Components.Fixed fixed1;
Modelica.Mechanics.Translational.Components.Fixed fixed2;
Modelica.Electrical.Analog.Basic.Ground ground1;
FlexCapac-parallel flexCapac-parallell(Q.start=-0.00000000031);
capac capac1(Q.start=0.00000000031);
equation
connect(flexCapac-parallell.flange-a,massl.flange-a);
connect(flexCapac-parallell.flange-b,fixed2.flange);
connect(flexCapac-parallell.n,groundl.p);
connect(flexCapac-parallell.p,capaci.n);
connect(capac1.p,inductor1.p);
connect(inductorl.n,groundl.p);
connect(mass1.flangeb,spring1.flange-a);
connect(springl.flange-b,fixedl.flange);
end Case4;
A.9
Case 5
model Case5
Modelica.Electrical.Analog.Basic.Inductor inductorl(L=4200000000.0);
Modelica.Mechanics.Translational.Components.Spring springl(s-relO=0, c=0.0036);
94
Modelica.Mechanics.Translational.Components.Mass massl(m=0.001, v.start=0.012);
Modelica.Mechanics.Translational.Components.Fixed fixed1;
Modelica.Mechanics.Translational.Components.Fixed fixed2;
Modelica.Electrical.Analog.Basic.Ground ground1;
FlexCapac-parallel flexCapac-parallell(Q.start=-0.00000000097);
capac capac1(Q.start=0.00000000097);
equation
connect(flexCapac-parallell.flange-a,massl.flange-a);
connect(flexCapac-parallel1.flange-b,fixed2.flange);
connect(flexCapac-parallell.n,groundl.p);
connect (flexCapac-parallell.p,capacl.n);
connect(capac1.p,inductor1.p);
connect(inductorl.n,groundl.p);
connect(mass1.flange-b,spring1.flange-a);
connect(springl.flange-b,fixedl.flange);
end Case5;
95
96
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