直流靜電式微型振動發電機之分析、設計與製作

國 立 交 通 大 學

電控工程研究所

碩士論文

直流靜電式微型振動發電機之分析、設計與製作

Analysis, Design, and Fabrication of a DC

Electrostatic Vibration-to-Electric Micro Power

Generator

研究生 ：張經富

指導教授：邱一 博士

直流靜電式微型振動發電機之分析、設計與製作

Analysis, Design, and Fabrication of a DC

Electrostatic Vibration-to-Electric Micro Power

Generator

研 究 生:張經富 Student: Chin Fu Chang

指導教授:邱一 Advisor: Yi Chiu

國立交通大學

電控工程研究所

碩士論文

A Thesis

Submitted to Department of Electrical Control Engineering

College of Electrical Engineering

National Chiao Tung University

In Partial Fulfillment of the Requirement

For the Degree of

Master of Science

In

Electrical Control Engineering

July 2010

Hsinchu, Taiwan, R.O.C.

中文摘要

由於近年來在低功率 VLSI 以及 CMOS 技術的發展，積體電路的能量需求 已經降低至數十μW 的程度，由環境獲取能量來驅動低功率的遠端感測器成為可 能。微機電系統是一個整合機械元件與電子電路的平台，在全世界的發展日漸流 行。藉由使用微機電系統科技，具有整合電子元件的機械能量獲取器已具有提供 低功率元件能量的能力。 本論文中，我們提出了對於一直流式振動至電能轉換器的新穎設計與分析方 法。在一平方公分的晶片面積及40V 輸出電壓的限制下，由 3.6V 電壓以及 120Hz 振動源驅動，最佳化的輸出功率在有外加4 克質量塊以及無外加質量塊的情況下 分別為40.5 μW 以及 0.87 μW。 本元件是在 SOI 晶圓上利用深矽蝕刻技術製作。與先前的元件做比較，應 用低壓氣相沉積技術所沉積的氮化矽薄膜提升了梳齒結構的絕緣能力。在利用高 精準度陰影遮罩的定義下，元件接腳以及開關側向包覆改由金作為沉積材料，解 決了先前使用鋁所造成的氧化問題，同時也減少了漏電電阻的產生。利用激振器 所測量的共振頻率與設計值相符。元件的可變電容以及寄生電阻皆已量測，其可 變電容範圍為 94pF 至 485pF，寄生電阻由 103MΩ提升為 156MΩ。機械式接觸 開關經由測試已經確認其可正常工作。然而，由於不垂直的梳齒結構，能量轉換 的過程中只產生較低的電壓。輸出功率的測量仍然在進行中。

Abstract

Energy scavenging from ambient environment becomes feasible to power independent remote sensors or portable devices due to low power VLSI and CMOS technology development. The requirement of power consumption has been reduced to a few tens of microwatts. Micro-Electro-Mechanical System (MEMS) is a platform that integrates mechanical devices with electric circuit. The development of MEMS is increasingly popular in the world. By using MEMS technology, a mechanical energy harvester with electric component is capable of supplying those low power devices.

In this thesis, the novel design and modeling method as well as the fabrication and measurement of a DC capacitive vibration-to-electric energy converter are introduced. With the device area constraint of 1_{cm , bias by a 3.6V battery, output} 2

voltage limited to 40V, and operation frequency of 120Hz, the optimum output power is 40.5μW and 0.87 μW for devices with and without a 4 gram external mass attachment, respectively.

The device was fabricated in SOI wafers with deep silicon etching technology. Compared with previous devices, LPCVD nitride improved the isolation of fingers. Gold pads and switch lateral coverage defined by the high precision shadow mask had reduced the aluminum oxidation problem and reduced the leakage resistance. The resonant frequency measured by a shaker agreed with our design value. Variable capacitance of 94pF to 485pF was measured. The parasitic resistance was improved from 103MΩ to 156MΩ. The mechanical contact switch was test and workable. However, due to non vertical combs fingers, lower voltage was generated in the conversion process. The output power measurement is still in progress.

致謝

在經歷許多無法想像的磨練以及挑戰之後，我的碩士生涯終於步入尾聲。首 先最感謝的就是我的指導教授邱一老師。老師在研究上一絲不苟的態度實在令人 敬佩。老師在我的研究中也時常替我發現問題並提供有用的建議。在嚴格的指導 下，這三年中學習到的東西是當初大學畢業的我想像不到的。老師在最後幫忙修 改論文的付出，也使得我的論文能夠變得更加流暢明瞭。希望在未來進入社會後 能將這段時間所受的訓練學以致用，創造自己的價值。 感謝口試委員方維倫老師以及施錫富老師。老師在繁忙中還安排時間前來， 並且對我的論文提出非常有用的建議。感謝實驗室的學長曾繁果以及陳弘諳的經 驗傳承，使我的研究能夠順利銜接。至於其他學長黃煒智、張子麟、吳昌修、揚 昇儒，也提供了寶貴的研究經驗，雖然相處時間不長但還是要謝謝你們。 感謝一路上與我一起打拼的同學林健安，你在實驗以及製程上的經驗比我多 上不少，也時常跟你請教製程中發生的問題。在研究生涯中也時常互相打氣，在 苦悶的研究生活中增添不少動力。希望你將來能夠鴻圖大展。 還有要感謝我的大學同學。雖然你們許多人都已離開學校很久了，但是先前 的情誼能夠在我碩士生涯的前中期提供給我精神上很大的支持，非常謝謝你們。 另外要特別感謝我的好友范澤崴，即使身處軍中依然時常關心我的情況。 另外還要感謝在方維倫老師實驗室提供儀器幫忙做實驗數據的量測，使得我 的研究時間能夠減少許多。實驗室碩二學弟妹劉鴻智、陳政安、莊哲明、陳姿穎、 劉俊宏及碩一的學弟李彥傑、吳彥霆，感謝你們在這段日子裡的陪伴，使得實驗 室的氣氛能夠歡樂不少，我由衷的希望你們能夠研究順利並且早日畢業。 還有要感謝我的母親以及姐姐，能夠支持我讀完碩士學位，將來一定要更加 努力使你們過更好的生活。最後要感謝的就是親愛的素華，即使距離遙遠，仍在 我人生最低潮的時候陪伴我度過，如果沒遇見妳我想我的人生還是一樣的平淡。 至於一路上所有幫助過我的人，在此致上我最高的謝意。

Table of Content

中文摘要...i

Abstract……..……….………...…...…………..…...ii

致謝……..……….………...…...…………..…...iii

Table of Content ...iv

List of Figures...vii List of Tables...x Chapter 1 Introduction...1 1-1 literature review ...2 1-1-1 Light exposure...2 1-1-2 Thermalelectric effect ...4

1-1-3 Human body movement ...5

1-1-4 Wind ...7

1-1-5 Ambient vibration ...8

1-1-6 Summary of power sources ...9

1-2 Ambient vibration energy conversion...9

1-2-1 .Electromagnetic energy conversion ...10

1-2-2 Piezoelectric energy conversion ...12

1-2-3 Electrostatic energy conversion ...14

1-2-4 Comparision of vibrational energy conversion technologies...16

1-3 Thesis Objective and organization...17

Chapter 2 Principle and Design...20

2-1 Characteristics of vibration sources ...20

2-2 Operation principle ...22

2-3 Device design...24

2-3-1 Auxiliary battery supply ...25

2-3-2 Variable capacitor design ...26

2-3-3 Dynamic analysis...29

2-3-4 Static analysis...35

2-4 Optimizing process ...37

2-4-1 Conditions for nprmal oscillation discussion...40

2-4-2 Optimum design...42

2-6 Summary ...55

Chapter 3 Fabrication Process...56

3-1 Fabrication process flow...56

3-2 Process issues and solutions...67

3-2-1 Non-ideal effects in the ICP etching ...67

3-2-2 Silicon nitride deposition ...69

3-2-3 Metal deposition issued...71

3-3 Fabricated device ...76

3-4 Summary ...79

Chapter 4 Measurement and Results ...81

4-1 Mechanical measurement...81

4-1-1 MEMS Motion Analyzer measurement ...81

4-1-2 Vibration measurement by our shaker ...84

4-2 Static electrical measurement...86

4-2-1 Leakage resistance ...87

4-2-2 Static measurement of SW1...92

4-3-3 Static measurement of SW2...94

4-3 Dynamic electrical measurement...95

4-4 Summary ...97

Chapter 5 Summary and Future work...99

5-1 Summary ...99

5-2 Future work...99

Reference ...102

List of Figures

Figure 1-1 Photovoltaic energy conversion [10]. ...3

Figure 1-2 Thermoelectric energy converter [12]...5

Figure 1-3 Components of suspended-load backpack [16]...6

Figure 1-4 Inverted pendulum model of human walking [16]...6

Figure 1-5 MEMS air flow harvester, with 10 pence coin for scale [19].. ...8

Figure 1-6 Electromagnetic energy converter [21] ...10

Figure 1-7 Three types of permanent magnet generation technologies [22] ...11

Figure 1-8 (a) PMG-17 energy harvester from Perpetuum Co. [25] and (b) VEH-3 energy harvest from Ferro solutions[26]...12

Figure 1-9 A two-layer cantilever beam piezoelectric energy converter [28]...13

Figure 1-10 A typical piezoelectric generator [28].. ...14

Figure 1-11(a) Gap-closing and (b) overlap in-plane variable capacitors [21]...15

Figure 2-1 Vibration spectra by Roundy [9]. ...21

Figure 2-2 Measurement of air purifier vibration. ...21

Figure 2-3 Vibration spectrum of an air purifier...22

Figure 2-4 Operation circuit of the electrostatic energy converter. ...22

Figure 2-5 Variable capacitor schematic...23

Figure 2-6 Capacitor charging and capacitance change by vibration ...23

Figure 2-7 Charge transfer and discharge process [36] ...24

Figure 2-8 Lithium-ion rechargeable battery...25

Figure 2-9 Top view of the in-plane gap closing variable capacitor topology. ...26

Figure 2-10 A generalized layout design. ...27

Figure 2-11 Single cell schematic...28

Figure 2-13 Displacement of shuttle mass versus time. ...31

Figure 2-14 Time shift for t and T scale...32

Figure 2-15 Boundary conditions between z1 and z2...34

Figure 2-16 Fingers at maximum displacement...35

Figure 2-17 Optimization flow chart ...38

Figure 2-18 Amplitude and power vs. spring constant...39

Figure 2-19 Amplitude and power vs. Frequency. ...40

Figure 2-20 System energy versus driven frequency...41

Figure 2-21 Contour of output power and voltage with external mass attachment for various Qr...44

Figure 2-22 Contour of output power and voltage without external mass attachment for various Qr.. ...48

Figure 2-23 Layout view of the previous device. ...56

Figure 3-1 Fabrication process.flow on the SOI device ...64

Figure 3-2 Notching effect (a) schematic and (b) SEM micrograph [42]...68

Figure 3-3 (a) Notching occurred on the finger’s bottom, (b) improved result [2] ...69

Figure 3-4 Schematic of poor step coverage...70

Figure 3-5 Shadow mask. ...71

Figure 3-6 (a) Gold deposition on SW1 and anchors (b) Gold deposition on SW2. ...72

Figure 3-7 (a) Upper part of a gold coated test finger, (b) close-up view. ...73

Figure 3-8 Broken silicon oxide at the big through-wafer hole...73

Figure 3-9 (a) Cross section of test structure, (b) close-up view. ...74

Figure 3-10 EDS result at square in Fig 3.9(b)...74 Figure 3-11 (a) Cross section view of test structure, (b) close-up view 130 μm below

wavelike etching profile...75

Figure 3-12 EDS result at square in Fig 3.11(c).. ...75

Figure 3-13 (a) Fabricated die, (b) central hole for mounting the external mass. ...77

Figure 3-14 Close-up view of (a) fingers, (b) serpentine spring...77

Figure 3-15 (a) Close-up view of Switch 1, (b) deposited gold on silicon...77

Figure 3-16 (a) Top view of Switch 2, (b) close-up view of Switch 2...78

Figure 3-17 (a) Backside silicon removed under Switch 1 (b) close-up view...78

Figure 3-18 Device with silicon nitride coating. ...79

Figure 3-19 (a) Device with the external mass on PCB, (b) top view. ...79

Figure 4-1 MMA system...82

Figure 4-2 Frequency response with external mass...82

Figure 4-3 Frequency response without external mass...83

Figure 4-4 (a) Mechanical measurement schematic (b) measurement setup...85

Figure 4-5 (a) Device at rest (b) device at resonance of 118 Hz (c) input acceleration waveform.. ...86

Figure 4-6 Leakage resistance measurement circuit...87

Figure 4-7 Variable capacitor measurement circuit [43]...88

Figure 4-8 RC discharge time constant measurement for LPCVD deposition...89

Figure 4-9 Measured variable capacitor versus displacement. ...90

Figure 4-10 Cross-section of non-parallel capacitor...91

Figure 4-11 Cactual/Cideal versus oblique angle. ...92

Figure 4-12 4.12 SEM photo (a) Close-up of non-parallel capacitance (b) rough surface of finger sidewall [47] ...93

Figure 4-13 DC model operation with mechanical switches...93

Figure 4-15 Charge Voltage. ...94

Figure 4-16 SW2 with (a) no applied voltage, (b) 98 V applied voltage (pull-in). ...94

Figure 4-17 Voltage on the variable capacitor. ...95

Figure 4-18 Voltage on the variable capacitor and input acceleration...96

Figure 4-19 (a) SW2 overview, (b) SW2 schematic circuit...96

Figure 4-20 Schematic of Switch2 control method. ...97

List of Tables

Table 1-1 Comparison of power sources ...9

Table 1-2 Comparisons of three types of vibrational energy converter [21] ...17

Table 2-1 Fixed parameters of optimizing process. ...37

Table 2-2 Free parameters of optimizing process ...38

Table 2-3 System Energy at different instances and positions...41

Table 2-4 Variable capacitor design parameters (with external mass)...47

Table 2-5 Variable capacitor design parameters (without external mass)...53

Table 2-6 Design parameter for previous device. ...54

Table 3-1 Thin film property of LPCVD and PECVD nitride [46]. ...71

Table 3-2 Element percentage at upper and lower part of sidewall...76

Chapter 1 Introduction

Micro-Electro-Mechanical System (MEMS) is a multi purpose platform which integrates several different fields such as mechanical, electronic, control, optical, chemical, and biomedical technology. Through combining different types of subsystems on a common substrate, the most significant advantage is reduced cost due to miniaturization. Quality, performance, and durability may also be improved dramatically in many different cases. The concept of systems on chip (SOC) is made possible by the integration of these subsystems. This state-of-the-art technology has already been applied in numerous occasions. There are many common applications such as sensors and actuators, micro optical systems, biomedical systems, and aerospace and defense systems. The best known MEMS device is the accelerometer mounted on the hand controller of video gaming machine.

Due to the advance of manufacturing technology and the requirements from consumers, portable electronic devices have received increasing interests in recent years. However, conventional power storage devices have limited energy capacity, causing power supply a primary concern [1]. Generally, energy is stored in storage devices such as traditional batteries [2], micro-batteries [3], micro-fuel cells [4], ultra capacitors [5], micro heat engine [6], and radioactive materials [7]. High efficiency and low power loss are necessary to prolong the active time of these storage devices. Researchers have attempted to increase the energy density in those storage devices, but still, finite lifetime and high maintenance cost remain problems.

Thanks to the breakthrough of low power CMOS VLSI (Very Large Scale Integrated circuit) technology and the low duty cycle characteristics, it is feasible to

systems. These devices have reduced power consumption in the level of tens to hundreds of microwatts [8]. Scavenging ambient energy from the environment to power these sensor nodes become a possible method. One can design a self-sustainable or self-renewable energy device by scavenging the environmental energy to supply part of or all of the consumed energy.

1.1 Literature review

Different technologies, such as light exposure, thermal gradients, human power, air flow, and vibration [9], can be used to scavenge or harvest energy from the environment. The environment is a sustainable energy supply compared with the common storage devices like batteries or fuel cells. Various approaches to convert energy from the environment to electrical energy to drive low power electronics are reviewed in this chapter. Due to the inexhaustible nature of scavenged energy, the performance of the energy devices is characterized by their power density, instead of energy density used for traditional storage devices.

1.1.1 Light exposure

Light exposure is a popular and mature method to scavenge energy. Solar or photovoltaic cells are the leading technology to convert solar energy directly to electricity with high efficiency. Solar cells can be manufactured by IC-compatible technologies with high quality, therefore causing its popularity. Such devices can supply low cost and pollution free energy.

Photovoltaic cells function by the photovoltaic effect [10]. When the cell is exposed to light, a light-induced voltage is generated. The photons of the incident solar radiation excite the electrons in the semiconductor, allowing the electrons to

move freely and thus cause an electric current flow through a load. The operation is shown in Fig. 1.1. For single crystal silicon, the device has efficiency ranging from 12% to 25%. The thin film polysilicon and amorphous silicon cost less than single crystal silicon cells but have lower efficiency [11].

In total, photovoltaic energy conversion can provide sufficient power and with a mature IC-compatible technology. However, the output power of photovoltaic devices depends heavily on the environmental circumstances. For example, the photovoltaic cells offer adequate power density up to 15 mW/cm2 if the device is placed outdoor and operated primarily during daytime. However, in regular indoor office lighting

conditions, the same photovoltaic cell will merely produce about 10 μW/cm2 [10].

Because of the environmentally dependent characteristics, photovoltaic cells are limited to certain applications.

Fig. 1.1 Photovoltaic energy conversion [10]

Photon incident Photon absorbed Photon absorbed Photon Reflected Electron excited

1.1.2 Thermoelectric effect

Temperature gradient is basically a power source which can be converted into electric power. This thermal-to-electric behavior is described by the Seebeck effect, or the thermoelectric effect [12]. Once two different metals are connected in a closed loop, a temperature variation in the loop will cause electrons to move and a voltage potential is built up between the two metals or semiconductor junctions.

According to the Seebeck effect, the developed voltage is proportional to the temperature difference between the high temperature and low temperature ends, and to the Seebeck coefficients of the two materials. Large Seebeck coefficients and high electrical conductivity can increase conversion efficiency and decrease power losses and, therefore, is beneficial to the thermal-to-electric energy conversion.

Materials typically used for thermoelectric energy conversion, including Sb Te , _{2 3}

2 3

Bi Te , Bi-Sb, PbTe, Si-Ge, polysilicon, BiSbTeSe compounds, and InSbTe, are not completely compatible to the IC process. In [13], annealing conditions have tremendous influence on the electrical resistivity of Bi-Sb and the thermoelectric

generator performances as a consequence. An output power density of 140 μW/_cm 3

for a 100 ˚C temperature difference is obtained but the temperature difference of this level is not commonly seen in a micro system [14]. So the output power is limited without large thermal gradients.

Fig. 1.2 illustrates the simplest thermoelectric generator comprising a p-type and a n-type thermoelements connected electrically in series and thermally in parallel. Heat is pumped into one side of the couple and rejected from the opposite side. An electrical current is produced, proportional to the temperature difference between the hot and cold junctions.

In general, connecting several thermocouple elements in series can achieve better performance. However, large series resistance increases ohmic power loss and thus

reduces the overall power conversion efficiency.

Fig. 1.2 Thermoelectric energy converter [12]

1.1.3 Human body movement

Human power is known as one of the most conventional energy sources. However, human-movement-to-electric power conversion has not been studied until recent years. The conversion principle is to transfer power from human activities to electric power. The activities include walking, breathing, body heat and so on. It is possible to power portable devices by harvesting energy from the human movement.

In recent years, needs of wearable electronic devices [15-17] have grown significantly. Many researchers focus their efforts on walking since this process seems a more practical energy source for wearable electronic devices. For example, a field scientist or explorer carrying heavy load can use a specially-designed power harvesting back pack to generate electric energy for his instruments such as GPS or notebook computers [16]. An average 7.37 W power output was measured from 6 participants who walked with speed ranging from of 4.0 to 6.4 km/hour and carried 20-, 29-, and 38 kg loads in addition to the fixed frame weighing 5.6 kg as shown in

Fig. 1.3 [16]. During walking, a person moves like an inverted pendulum, as shown in Fig. 1.4, causing the hip to move up and down by 4 to 7 cm, a considerable amount of mechanical energy must be transferred if the load is heavy.

Fig.1.3 Components of suspended-load backpack [16]

“Heel-strike” devices are another walking power harvester [17]. However, the energy level of generation is relatively small (10 to 20 mW). This energy could be used in a variety of low-power applications, such as health monitors, self-powered emergency receivers, and radio frequency identification tags. The application is limited by the piezoelectrics and IC integration issues as well as power delivery issues. The piezoelectric shoe inserts offer a good solution for specific requirement such as RFID tags or other wireless devices worn on the foot.

1.1.4 Wind

Wind power is a renewable power generation technology to become a mainstream alternative for generation capacity expansion in the twenty-first century. This idea is to convert wind energy into a useful form like electricity by wind turbines or windmill. Due to the rotating characteristic of wind blades, the majority type of wind energy converter is electromagnetic conversion. Wind generated energy is also environmentally dependent which is similar to solar energy.

The power from wind is related to the air velocity. With slow wind at 3 m/s

velocity, the average power is about 80 μW/cm3. The maximum average power

density of 1060 μW/cm3 at 12 m/s air velocity was produced from a strong wind [18]. This indicates more usable power can be generated from high-velocity wind. However, wind power generation should be at large scale in order to obtain large amount of energy. Wind energy generators should be placed at locations where a sustainable and stable air flow is present. Therefore, wind power is a suitable energy source for wireless sensors for where a suitable wind source exists.

Few small-scale air flow harvesters have been proposed to date. One device based on MEMS technology is shown in Fig. 1.5 [19]. It comprises a 12-mm-diameter

output power of 1 mW could be delivered at a volume flow of 35 l/min and a pressure drop of 8.4 mbar. For operation in a free stream, the same output power would be expected at a flow speed of around 40 m/s which is rarely seen in practical use.

Fig. 1.5 MEMS air flow harvester, with 10 pence coin for scale [19]

1.1.5 Ambient Vibration

Vibration can also serve as an energy source. Ambient vibration can be observed in many environments. Most sources of vibrations are at low frequencies ranging from 60 to 200 Hz [9]. Different levels of mechanical vibration occur in exterior windows, aircraft, automobile, industrial equipments, and many small household appliances. Generally, the maximum power is extracted at resonance with the ambient vibration source. Theory and experiments show that the power density of more than

300 μW/cm3 can be generated [20]. A more detailed discussion of this harvesting

1.1.6 Summary of power sources

Table 1.1 shows the comparison of the several power sources for portable devices. The values in this table are estimates taken from literatures or analysis based on the survey in the previous sections. Vibration is chosen as the source of energy scavenging in this study because of its ubiquity and sufficient power density.

Table 1.1 Comparison of power sources

Power sources Power density Commercially available?

Solar (outdoors) [10] 15, 000 μW/cm2 Yes

Solar (indoors) [10] 10 μW/cm2 Yes

Temperature gradient [13] 140 μW/cm3 at 100˚C gradient Soon

Human power [15, 16] 330 μW/cm2 No

Wind energy [18] 1060 μW/cm3

at 12 m/s velocity No

Vibration [20] 375μW/cm3 at 120Hz, 2.5m/s2 Yes

1.2 Ambient vibration energy conversion

Ambient vibration can be converted into electricity based on the overview described in previous sections. Vibration-driven harvesters will be introduced and discussed in this section. Three types of methods can be utilized to generate electricity from vibration sources.

In conventional macroscale engineering, electrical generators are based on electromagnetic transduction. In small-scale energy harvesting, two main techniques are added. Piezoelectric transduction is generally impractical for rotating systems but is well suited to the reciprocating nature of the motions typically used for harvesting. Electrostatic transduction, which is both impractical and inefficient for large machines,

becomes much more practical at small size scales and is well suited for MEMS implementation.

1.2.1 Electromagnetic energy conversion

As described by Faraday’s low of induction, a change of magnetic flux linkage with a coil induces a voltage in the coil, driving a current in the circuit. The combined force on the moving charges in the magnetic field acts to oppose the relative motion, as described by Lenz’s low. The mechanical work done against the opposing force is converted to energy in the magnetic field associated with the circuit inductance. A typical electromagnetic energy converter is shown in Fig. 1.6 [21]. Mechanical acceleration is produced by vibrations that cause the mass to oscillate. A coil is attached to the mass and moves through a magnetic field built by a permanent magnet. The induced voltage was produced by the change of magnetic flux. Thus, the output power is proportional to the magnetic field and coil number.

Microscale electromagnetic generation technologies can be broadly classified into three categories: rotational, oscillatory, and hybrid devices [22], as shown in Fig. 1.7. Rotational generators imitate the operation of macroscale motor/generators and have been designed to operate using rotational power from miniature turbines or heat engines. They are designed for continuous rotational motion under a steady driving torque. In contrast, oscillatory generators operate in a resonance mode, usually relying on relatively small displacements between a permanent magnet and coil to acquire power from environmental vibrations. Lastly, hybrid devices rely on vibrations, but convert linear motion into rotational motion using an imbalanced rotor.

Fig1.7 Three types of permanent magnet generation technologies [22]

The most common issue for electromagnetic energy conversion is the relatively low induced voltage. Methods to increase induced voltage include increasing the number of turns of the coil or increasing the permanent magnetic field. However, it is difficult to fabricate large number of coils with planer thin film processes. Thus the power density of electromagnet converter is lower than other types of device.

The first microscale implementation of this type of vibration-driven harvester

was reported in 1995 by Williams et al. [23]. The 25_{mm device demonstrated a} 3

peak power of 0.3 μW for a 0.5μmvibrations at 4400Hz. Ching et al. used a small

NdFeB magnet supported by a laser-micromachined Cu spring structure [24]. A

coil coil coil Magnetic Rotor Magnet Eccentric Magnet

and the devices successfully powered low data rate infrared and RF wireless communication modules.

There are also commercial products that utilize resonant magnetic power

generation schemes. Perpetuum Co. Ltd. markets a 130 _{cm vibration energy} 3

harvester tuned to 100 or 120 Hz vibrations that delivers 3.5 mW for 0.1 g vibrations or 40 mW at 1 g [25]. Ferro Solutions offers a similar 87_{cm product that can produce} 3

10.8 mW for 0.1 g vibrations at 60 Hz [26].

(a) (b) Fig. 1.8(a) PMG-17 energy harvester from Perpetuum Co. [25]

and (b) VEH-3 energy harvest from Ferro solutions[26]

1.2.2 Piezoelectric energy conversion

Piezoelectric effect is a phenomenon whereby a strain in a material generates an electric field in that material, and inversely an applied electric field generates a mechanical strain [27]. The former can be used to convert energy. When an external force is applied, some of the work done is stored as elastic strain energy, and some in the electric field associated with the induced polarization of the material. If there is an external conduction route through a load, a current is generated to neutralize the net

charge produced as a result. Piezoelectric materials with high electromechanical coupling coefficients are generally ceramics, with lead zirconate titanate being the most common. The most common geometry is to place the piezoelectric material as a thin layer on a cantilever beam from which the proof mass is suspended.

Fig.1.9 A two-layer cantilever beam piezoelectric energy converter [28]

The first reported piezoelectric microgenerators appear in the patent literature. Snyder described the use of a piezoelectric generator mounted on the wheels of a car to power the tire pressure sensors [29, 30]. The device would be powered from the wheel vibration during driving. If an abnormal tire pressure is detected, the signal could be reported to the driver via a low-power radio link. This kind of device is mounted on all newly produced cars in the United States due to the government regulation.

Roundy et al. described an RF beacon powered by both a solar cell and an

optimized piezoelectric generator [28]. A power level of 375 Wμ was generated from

Fig. 1.10 A typical piezoelectric generator [28]

The difficulty of using piezoelectric material is the incompatibility with MEMS and CMOS processes. Another drawback of the piezoelectric converter is the requirement of additional circuitry to rectify the AC current. The supplementary circuitry has power losses and decreases the efficiency of the conversion. Most researches so far utilize bulk materials, which is still not suitable for integration with microsystem technology.

1.2.3 Electrostatic energy conversion

Electrostatic energy converters mainly use the change of capacitance of a mechanically driven variable capacitor as shown in Fig. 1.11. Mechanical forces from vibration are utilized to do work against the attraction of opposite charged parts. For a typical electrostatic energy converter, the variable capacitor is initially biased from a voltage source and disconnected instantly after the capacitor is fully charged. In the

constant charge process, when the capacitance decreases due to vibration, the voltage

on the variable capacitor increases (Qconstant=CV) and thus the mechanical kinetic

energy is transfered into electrical potential energy.

Fig. 1.11 (a) Gap-closing and (b) overlap in-plane variable capacitors [21]

MEMS variable capacitor can be fabricated through mature silicon-based micromachining process such as deep reactive ion etching. Therefore, it is relatively suitable for IC processes as mentioned earlier in this thesis. The converter also provides relatively high output voltage levels and adequate power density compared with electromagnetical counterparts. However, the disadvantage of the converter is the necessity of an auxiliary voltage source Vin used to initiate the conversion cycle.

The lifetime of the voltage source is unfortunately limited. One solution proposed by Bernard et al. [32] was to use an inductive flyback circuitry to constantly feedbacks the temporary storage energy back to the external voltage supply for further usage. Another solution was proposed by Ingo Kuehne et al. [33] where the build-in voltage caused by the work function difference between two conductors was utilized as a bias source and thus no outer bias source is needed. In addition, one can also employ a moving layer of permanently embedded charge, or electret, to carry out electric energy harvesting [34], although such devices currently suffered from low power

device is AC signal which needs to be rectified and thus the power loss is inevitable. A capacitive energy converter was implemented by Meninger et al. [35]. The comb finger structure in MEMS technology was used with silicon on insulator (SOI) wafers. The simulation showed an output power of 8.6 μW with a device size of 1.5 cm × 0.5 cm × 1 mm from the vibration of 2.52 kHz. Another design was proposed by

Roundy [9], which could achieve an output power density of 110 μW/cm3 under

vibration input 2.25 m/s2 and frequency of 120 Hz.

In electrostatic capacitive energy conversion, the switch timing control should be controlled accurately to achieve maximum conversion efficiency. A prototype circuitry with the two ideal diodes was proposed by Roundy [9]. The experiment showed excessive power reduction due to the far from ideal operation of the diodes. The power consumption by the electronics or parasitic capacitive and resistive coupling still existed. Therefore, improved switch design is critical for better energy conversion efficiency.

1.2.4 Comparison of vibrational energy conversion technologies

From the above literature overview, the three types of vibrational energy converters are listed in Table 1.2 [21]. According to the characteristic comparison of these energy converters, electrostatic capacitive vibration-to-electric energy conversion is suitable for scavenging ambient energy because of its ubiquity in the environment and sufficient output power density. The fabrication technologies for electrostatic converter are very mature in MEMS system. The materials and process are compatible with IC process.

Table 1.2 Comparisons of three types of vibrational energy converter [21]

Converter Power density Advantages Disadvantages

Piezoelectric ~200 μW/_cm 3

High power density No bias source

AC output Process integration

Electromagnetic < 1 μW/_cm 3

No bias source AC output

Low power density Process integration

Electrostatic 1-100 μW/_cm 3

DC output Process integration High power density

Bias souce

1.3 Thesis objectives and organization

Most electrostatic energy converters use switching devices such as diodes, MOSFET, or integrated mechanical switches for the control of conversion cycles. This kind of operation results in a nonlinear system, especially in the movement of mechanically variable capacitor. Many researchers [33, 36, 37] used simulation tools to model their devices. However, it is difficult to perform systematic analysis by using simulation tools. Oftentimes some parameters have to be assumed in order to run the simulation. For example, Roundy et al. assumed the minimum gap of their in-plane gap closing type converter to be 0.5μmor 0.25μm[38]. In fact, this value should not be chosen arbitrarily because it is critical for determining the maximum electrostatic spring constant, which will significantly affect the stability of the dynamic system. In our study, the minimum gap is determined by the initial gap and the amplitude of the oscillating mass. Both the amplitude and the initial gap are parameters in the

calculation. If the minimum gap is too small, the total effective spring constant will decrease dramatically. The pull-in effect can happen when the effective spring constant becomes to zero. The detail discussion will be presented in chapter2.

Another simplification is about the system nature frequency; many studies [36, 37, 39] ignored the interaction between the electrostatic attraction and the mechanical restoring force and simply designed the system nature frequency from mass and spring. In reality, during oscillation, the effective spring constant would not be simply the same as that of the mechanical system. In this study, the mechanical spring constant is also a varying parameter in our calculation.

In our previous work on capacitive energy conversion with integrated mechanical switches [39], Simulink was used to simulate the time domain behaviors of the device. The problem of time domain simulation was the difficulties to find the physics behind phenomena. Another problem of the previous device was the oxidation of the aluminum coating on the integrated mechanical switches. This results in high contact resistance. Moreover, even the improvement of fabrication processes and electrical power measurement were conducted, the leakage resistance still existed in the variable capacitance.

Based on our previous work on the electrostatic generator with integrated mechanical switches [39], the goals of this thesis include:

(a)..Design and analysis of the optimal power generation for the electrostatic converter with or without the external mass attached,

(b) Novel method for modeling the dynamics of the energy converter, (c) Solving the mechanical switch contact problem.

The organization of this thesis is as the following. The principle, design and optimization of the energy converter are discussed in Chapter 2. Fabrication processes and results are described in Chapter 3. The measurement results and discussions are

presented in Chapter 4. Finally, conclusion and future work will be discussed in Chapter 5.

Chapter 2 Principle and Design

In this chapter, we will discuss the fundamentals of micro electrostatic capacitive energy converters. The energy generation depends on the change of capacitance of a variable capacitor caused by vibration. Kinetic energy is converted into electrical energy during this process. The variable capacitor is initially charged by an external power supply such as a battery. The charge-discharge cycles are controlled by mechanical switches. The optimal design and analysis are presented in this chapter. A theoretical model was established to analyze the device characteristics; the results are compared with simulation in our previous work [39].

2.1 Characteristics of vibration sources

Vibration sources are generally more ubiquitous. The output power of a vibration driven converter depends on the nature of the vibration source, which should be known in order to estimate the power generating capability. There are various kinds of vibration sources in the environment. Measurement of different vibration sources was conducted by Roundy [9], as shown in Fig. 2.1. From the low level vibration sources, two characteristics can be observed. First, fundamental peaks occur at a common low frequency. Second, the high frequency vibration modes are lower in acceleration magnitude than the low frequency fundamental modes. Low level vibration in typical households, offices, and manufacturing environments is considered also as a possible power source for wireless sensor nodes.

Our measurement of the vibration spectrum of an air purifier is shown in Fig. 2.2. A piezoelectric accelerometer (PCB Piezotronics model 353B17) was put on the air purifier and the data was collected by an oscilloscope. The fundamental vibration mode was at about 120 Hz, as observed by Roundy. The peak acceleration of the air

purifier is about 2.2m/s2 at about 120 Hz, as shown in Fig. 2.3. These results will be used as our targeted input vibration source due to its common existence.

Fig. 2.1 Vibration spectra by Roundy [9]

Fig. 2.2 Measurement of air purifier vibration Air purifier

To oscilloscope

Fig. 2.3 Vibration spectrum of an air purifier

2.2 Operation principle

The main component of the electrostatic energy converter is a variable capacitor

Cv [36]. A schematic circuit of the energy converter is shown in Fig. 2.4. It is

composed of an auxiliary battery supply Vin, a vibration driven variable capacitor Cv

and an output storage capacitor Cstor, which is connected to the load RL. Two switches,

SW1 and SW2, are used to connect these components and control the charge-discharge conversion timing [39].

Fig. 2.4 Operation circuit of the electrostatic energy converter

A B C D Vin Cv Cstor RL VL SW1 SW2 20 70 120 170 220 Frequency (Hz) 1 10 0.1 Acceler ation ( m/s 2 ) 0.01 0.001 100 2.2 m/s2at 120 Hz

A more detailed schematic of the energy converter is shown in Fig. 2.5. The change of the capacitance is driven by the external vibration source. SW1 is implemented by a contact mechanism between nodes A and B. SW2 is actuated by the electrostatic pull-in force between nodes B and D. When node B reaches the pull-in voltage, it will be attracted by node D and touch node C.

Fig. 2.5 Variable capacitor schematic

In the energy conversion cycle, the variable capacitor Cv is first charged by the

auxiliary voltage supply Vin through the switch SW1 when Cv is at its maximum Cmax,

as shown in Fig. 2.6. After Cv is charged to Vin, SW1 is opened and the capacitance

changes from Cmax to Cmin due to the electrode displacement by vibration. In the

process, the charge Q on the capacitor remains constant (SW1 and SW2 both open).

Therefore, the terminal voltage across the capacitor Cv is increased and the

mechanical energy is converted to the electrical energy stored in the capacitor.

Fig. 2.6 Capacitor charging and capacitance change by vibration VL Cmax Cstor Vin SW2 RL SW1 VL Cmax Cstor Vin SW2 RL SW1 Cmin SW1 B Displacement C D

SW2 Cstor Pull-in electrode (GND)

RL

Cmin Cstor RL VL SW2 VL(t) SW2 open (Discharging) SW2 close (Charging) ‧‧‧ Vsat ‧‧‧ Vo tn tn-1 Δt

When the capacitance reaches Cmin and terminal voltage reaches Vmax, SW2 is

closed and the charge redistributes between Cv andC_stor with balanced voltage Vo, as

shown in Fig. 2.7 [36]. The energy stored in the variable capacitor Cv is transferred to

the the storage capacitor Cstor. SW2 is then opened and Cv varies back to Cmax,

preparing for the next conversion cycle. We notice that there are two conversion cycles in one oscillation cycle since the period of oscillation contains two headings. The charge on Cstor is dissapated through the load resistance RL with a time constant τ

= RLCstor before it is charged again by Cv. The output voltage VL will eventually

reach the steady state when the initial and final voltages of the charge-discharge process become equal.

Fig. 2.7 Charge transfer and discharge process [36]

2.3 Device design

The variable capacitor is the main component of the converter. In this study, capacitors with and without an external mass are both considered. To meet the 120Hz vibration source, we used an external mass mounted on the device to adjust its resonance frequency. In the case without external mass, the 120Hz requirement results in a very small mechanical spring constant. Because of the small mass, the kinetic energy in the system is inevitably small, causing very low output power compared to the case with an external mass. However, the vertical force caused by the weight of the mass is also reduced so that damage caused by the external mass can be alleviated.

The output power strongly depends on the maximum capacitance Cmax. But the

electrostatic force also increases with the Cmax, indicating that it may influence the

movement of the mass for large Cmax. Therefore, the variable capacitor must be

designed by considering the 1 _{cm device area constraint, the maximum} 2

capacitanceC_maxand the associated electrostatic force.

2.3.1 Auxiliary battery supply

The auxiliary battery supply is used to pre-charge the variable capacitor through SW1. Typical energy storage devices include capacitors, inductors and batteries. Capacitors and inductors have lower energy density. They often serve as short-term energy storage cells. Batteries, such as NiZn, NiMH, NiCd, and Lithium-ion (Li-ion), store energy chemically and are rechargeable. Among these types, Li-ion batteries (Fig. 2.8) offer the best performance with high energy density, high discharge rate, high cell voltage, long life span, and no memory effect. In this study, LIR1620 (3.6 V, Φ 16 mm, H 2.2 mm, 1.2 g) and LIR2016 (3.6 V, Φ 20 mm, H 1.8 mm, 1.6 g) Li-ion cells can be used as the auxiliary battery supply. Moreover, the battery can act as part of the external mass if it is well bonded on the device.

2.3.2 Variable capacitor design

SOI wafers with highly doped thick device layers and deep silicon etching technology were used to fabricate the devices. An in-plane gap-closing comb structure is used for the variable capacitor, as shown in Fig. 2.9. Compared with the in-plane overlap type comb structures, this topology has the advantage of larger capacitance change for smaller displacement. Compared with the out-of-plane gap closing capacitors, this topology has the advantage of lower mechanical damping loss and possibility to incorporate minimum gap control designs.

Fig. 2.9 Top view of the in-plane gap closing variable capacitor topology The symbols used in the following discussion are listed below:

d: initial gap between comb fingers dmin: minimum air gap between fingers

t: comb finger width

Lf: overlap length of comb fingers

h: thickness of device layer

z : relative displacement between the movable and fixed electrodes Ng: number of variable capacitor finger cells

Relative motion Spring Fingers Proof mass Anchor Anchor

S

S’

⇒

0

ε : permittivity of free space ( -12

0= 8.842 10

ε × F/m)

μ: viscosity of air at 1 atm (_{μ =1.82 10 ×} -5 _Pa-sec)

α: damping coefficient depending on effective region (α ≈ 1.74)

The total variable capacitance between the comb fingers is [36] C (z)=_v 2Ng 0₂ L hd₂f

d -z

ε

(2.1) From this equation, the variable capacitance strongly depends on the comb finger structure. Layout design under restricted area directly affects the variable capacitance. A general model of the comb finger layout is shown in Fig. 2.10. The total layout area is limited to 1_{cm . A number of free parameters can be adjusted to obtain the optimal} 2

design. In Fig. 2.10, the layout can be divided into three parts. First is the gray portion for the support of the fixed comb fingers. The second part is the rest of the layout area which is defined as S. The third part is the area occupied by the finger cells in S, which is defined as S', the area ratio is defined as

S =_r S'

S (2.2)

For the movable part of the variable capacitor, the area S' occupied by fingers can be divided into cells, as marked by the rectangle in Fig. 2.11. The air gaps around the fingers are equal to maintain consistent etching rate in the ICP deep RIE process.

Fig 2.11 Single cell schematic

The number of fingers is the finger area S'in layout divided by the finger cell area r g f SS N 2(d t)(L 2d) = + + (2.3) The mass of movable plate is

m₀ =ρh S 1-S⎡_⎣

(

_r

)

+N t L_g

(

_f +d

)

⎤_{⎦ (2.4)} In these two equations, two free parameters, Sr and d, are utilized to optimize the

layout design.

The device will be fabricated on SOI wafers. The thickness h is chosen as 200 μm to have large capacitance. The finger width t of 10 μm is restricted by the aspect ratio of up to 20:1 in the deep reactive ion etching process. A maximum device area of 1cm is set as the device size constraint. 2

d d

t

f L 2(d + t) Fixed plate Moving part d

2.3.3 Dynamic analysis

The dynamic analysis is performed to decide the mechanical spring constant k under certain proof mass m in order to achieve the theoretical maximum displacement under the targeted input vibration. The electro-mechanical dynamics of the variable capacitor can be modeled as a spring-damper-mass system, as shown in Fig. 2.12.

Fig. 2.12 Schematic of the dynamic model

The equation of motion is

mz + b z + (k +k )z = -my _m _{0 e}  (2.5) where y (=Y0sinωt) is the displacement of device frame caused by the vibration, z is

the relative displacement between movable and fixed plates, k is the mechanical ₀

spring coefficient, b z_{m is the mechanical damping force, and} k z is the _e

electrostatic force caused by the charge on the capacitor, which acts as a negative spring force with ke as the electrostatic spring constant.

The mechanical damping can be determined from experimental data. From our previous MMA (MEMS motion analyzer) measurements, the quality factor is approximately 6 to 8. The corresponding mechanical damping is 0.38 to 0.52 based on

m bm ke k z(t) y(t)

₂

m

km Q =

b (2.6)

The electrostatic force induced by constant charge Q on Cv is [36]

_{e e} 2 g o f -Q F =k z= z. 2N ε L hd ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (2.7) The force can be viewed as a negative spring force with a negative electrostatic spring constantk . The electrostatic spring constant is determined by the charge Q on the _e

variable capacitor which alternates betweenQ_maxand Q_minin the charge-discharge

process. The corresponding electrostatic spring constants are defined as k_e,max and

e,min

k . A newly introduced parameterQ is defined as the charge ratio between _r

min Q toQ_max min r max Q Q Q = (2.8) The charge ratio is related to the output characteristics. Further discussion of this parameter will be presented in the next section.

The relationship between the electrostatic spring constant and the dynamic activities of the system is

1 0 e,max 2 0 e,min k = k + k ; z(t)z(t) < 0 k(t) = k = k + k ; z(t)z(t) > 0 ⎧⎪ ⎨ ⎪⎩   (2.9) The system becomes a piecewise linear system described by

1 m 1 1 1 2 m 2 2 2 mz + b z + k z = -my; z(t)z(t) < 0 mz + b z + k z = -my; z(t)z(t) >0         (2.10)

where z is the relative displacement between fingers after charging at ends and ₁

before discharging at center with Q = Q_max , z is relative displacement after ₂

discharging at center and before charging at ends withQ = Q_min. In Fig. 2.13, the

respectively. The solutions to Eq. 2.10 are composed of homogeneous solution and particular solution [40],

(

)

( )

(

)

( )

1 1 2 2 -ζ ω t 1 1 d1 2 d1 p1 -ζ ω t 2 3 d2 4 d2 p2 z = e C cosω t + C sinω t + z t z = e C cosω t + C sinω t + z t (2.11) where 2 2 m m 1 2 1 2 1 2 d1 1 1 d2 2 2 1 2 b b k k = , = ; ω = , ω = ; ω = ω 1- , ω = ω 1-2mω 2mω m m ζ ζ ζ ζ ;and p1

z and z are the particular solutions of the equations as shown below _p2

(

)

(

)

(

)

(

)

2 2 2 0 0 m 0 p1 2 1 0 0 0 2 2 2 m 0 1 0 2 2 2 0 0 m 0 p2 2 2 0 0 0 2 2 2 m 0 2 0 Y ω b ω z (t) = ω - ω sinω t - cosω t m b ω ω - ω -m Y ω b ω z (t) = ω - ω sinω t - cosω t m b ω ω - ω -m ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (2.12)

Fig.2.13 Displacement of shuttle mass versus time

Due to the difference in total spring constants, the time needed for the mass to move from ends to center is not equal to the time needed from center to ends. Therefore a parameterα is defined as the portion of time occupied by maximum charge onC . _V α ₌ t1 _(2.13) max Q max Q min Q min Q 1 z ' 1 z ' 2 z 2 z

wheret₁ andt represent the time occupied by the ₂ k and₁ k equations in Eq. 2.10, ₂ respectively. Duringt , the charge on₁ C is_V Q_max; duringt , the charge on₂ C is_V Q_min.

Fig. 2.14 Time shift for t and T scale

A new time scale is also defined to describe the system behavior. At T = 0, the mass is at the maximum displacement, as shown in Fig. 2.14. The delay between T and t is TΔ . If T t - T= Δ is substitute into Eq. 2.11, the solution or the displacement function can be rewritten as

(

)

(

)

1 1 2 2 -ζ ω (T+ΔT) 1 1 d1 2 d1 p1 -ζ ω (T+ΔT) 2 3 d2 4 d2 p2 z (T) = e C cosω (T + ΔT) + C sinω (T + ΔT) + z (T + ΔT) z (T) = e C cosω (T + ΔT) + C sinω (T + ΔT) + z (T + ΔT) (2.14)

The velocity is the derivative of the displacement functions

(

)

(

)

1 1 2 2 -ζ ω (T+ΔT) 1 1 1 d1 1 2 d1 1 p1 -ζ ω (T+ΔT) 2 2 3 d2 2 4 d2 2 p2 z (T) = -ω e C sin ω (T + ΔT) + - C cos(ω (T + ΔT) + ) + z (T + ΔT) z (T) = -ω e C sin ω (T + ΔT) + - C cos(ω (T + ΔT) + ) + z (T + ΔT) φ φ φ φ ⎡ ⎤ ⎣ ⎦ ⎡ ⎤ ⎣ ⎦     (2.15) ΔT t T z(t) y(t)

-1 1 -1 2

1 ₂ 2 ₂

1 2

ζ ζ

where = tan , = tan

1- ζ 1- ζ

φ φ

The boundary conditions between z and ₁ z can be found from the stable ₂

oscillating dynamics. At the ends of its travel range indicated in Fig.2.13, the velocity of the moving mass is equal to zero; at the center of travel, the velocity of z and ₁ z ₂ are continues and the displacement is equal to zero. The next conversion cycle has the same dynamic characteristics; however, the direction is opposite to the previous one. Therefore, there are two conversion cycles in one oscillation cycle. The boundary conditions used in this analysis can be summarized as

z (0) = 0₁ , (2.16a) z (₂ 1 ) = 0 2f  (2.16b) z 0 = -z₁

( )

₂ 1 2f ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (2.16c) z 0 = A₁

( )

(2.16d) z₁ α = 0 2f ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (2.16e) z₂ α = 0 2f ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (2.16f) z₁ α = z₂ α 2f 2f ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠   (2.16g)

Eqs. 2.16a and b mean the velocity at both ends are equal to zero; Eqs. 2.16c and d show the amplitude equals to A; Eq. 2.16e and f show that the displacement at center is zero; Eq. 2.16g is the continuity of velocity at center. The next conversion cycle has identical behavior as the previous one except for the opposite direction. The boundary conditions are depicted in Fig. 2.15.

Fig. 2.15 Boundary conditions between z1 and z2.

The amplitude A solved from above equations is directly related tok_e,max. The amplitude A and the initial gap d of the comb fingers are depicted in Fig. 2.16. The capacitance is maximum as the shuttle mass reaches its maximum displacement A. The electrostatic spring constant k_e,max is related to A as

2max 2max in2 0 f in2 e,max 2 2 2 0 f 0 f -Q -C V -2Nε L hdV k = = = 2Nε L hd 2Nε L hd (d -A ) (2.17)

The electrostatic constant k_e,min is also a function of amplitude because

min 2 2

e,min e,max r e,max max

Q

k = ( ) k = Q k

Q (2.18)

Therefore, the parameters related tok such as the nature frequencies and damping _e ratios can also be expressed as functions of the amplitude A. The seven unknowns

1 2 3 4

C ,C ,C ,C , , T, and Aα Δ can be solved from the seven boundary conditions in Eq.

1 1 z (0) 0 z (0) A = ⎧ ⎨ ₌ ⎩  2f α 0 1 2f 1 f 1 2 2 1 z (0) -z ( ) 2f 1 z ( ) 0 2f ⎧ ₌ ⎪⎪ ⎨ ⎪ ₌ ⎪⎩ 1 2 z ( ) 0 2f z ( ) 0 2f α α ⎧ ₌ ⎪⎪ ⎨ ⎪ ₌ ⎪⎩ T z(t) 1 2 z ( ) z ( ) 2f 2f α ₌ α   A z1 2 z

2.16. Since they are complicated set of nonlinear equations, a MATLAB solver is used to find the numerical solution.

Fig. 2.16 Fingers at maximum displacement

2.3.4 Static analysis

Static analysis can be used to obtain mathematical guidelines for overall system design including layout and circuit. Once the amplitude is decided from the dynamic analysis, parameters such as initial gap and mechanical spring constant can be determined directly. At the output node of the variable capacitor, the charge ratio before and after SW2 is switched and can be expressed as

min min o min o o

r

max max min min max max

C V C V V

Q

Q = = = =

Q C V C V V , (2.19)

where V_{max o}and V are the capacitor voltage before and after discharge by SW2, as discussed in section 2.2. Assume the storage capacitorC_storis large and the output ripple can be ignored, the charge flow through output load is

o out L V Q = 2fR (2.20) The charge flow into the variable capacitor is

Q = C V (1- Q ) (2.21) _{in max in r} m

d

o min max r

L o min max min o

L o min L min r max L min min L V C V (1- Q ) = 2fR V C V - C V = 2fR V C 2fR C Q = = = (2.22) 1 V _{+ C} 1+ 2fR C 2fR ⇒ ⇒

Eq. 2.22 can be rearranged to the following form r L min r Q R = 2fC (1- Q ) (2.23) From Eq. 2.22, we know that the charge ratio is determined by the frequency of vibration, the minimum capacitance and the output load. If the voltage ripple of

stor

C needs to be considered, the charge ratio is L stor min r _-1 2fR C stor min C Q = C ⎛⎜_⎜1- e ⎞⎟_⎟+ C ⎝ ⎠ (2.24)

The approximation of Eq. 2.24 for largeC_storis the same as Eq. 2.22.

The output power can also be determined as we know the amplitude of the shuttle mass displacement. For largeC_stor, the output power is

2 o out L V P = R (2.25) Output power can also be expressed in following form from Eq. 2.22 and Eq. 2.23 2

out min max r r

P = 2fC V Q (1- Q ) (2.26) It is worthy to mention that an optimum load based on a similar system analysis was derived in [41] by finding the derivative of output power with respect to the output load. The optimal load is [41]

_L,opt

min

1

R =

2fC (2.27) However; the result in Eq. 2.27 was based on an assumption of fixed amplitude. The same result can also be derived by finding the derivative of output power respect to

charge ratio in Eq. 2.26 if we regard other parameters as constants. The maximum output power occurred atQ_r =0.5. As we substituteQ_r =0.5 into Eq. 2.23, we can obtain an identical result as Eq. 2.27.

However; in this research, the charge ratio is related to the electrostatic force and thus the amplitude. Therefore, we should not assume a fixed amplitude arbitrarily. The assumption ignored the electromechanical coupling effect in the

mass-spring-damper system. We treat Q as an operation parameter instead of a _r

constant.

2.4 Optimizing process

Because Eqs. 2.16 are highly nonlinear, we use “fsolve” in Matlab to find numerical solution. The “fsolve” solver needs an initial guess to find the solution of the set of equations. At low amplitude, the system can be regarded as linear. The starting guess was therefore found by the approximated linear results. For the next calculation the previous solution was used as new starting guess. After repeating these steps, the maximum amplitude can be found.

The dimensions of the device are partially fixed as shown in Table 2.1. Four free parameters are used to find the optimal condition under restricted area, as listed in Table 2.2. The detailed optimization flow chart is shown in Fig. 2.17.

Table 2.1 Fixed parameters of optimizing process

Parameter Description of constants Value

h Device thickness 200μm

Table 2.2 Free parameters of optimizing process

Variable Description of variables Range

0

k Mechanical spring constant 1~3000N/m

r

Q Charge ratio 0.1~0.9

r

S Surface ratio 0.05~0.95

d Initial gap 1μm~70μm

Fig. 2.17 Optimization flow chart

Qr Sr d k0 N Y N Y N Y N Y

Max. range ? Max. range ?

Max. range ? Max.

amplitude ? Save amplitude Change next d Change next k0 Change next Qr Change next Sr END

Mechanical spring constant k0 (N/m) 50 40 0 30 20 10 Amplitud e ( μm) 50 40 30 20 10 Output power ( μW) 0 Amplitude Output power 2 r r a 2.25 m / s d 40 m Q 0.5 S 0.8 f 120 Hz m 4 gm μ = = = = = = at 2392 N/m, Amax = 39.35 μm Pmax= 45.8 μW

The oscillation amplitude A was calculated by the “fsolve” solver for each set of free parameter values in the ranges in Table 2.2. For example, Fig.2.18 shows the calculation results for the initial gap of 40 mμ , charge ratio of 0.5 and area ratio of 0.8

as a function of k0. At k0= 2392N / mthe amplitude reaches the maximum of

39.35 mμ . The corresponding output power is 45.8μW and output voltage is 68.62V.

Similar results can also be obtained in frequency domain. Fig.2.19 shows the

amplitude and power versus frequency for k0=2392N/m, which corresponds to the

maximum power in Fig 2.18.

Fig 2.18 Amplitude and power vs. spring constant

It is interesting to notice that there is a range of design or operation parameters where no solution of the oscillation amplitude A exists in Figs. 2.18 and 2.19. This phenomenon is discussed in the next section from an energy point of view.

50 40 30 20 10 0 50 40 30 20 10 Am plitud e (μ m ) 0 Output power ( μW) Amplitude Output power at 120 Hz, Amax = 39.35 μm Pmax = 45.8 μW Frequency (Hz) 2 r r 0 a 2.25 m / s d 40 m Q 0.5 S 0.8 k 2392 N / m m 4 gm μ = = = = = =

Fig. 2.19 Amplitude and power vs. Frequency

2.4.1 Conditions for normal oscillation discussion

At maximum displacement, the energy in the system is the sum of the mechanical spring potential energy and the capacitor electrostatic energy. The capacitor energy increases after charging by the battery. At center, the system energy is the sum of the kinetic energy and the capacitor energy. The capacitor energy decreases after discharging to the load. The system energy at different instances and positions is listed in Table 2.3. Fig. 2.20 shows these calculated system energies at different frequencies. The interaction between the external force and the restoring force in the system cause two conditions for which the energy can not be balanced and no normal oscillation can be found, as discussed next.

at center before discharging at center after discharging

at max displacement after charging at max displacement before charging System en er gy ( μJ) 1 2 2.5 0.5 1.5 90 100 110 120 130 140 150 1 f f₂

Table 2.3 System Energy at different instances and positions

Position At center before discharging At center after discharging At ends before charging At ends after charging Kinetic energy 1mz2 2  2 1 mz 2  0 0 Spring energy 0 0 0 2 1 k A 2 2 0 1 k A 2

Capacitor energy 2max

max Q 2C 2 min max Q 2C 2 min min Q 2C 2 max max Q 2C Frequency (Hz)

Fig. 2.20 System energy versus driven frequency

In Fig 2.20, we can see the energy at center after discharging crosses the energy at ends after charging at 120Hz. As the system approaches resonance at f2, the output

reaches maximum and the energy left in the system after discharging at center becomes insufficient for the mass to move to maximum displacement. Similar observation can be found at f1. As the system approaches the frequency at f1 the energy

the mass to maintain stable operation.

2. 4 .2 Optimum design

The output voltage is limited to 40V for further integration with power

management circuits. The calculated power can be plotted versus area ratio Sr and

initial gap d for each charge ratio Qr as shown in Fig. 2.21 and Fig. 2.22 for devices

with and without attached external mass, respectively. From Eqs. 2.23 and 2.26, we know that a larger surface ratio has benefits of lower load resistance and higher output power. For the device with attached mass, the robustness of the devices and the area reserved for mass attachment must be considered. Therefore, a maximum value of surface ratio of 0.8 is chosen in our consideration. For the device without attached mass, it is simpler and there is no such restriction.

The optimal design under 40V output voltage is obtained from Fig. 2.21 and Fig. 2.22. The parameters used in this design are optimized for maximum power and the results are summarized in Table 2.4 and Table 2.5. The optimum design with external mass attachment can generate 40.53μW. In the case without external mass attachment, 0.87μW can be generated.

Since the battery provides the initial charge for the energy conversion, the net output power is slightly less than the values calculated above. Because the current that flows into the variable capacitor from the battery is equal to the current that flows out the variable capacitor to the load, the percentage of net power generation is about 91%, as calculated in Eq. 2.29.

out out out out in in in in P I V V = = P I V V (2.28) in net out P %P = 1-P = 91% (2.29)

We can notice that in the device with the external mass, the output power is less dependent on the area ratio because the difference of silicon mass caused by the area ratio is minor compared to the external mass. On the other hand, in the device without external mass, the output power is related to both area ratio and initial gap. An obvious trend is the less the area ratio is, the higher output power is.

With external mass attached

Fig. 2.21 Contour of output power and voltage with external mass attachment for various Qr r r out optimal design: Q = 0.3 S = 0.8 d = 48 μm P = 40.53 μW

Fig. 2.21 Contour of output power and voltage with external mass attachment for various Qr (continued)

Fig. 2.21 Contour of output power and voltage with external mass attachment for various Qr (continued)

Table 2.4 Variable capacitor design parameters (with external mass)

Variable Description of variables Designed value

h Device thickness 200 μm

Ng Number of variable capacitor cells 1539

Wf Finger width 10 μm

Lf Finger overlap length 400 μm

d Finger initial gap 48 μm

Zmax Maximum displacement 47.34 μm

Qr Charge ratio 0.3

Cmin Minimum value of capacitance 45.4 pF

k Mechanical spring const. 2305 μN/μm

m Mass of movable plate 4.03g

RL Driven load resistance 39.3MΩ

Cstor Output temporary storage capacitor 10.6 nF

Vout Output voltage (steady state) 40 V

Pout Output power (steady state) 40.53 μW

Without external mass attached

Fig. 2.22 Contour of output power and voltage without external mass for various Qr(continued)