Dynamic analysis of an electrostatic energy harvesting system

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

OF TECHNOLOGY

at the

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0 8 2013

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

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LIBRARIES

June 2013

©

Massachusetts Institute of Technology 2013. All rights reserved.

A u th o r ...

...

Department of Civil and Environmental Engineering

May 23, 2013

Certified by ...

....

....

Konstantin Turitsyn

Esther and Harold E. Edgerton Assistant Professor

Thesis Supervisor

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

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 en-ergy harvesting from low frequency vibrations. Several recent studies have indicated that introduction of nonlinearity can significantly improve the efficiency of such sys-tems. 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 trans-fer 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

Acknowledgments

I would like to extend my gratitude to the many people who helped to bring this re-search 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 en-couragement 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.

Contents

2 Physical model and linearized analysis 41 2.1 Physical model . . . . 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

2.2 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

3 Numerical analysis

3.1 Modelica language for modeling

3.2 Modelica model . . . . 3.2.1 The constant capacitor

3.2.2 The variable capacitor

3.3 Test the Modelica model...

3.4 Numerical studies of different cases 3.4.1 Base . . . .. 3.4.2 Case 1 . . . .. 3.4.3 Case 2 . . . .. 3.4.4 Case 3 . . . . 3.4.5 Case 4 . . . . 3.4.6 Case 5 . . . . 67 . . . . 67 without external . . . . 68 . . . . 68 . . . . 69 . . . . 70

input and resistor 71 . . . . 73 . . . . 74 . . . . 76 . . . . 77 . . . . 78 . . . . 79 3.5 Forced vibration simulation

4 Conclusions A Modelica script

A.1 Constant capacitor . . . . A.2 Variable capacitor . . . . A.3 Model used to test energy conservation

A.4 Model Base . . . .

A.5 Case 1 . . . . A.6 Case 2 . . . . A.7 Case 3 . . . . A.8 Case 4 . . . . A.9 Case 5 . . . . 81 85 89 89 89 90 91 91 92 93 94 94

List of Figures

1-1 Typical schematics of electromagnetic generators [4] . . . . 17

1-2 (Upper left) isometric, (upper right) side, and (lower) schematic views [17] . . . .. 17

1-3 Schematic design by Rajeevan and Anantha [1] . . . . 18

1-4 Schematic of Prez-Rodrguez, Serre, Fondevilla, Cereceda, and Morante's design [14] . . . . 19

1-5 Generator by Williams, Shearwood, Harradine, Mellor, Birch and Yates [2 3] . . . .. 19

1-6 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-abe [24] . . . .. . . . . 24

1-12 Two approaches to harvest piezoelectric energy in shoes [18] . . . . . 25

1-13 Curved PZT unimorph excited in 3-1 mode [25] . . . . 26

1-14 Schematic of PZT model [7] . . . . 26

1-15 Three types of electrostatic generators [4] . . . . 28

1-16 Controller architecture [9] . . . . 28

1-17 Schematic drawing of an electrostatic transducer [9] . . . . 29

1-19 Schematic design of electrostatic generator with electret [2] . . . .

1-20 Schematic design of an electret based in-plane overlap electrostatic

1-21 1-22

1-23

1-24

generator [19] . . . . Design by Miyazaki, Tanaka and Ono [11] . . . .

Design by Tashiro and Kabei [20] . . . . Design by Meninger, Mur-Miranda and Amirtharajah [10] Schematic design by Cottone, Vocca and Gammaitoni [6]

1-25 Envelope modulation 6(t) [22] . . . . 1-26 Three types of nonlinear energy sinks [16] . .

2-1 Configuration of the variable capacitor . . . .

2-2 Charge trapping configuration . . . .

2-3 Configuration of mechanical system . . . .

2-4 Configuration of electrical subsystem . . . . .

2-5 Configuration of the energy harvesting system 2-6 Electrical energy generation . . . . 2-7 Analytical analysis model . . . . 2-8 Plots of 1 - a2- . . . .

2-9 Plots of (1 - a2)-E . . . .

3-1 Energy Plots . . . . 3-2 Numerical analysis model . . . . 3-3 Numerical results of model Base . . . .

3-4 Numerical results of model Case 1 . . . .

3-5 Numerical results of model Case 2 . . . . 3-6 Numerical results of model Case 3 . . . . 3-7 Numerical results of model Case 4 . . . . 3-8 Numerical results of model Case 5 . . . . 3-9 Model with external input and resistor . . . .

3-10 Numerical results of model with external vibration input

31 . . . . 32 . . . . 33 . . . . 33 . . . . 34 . . . . 36 . . . . 37 . . . . 38 . . . . 42 . . . . 43 . . . . 44 . . . . 45 . . . . 46 . . . . 47 . . . . 47 . . . . 64 . . . . 64 82 . . . . 7 0 . . . . 7 1 . . . . 7 4 . . . . 7 5 . . . . 7 6 . . . . 7 8 . . . . 7 9 . . . . 8 0 . . . . 8 1

List of Tables

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. Even-tually 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. Vi-brations 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

lower the cost.

This thesis is structured as follow. In Chapter 1, different types of energy gener-ators 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 elec-tromagnetic, piezoelectric, or electrostatic technologies that converts mechanical ergy to electrical one. Electric circuit that transfers or makes use of electrical en-ergy. 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 gen-erated 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 microelectromechan-ical system based electromagnetic vibration-to-electrmicroelectromechan-ical power generator has been

B B Din 00 Motgon a usMain 41) CON Wo -0 N "

Figure 1-1: Typical schematics of electromagnetic generators [4]

MIU ~r Amaent e wswsen my~ ein 14b PwineCus Dw"wqm

Rwg Mal I WS 4 NI Idbago, u

capi

Parw suedtwe Re fthq cyupcvie

Woontamu byli nuif

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 ex-ternal 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 2.5mm3, consisting of 20 can-tilevers 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.

---sprng,k VCM loomutrp SOOT swo

mass, M sw

* ow i r e c ,t I --- -- ... .. ..---- .---

C-prmanent Criica Pah S

magneto B

(b) Detailed Block Diagram of Self-powered (a) Generator Mechanical Schematics 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 pow-ered 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 envi-ronment.

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 1cm2, and the cross-section shape is nearly circular.

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 mate-rial 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.

pinr 0 A3os

Figure 1-5: Generator by Williams, Shearwood, Harradine, Mellor, Birch and Yates

[23]

Williams, Shearwood, Harradine, Mellor, Birch and Yates' [23] design of the proto-type micro-electromagnetic inertial generator is shown in Figure 1-5. The generator

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 gen-erator.

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 separa-tion 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

gener-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.

Permanent man ico-cantilevers

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 can-tilever 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 four-magnet arrangement. The coil is designed to move laterally relative to the four-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.

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

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 Free (Ms6) FallI Heightlh

f ,

11A

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

steel _perimentpipe

Figue 111:Schmatc drwin ofexprimnt b Xu Akyam, Nnak Free Waan-storage oscilloscope

Insulator

R Pt

Si3 N4R

PC _{Steel PC Al203}

(a) Schematic Drawing of constant rate load- (b) Schematic Drawing of impact loading

ex-ing experiment periment

Figure 1-11: Schematic drawing of experiment by Xu, Akiyama, Nonaka and Watan-abe [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

ce-ramics 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.

PZT PZT unimorph

dimorph _PVDF

Alo stave

MWtI 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 genera-tor in shoe sole. Two methods have been used, explained in Figure 1-12, of piezoelec-trically 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 10~7A. After trying different methods, they finally developed an offline, forward-switching 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 effi-cient 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

Figure 1-13: Curved PZT unimorph excited in 3-1 mode [25]

the optimal design parameters. Nine samples have been fabricated to test their per-formance 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 vi-brations. It was shown that this kind of PZT device could be used to harvest kinetic energy, such as human walking.

Fin mechanical port X1 M Fr n R e electrica b Fr nt t port PZT stack dynamice

Figure 1-14: Schematic of PZT model

[71

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 fre-quency 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 prop-erty is the vertical structure (homogeneous bimorph, and symmetric or asymmetric heterogeneous bimorph). The other property is the support: cantilever with a tri-angular 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 dif-ferent 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

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

Figure 1-15:

(b) In-plane gap closing

(c) Out-of-plane gap clos-ing

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 voltage-constrained cycles provide more energy than those in charge-voltage-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

mechan-7pmmetd

Anchor

11 1jU

Devioe Wsa

Howdl Wahr

SteadayComb SUlcon C~] Oxdde X Aluminum

(a) Transducer Plan View (b) Transducer Cross-sectional 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 perfor-mance 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, conse-quently, 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

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 require-ments: 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 ma-terials 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/m2.

The experimental results show that 6mW power could be generated when the gen-erator 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.

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

in-tegrate 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 exper-iment has shown that a vibration source of 2.25m/s2 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.

MMEU

Adhbsive 5 Movable E de

bondina Electrode Glass

Figure 1-20: Schematic design of an electret based in-plane overlap electrostatic gen-erator [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 stud-ied 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 trans-porter 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

PMOS

a[J=z

IFL

(a) Energy harvesting sys-tem

Pr

(c) Externally powered (b) Variable capacitor 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

VC Resomaw Peronl fix Dat Rcrdtoeie

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. Con-troller 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 sys-tem. It has been found that the output energy could be easily increased by designing the system for higher voltage operation.

Am%

=silicon

(a) MEMS device plan view

Device Wafer

Hanle WSlW

..1Oxide =m liriw

(b) MEMS device side 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

10

fi=

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.

wim &V*

CM*

Both wind load and seismic load can excite vibrations of a bridge. There are sev-eral 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 cal-culated using equation

C

fi=-Hz

L

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 con-crete 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 satisfac-tory 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.

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 oscilla-tions, 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 I X 1

xyzstag

Figure 1-24: Schematic design by Cottone, Vocca and Gammaitoni [6]

Cottone, Vocca and Gammaitoni [6] proposed a new method, exploiting the dynami-cal features of stochastic nonlinear oscillators. Figure 1-24 shows the schematic design

of an energy harvesting system, consisting of a piezoelectric inverted pendulum, de-signed 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 conve-niently 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 reso-nant 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 max-imum 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

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 spring Linear sprng ~ *inear

Limear spring Linear aper Lin Iear damper s--- damper

(a) (b) (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 non-linear energy sinks (NESs). Three types of NESs, shown in Figure 1-26, have been

designed and their effects on the stiffness and damping properties of the linear struc-ture have been studied via (local) instantaneous and (global) weighted-averaged ef-fective 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 struc-tural modes, redistributing nonlinear energy between strucstruc-tural 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.

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 de-scribe their dynamics. We dede-scribe the analytical studies aimed to understand how the system parameters affect the response of the system.

2.1

Physical model

2.1.1 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 be-tween 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,

Id

(b) Variable capacitor (c) Variable capacitor mov-(a) Variable capacitor ing 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, result-ing 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 ca-pacitor is given by

C _o(Lo - |xi)w (2.1)

d

where co is the dielectric constant(8.854 x 10-1 2F/m), LO is the length of the fixed electrode, x is the relative displacement between those two electrodes of the capac-itor, 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

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.

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

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 con-stant 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

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.

massi1 spring1

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

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.

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 cir-cuit 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

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 pow-ering 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.

-v v

External *--I vibration

source

~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 sub-system.

This electromechanical system is a two degree-of-freedom system, and x and q are used as the generalized coordinates.

The magnetic coenergy W*, accounting for all the inductance in the system, is ex-pressed as

W*, = Lidi = Li2 = L412 (2.6)

The electrical energy We, accounting for all the capacitance in the system, is ex-pressed as

2 2

We = + q2(2.7)

C(x) 2Co

where C (x) = _d -o(Lo-') and q2 = q + Qo.

The Lagrangian is L(x,tjq1,j1 ) = T-V+W -We =m c2 _ kx+ _-L1 2 2 2 C (x) (q + QO) 2 2Co (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 (t ) d ((L dt 82 aL -q = qi DL x -(2.9) (2.10)

where E and -E, are generalized external forces. --( )L = Ld -( j )H

z

1

Qo

aL

d

ax 2cow (Lo - xIl) 2

Therefore, substitution of Equation 2.8 into Equations 2.9 and 2.10 gives equations of motion as below

Li + ( + -)qi + - = 0 (2.14)

mz5 + kx + q d sign(x) -0

1 2cow (Lo-IXI)2

-The existence of nonlinear terms d1 and q si_" 2 in equations of motion

eow(Lo-IxI) 2cow (Lo-IXI)2 i qain fmto

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 0 (2.15)

z

Thus, the equilibrium position is qi = - x - 0. In other words, the system will oscillate around qi = , x = 0. Any initial conditions differing from this

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, Equa-tion 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 redistribu-tion 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 equa-tions 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

L4 +( _ + -)(q- )+G =0 (2.18)

mz+kx + (q-)2 d sign(x) 0

2

eow (Lo-\xi)2

Assuming that |xi

<

<

Qo,

d _{1 |x|}

~(Ld - + _(2.19)

(q - 9 )2_{d sign(x)} 2eow (Lo - |x\)2 Qod Q20d sign(x)jxI 2eowgL20 4eowL20 Lo Q0 sign(x) Q2 0 (2.20) 2Co Lo 4C0L2(

Thus, substitution of Equations 2.19 and 2.20 into Equation 2.18 gives linearized

equations of motion:

Lij+ g- 2C0Lox =0 (2.21)

m + kx - q + 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 (2.22)

where <D is the response column vector

<P = 4(2.23)

M is the mass matrix

L 0 0 m J

(2.24) K is the stiffness matrix

2 K - o T Q0

L

I

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

T2C₀L0

= 0 (2.27)

T 2COLO _2C~o k + _4CoL4 20 2 - mw2

o m

which leads to the quartic equation

2 - L2) (k + 0 - mw2)

CO 4C02L20

Q 2_{0 0}

4CoL2 0

with two solutions wi and w2.

Setting We = , Wm = k and e = 4c 2 and substituting into Equation

2.28 give

w2 (1- _We2

2

~ 2 _Wm + ) ₂ =0 (2.29)

Q02

e 4 02 can be rewritten as E 4C₀ 2. We can easily tell from this equation

6-4CoLo k can be rertenaL

that E is a fraction of the electrical energy over twice of the spring potential energy of the system.

To simplify Equation 2.29, defining q = (g)2, and ( = (''a)2 _{and substituting them}

into Equation 2.29 gives the governing equation

(1 - )(1 + 6 - n) - = 0 (2.30)

The solutions of this equation are

2 2

(2.31)

According to the matrix form of equations of motion,

(K - MW2_)<=O

0,

substitution of mass and stiffness matrices gives

[

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.

Mode 1: q, Q0 x1 (2 LCow12)Lo Q0

2~1

(2.34)

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.

<b+ = <k+Ct cos(wit + #1) + <I2c2jcos(w2t +

#2)

where c+, c+,

#1,

#2

#

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

1